“There is geometry in the humming of the strings, there is music in the spacing of the spheres.” — Pythagoras
Counting, rhythm, scales, intervals, patterns, symbols, harmonies, time signatures, overtones, tone, pitch. The notations of composers and sounds made by musicians are connected to mathematics. The next time you hear or play classical, rock, folk, religious, ceremonial, jazz, opera, pop, or contemporary types of music, think of what mathematics and music have in common and how mathematics is used to create the music you enjoy.
With apologies to the musical Grease, mathematics and music go together like rama lama lama ke ding a de dinga a dong. You need to look no further than Plusto see how the links between the two have fascinated researchers for centuries — see all Plus articles tagged with the mathematics and music tag.
Their article Geometrical Music Theory, published in the April 18 edition of Science, outlines their theory that musical operations, such as transpositions, can be expressed as symmetries of n-dimensional space.
They categorise sequences such as chords, rhythms and scales into mathematical “families”. The families can be represented by points on the complex plane, and different types of categorisation produce different geometrical spaces. The authors argue that through this method researchers will be able to analyse music more deeply and understand how music has changed over time. They also say that the theory will allow the comparison of many kinds of Western music, although only some non-Western styles. This is because the theory is based on concepts such as the “chord”, which are present in nearly all Western styles, but not all non-Western styles.
The basis of geometrical music theory is that it provides a unified mathematical framework for musical events that are described differently depending on the scenario, but are fundamentally the same. For example, a “C” followed by the “E” and “G” above it may be described as a “C major chord,” “an ascending C major arpeggio,” “a major chord” and the list goes on. The authors describe five different methods of categorising such collections of notes. These are called “OPTIC symmetries,” with each letter of OPTIC representing a different categorisation method that may, for example, look at what octaves the notes are in, their order of play, or how many times each note is played. Each categorisation focuses on one aspect of the music, ignoring the others. The five symmetries can then be combined together to produce different musical concepts, some familiar, some new. Three-note chords end up on a triangular doughnut while other chord types live on the surface of a cone.
Tymoczko believes that their theory can be used to investigate the differences between musical styles. “Our methods are not so great at distinguishing Aerosmith from The Rolling Stones,” he said. “But they might allow you to visualise some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.”
The authors even hope that through their work, new musical instruments may be developed. “You could create new kinds of musical instruments or new kinds of toys,” said Tymoczko. He also envisaged new visual shows that could accompany the music. “Imagine going to a classical music concert where the music was being translated visually.”
So next time you go see a visually spectacular Madonna or Kylie show, look at the big screens — you might learn some maths!
The new shape of music: Music has its own geometry, researchers find
The figure shows how geometrical music theory represents four-note chord-types — the collections of notes form a tetrahedron, with the colors indicating the spacing between the individual notes in a sequence. In the blue spheres, the notes are clustered, in the warmer colors, they are farther apart. The red ball at the top of the pyramid is the diminished seventh chord, a popular 19th-century chord. Near it are all the most familiar chords of Western music. Credit: Dmitri Tymoczko, Princeton University
The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.
And the so-called musica universalis or “music of the spheres” emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies — the sun, moon and planets — could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University — have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
Writing in the April 18 issue of Science, the trio has outlined a method called “geometrical music theory” that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into “families.” They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way “x” and “y” coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph’s University in Philadelphia. In an accompanying essay, she writes that their effort, “stands out both for the breadth of its musical implications and the depth of its mathematical content.”
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like “chord” are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
“The music of the spheres isn’t really a metaphor — some musical spaces really are spheres,” said Tymoczko, an assistant professor of music at Princeton. “The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn’t done before.”
Like what?
“You could create new kinds of musical instruments or new kinds of toys,” he said. “You could create new kinds of visualization tools — imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas.”
“But to me,” Tymoczko added, “the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries.”
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle “C” on a piano, followed by the note “E” above that and the note “G” above that. Musicians have many different terms to describe this sequence of events, such as “an ascending C major arpeggio,” “a C major chord,” or “a major chord.” The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the “OPTIC symmetries,” with each letter of the word “OPTIC” representing a different way of ignoring musical information — for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
“Have Western composers already discovered the essential and most important musical objects?” Tymoczko asked. “If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization.”
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
“Our methods are not so great at distinguishing Aerosmith from the Rolling Stones,” Tymoczko said. “But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.”
The deep connection between music and mathematics was recognized at least as early as the time of Pythagoras. Now, Ian Quinn, an assistant professor in Yale’s music department and its cognitive science program, and his colleagues have devised a new mathematical means of understanding music. This “geometrical music theory” can translate the language of music theory into that of contemporary geometry and create visual representations of music’s underlying mathematical structure.
In the April 18 issue of Science, they describe five ways (“symmetries”) of categorizing groups of notes that are similar but not identical: the same note in different octaves, or the same group of notes in a different order. Then they show how these symmetries can be combined to map musical works in coordinate space where, for instance, two-note chords take the shape of a Mobius strip, three-note chord types take the shape of a three-dimensional cone, and four-note chord types somewhat resemble a pyramid.
“We can put any music into the model,” Quinn says, “and visualize the structure behind similarities and differences among musical styles—why Chopin, for instance, sounds different from Mozart.” Or Lennon from McCartney.
The translation of music theoretical terms into precise geometrical language provides a framework for investigating contemporary music-theoretical topics, Quinn says. It can also be useful in analysis, composition, pedagogy, and even the design of new kinds of instruments. Adds Quinn, “My students have used the models to write in the styles of various composers. Somewhat to my surprise these complex topics are fairly easily taught.”
Geometry shapes sound of music, FSU professor says
Through the ages, the sound of music in myriad incarnations has captivated human beings and made them sing along, and as scholars have suspected for centuries, the mysterious force that shapes the melodies that catch the ear and lead the voice is none other than math.
The space of three-note chord types is a cone. Numbers refer to pitch classes, with 0 = C, 1 = C#, etc. Points represent equivalence classes of transpositionally related chords. Thus, (C, D, E) and (D, E, F#) are both instances of 024. (Image made with Dmitri Tymoczko’s “ChordGeometries” program available here.)
It’s geometry, to be more precise, and now, a trio of 21st-century music professors from Florida State University, Yale University and Princeton University have analyzed and categorized in brand-new ways the mathematics intrinsic to musical harmony. Their cutting-edge collaboration has produced a powerful tool they call “geometrical music theory,” which translates the language of music theory into that of contemporary geometry.
The research is described in the April 18 issue of the journal Science, where the publication of work by music theorists and composers is rare if not unprecedented, said Clifton Callender, an assistant professor of composition in FSU’s College of Music. Callender is co-author of the paper “Generalized Voice-Leading Spaces” with Ian Quinn of Yale and Dmitri Tymoczko of Princeton.
“Our research offers a variety of tools for understanding and exploring music by drawing upon contemporary mathematics in natural and musically relevant ways,” Callender said. “It also provides a way to compare chords, and represents all possible combinations of pitches, including those found in non-Western music and avant-garde works that don’t conform to the traditional scales of Western music.”
As a result, composers could explore all sorts of uncharted musical possibilities; musicians may well be trained differently; new types of toys and musical instruments might be created; and music could be manifested visually (and geometry manifested aurally) in previously unimagined ways.
Geometrical music theory represents a culminating moment in the longstanding marriage of music and math. That marriage began when Pythagoras described pleasing musical intervals with simple mathematical ratios more than 2,600 years ago and further evolved during the Middle Ages when deep thinkers used those same ratios to model the “music of the spheres”—what many at that time believed to be the literally harmonious movements of the sun, moon and planets.
Understanding and interpreting music, say the authors of the study, is a process of discarding information—which in turn is the key to discovering its underlying mathematical structure.
A chord, for example, may be variously described as “the opening chord of Bach’s G minor Sonata for Unaccompanied Violin,” “G minor triad,” “minor triad” or simply “triad.”
“Each of these terms can refer to the same musical object at different levels of abstraction,” Callender said.
“We also experience a sense of distance when moving from one chord to another,” he said. “Changing one note just a little feels like a small motion between similar chords, while changing many notes by large amounts feels like a large motion between dissimilar chords.
“So, building on my own research and that of my Princeton colleague, our research modeled these spatial intuitions about chords at various levels of abstraction geometrically, using what mathematicians call ‘quotient spaces.’ Most of those spaces are warped and twisted such that they contain multiple ‘straight’ paths connecting any pair of points,” Callender said.
“Imagine being near the peak of a mountain and needing to get to the immediately opposite location,” he said. “You could proceed clockwise around the peak, counter-clockwise, or directly over the peak. These same three paths represent unique types of motions between major and minor triads in the space of three-note chord types, which is a cone. In fact, these motions and chords have been ubiquitous in Western music since medieval times to the present day.”
At each level of abstraction, musical objects are grouped into families of chords or melodies. Mathematical structure is assigned to the “families” so that they can be represented as points within complex geometrical spaces in much the same way that “x” and “y” coordinates correspond to points on a two-dimensional plane in simple high school algebra. The different families produce an exotic maze of diverse geometrical spaces such as twisted triangular donuts and pinched cones—and even some spaces that mathematicians haven’t dreamed up names for yet.
“My fellow researchers and I have found it thrilling to discover unexplored areas of mathematics in the course of solving musical problems,” Callender said.
“Professor Callender and his colleagues at Yale and Princeton are working at the forefront in this rarified area of music theory,” said Don Gibson, dean of the FSU College of Music. “Their research—and its publication in Science—represents a signal achievement in the discipline.”
A few years ago, Princeton University music theorist and composer Dmitri Tymoczko was sitting in the living room of his home playing with a piece of paper. Printed on the sheet were rows and columns of dots representing all the two-note chords that can be played on a piano — AA, AB b , AB, and so on for the rows; AA, B b A, BA, and so on for the columns. It was a simple drawing, something a child could make, yet Tymoczko felt that the piece of paper was trying to show him something that no one ever had seen before.
Suddenly Tymoczko (pronounced tim-OSS-ko) realized that if he cut two triangles from the piece of paper, turned one of the triangles upside down, and reconnected the two triangles where the chords overlapped, the two-note chords on one edge of the resulting strip of paper would be the reversed versions of those on the opposite edge. If he then twisted the paper and attached the two edges, the chords would line up. “That’s when I got a tingly feeling in my fingers,” he says.
Tymoczko had discovered the fundamental geometric shape of two-note chords. They occupy the space of a Möbius strip, a two-dimensional surface embedded in a three-dimensional space. Music is not just something that can be heard, he realized. It has a shape.
He soon saw that he could transform more complex chords the same way. Three-note chords occupy a twisted three-dimensional space, and four-note chords live in a corresponding but impossible-to-visualize four-dimensional space. In fact, it worked for any number of notes — each chord inhabit ed a multidimensional space that twisted back on itself in unusual ways — a non-Euclidean space that does not adhere to the classical rules of geometry. A physicist friend told him that these odd multidimensional spaces were called orbifolds — a name chosen by the graduate students of Princeton mathematician William Thurston, who first described them in the 1970s. In the 1980s, physicists found a few applications for orbifolds in arcane areas of string theory. Now Tymoczko had discovered that music exists in a universe of orbifolds.
Tymoczko’s insight, made possible through a research collaboration with Clifton Callender from Florida State University and Ian Quinn from Yale University, has created “quite a buzz in Anglo-American music-theory circles,” says Scott Burnham, the Scheide Professor of Music History at Princeton. His work has “physicalized” music. It provides a way to convert melodies and harmonies into movements in higher dimensional spaces. It has given composers new tools to write music, has revealed new ways to teach music students, and has revealed surprising musical connections between composers as distant as Palestrina — the Italian Renaissance composer — and Paul McCartney.
In a book to be published in March by Oxford University Press, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, Tymoczko uses the connection between music and geometry to analyze the music of the last millennium and position modern composers in a new landscape. He rejects the idea that music can be divided into distinct genres. As Tymoczko sees it, medieval polyphony, the high classical music of Beethoven and Mozart, the chromatic romanticism of Wagner and Debussy, the jazz improvisations of Bill Evans, and the Beatles’ Sgt. Pepper’s Lonely Hearts Club Band all are built on the same handful of principles. Tymoczko writes in the preface: “It would make me happy to think that these ideas will be helpful to some young musician, brimming with excitement over the world of musical possibilities, eager to understand how classical music, jazz, and rock all fit together — and raring to make some new contribution to musical culture.”
The link between geometry and music has deep roots. Sometime between 530 and 500 B.C., in the town of Kroton on the rocky southern coast of Italy, Pythagoras and his followers made one of the most consequential discoveries in the history of science. If the string of a harp is shortened by half, it creates a tone one octave above that of the unshortened string. If the original string is shortened by two-thirds, the resulting tone is separated from the octave tone by a euphonious interval we know today as a fifth. Further experimentation showed that dividing the string into four parts produces intervals now known as fourths, with fur ther divisions of the string producing the familiar 12-note chromatic scale that the Greeks bequeathed to history.
Pythagoras and his followers thought big. The rational division of the musical scale was not just beautiful or pleasing — it was a sign that the universe was constructed on a rational basis and could be understood. “It was the first consistent realization that there is a mathematical rationality in the universe and that the human mind can make sense of that rationality,” says Kitty Ferguson, the author of The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space.
Two-and-a-half millennia later, fifths and fourths are still the basis not just of three-chord rock-and-roll, but of much basic music theory. Students learn how to recognize intervals and relate those intervals to different kinds of scales, like major and minor scales. They practice transposing, inverting, and modulating melodies and chords. They absorb, perhaps without fully realizing it, the mathematician Gottfried Leibniz’s injunction that music is the “unknowing exercise of our mathematical faculties.”
Tymoczko falls squarely into the mathematical tradition in music. His father, Thomas, was a well-known philosopher of mathematics at Smith College who was fascinated by the use of computers in mathematics. His sister, Julianna, is a mathematician specializing in algebraic geometry at the University of Iowa.
But Tymoczko, growing up in the 1980s in Northhampton, Mass., spent more time listening to the Talking Heads, John Coltrane, and Brian Eno than solving equations. He swapped his piano lessons for guitar lessons and began playing in bands. He entered Harvard intending to study music, but the abstract and atonal music his professors preferred left him cold, and he switched from composition to philosophy. After studying philosophy at Oxford on a Rhodes scholarship, he kicked around Harvard for a few years as a teaching assistant, composing on the side and dabbling in journalism. Finally he decided to become serious about music again and enrolled in music graduate school at the University of California, Berkeley.
Tymoczko got interested in the mathematics of music for what he calls a “selfish reason” — he wanted his music to sound better. His music is distinctly modern but also “tonal” — a term that he defines in his new book as music that adheres to several basic principles. In compositions such as “Four Dreams” and “The Agony of Modern Music” (termed “brilliant” by music critic Alex Ross), he combines the free-flowing vibe of jazz, the strangeness of 20th-century music, and the raw exuberance of the rock ‘n’ roll he listened to growing up. Combining such different kinds of music is a delicate balancing act, and to do it well, Tymoczko decided that he needed to understand music at a more fundamental level.
Early on, Tymoczko recognized that tonal music has two major distinguishing features. First, melodies tend to move short distances from note to note — a characteristic known in music theory as “efficient voice leading.” Think of the tunes you can hum off the top of your head. They probably have notes that are fairly close to each other, rather than strings of notes that jump wildly up and down. Melodies with close-together notes are easier to sing and to play on most instruments, and they’re easier to listen to.
A second important feature of tonal music is that it uses harmonies that sound good rather than bad. For reasons that aren’t entirely clear, humans (including infants) tend to prefer certain kinds of chords to other kinds of chords. Echoing the findings of Pythagoras, we tend to like chords that divide the octave almost, but not completely, evenly. Triads, seventh chords, ninths, elevenths — the archetypal chords of everything from early polyphony to jazz — divide the octave into ever-smaller but approximately equal-sized segments, much as the vertical lines on a ruler divide feet and inches into equal-sized intervals. In contrast, notes that are next to each other sound harsh and dissonant when played together, like a child banging his fists on a piano.
These two properties may seem to be unrelated. But the “amazing and mysterious” thing about music, Tymoczko says, is that each requires the other. Three singers can go from a pleasing C-major chord to the complementary and more plaintive A-minor chord by moving just one note: changing from CEG to CEA. Someone playing “Hey Jude” on the piano can move his or her fingers very little while moving from one sonorous chord to another. “Miraculously, the chords that sound good together and the ones that produce efficient voice leading are the same,” Tymoczko says.
Tymoczko and other music theorists knew that these obser vations must have a mathematical representation, and previ ous theorists had captured some of these properties using geo metric ideas. He developed an intense e-mail relationship with Quinn, Callender, and several other theorists who were working on the same general problem: How can music be captured mathematically in the most powerful possible way? “People had been coming up with ad hoc solutions,” says Yale’s Quinn. “We wanted to do something general.”
Initially, Tymoczko took an algebraic approach, representing chord progressions by equations. But a critical contribution came from Callender. In a paper published in Music Theory Online, he explained how you could arrange two-note chords on a two-dimensional surface, so that changes from one chord to another could be represented by movements of a point on the surface. However, this surface had some very odd properties. For example, if a point moving on the surface hit an edge it would essentially bounce off, like a billiard ball careening off a bumper. In mathematical terms, the surface had a discontinuity or “singularity.”
Callender, Quinn, and Tymoczko all knew that the unusual properties of the space meant that it somehow harbored additional, non-Euclidean dimensions. But how were those dimensions configured? This is where Tymoczko’s checkered academic history came in. From his time at Harvard and Berkeley, he had lots of mathematician and physicist friends. One suggested making a physical representation of the space and playing with it. That’s the piece of paper he printed out, stared at in his living room, and used to discover the Möbius strip representation of two-note chords. Tymoczko’s scientist friends also told him that when they make a big discovery they submit the work to Science. A 2006 paper by Tymoczko and a 2008 paper by all three collaborators became the first on music theory in Science’s 129-year history.
Tymoczko’s discovery of the orbifold structure of musical space immediately had an important consequence: It explained his earlier observations about efficient voice leading and euphonious chords. When orbifolds are used to represent musical sounds, the chords that most evenly divide the octave reside in the central regions of the space. For example, in the Möbius strip representation of all two-note chords, fourths and fifths occupy the central area of the strip, while dissonant chords of closely spaced notes huddle near the edge of the strip. Composers can move from one euphonious chord to another while moving short distances in the central region of a musical space. Movements of short distances correspond to notes that are close together, producing singable melodies.
The implications for composers are momentous, says Tymoczko. “Imagine that you’re a blind man in a city who knows how to get from your home and to a coffee shop. Suddenly your vision is restored. You realize that not only are the post office and coffee shop very close to each other, but that you can get there many other ways, and many of them are better than the way you knew.”
The geometric representation of music also provides a powerful way to analyze past compositions. In his new book, Tymoczko uses geometric concepts to “retell the history of Western music.” For example, on the Möbius strip representation of two-note chords, medieval composers of two-voice polyphony tended to remain near the center of the space, with occasional leaps to its edge whenever the two singers were to voice the same note. By the high classical period of Bach, Mozart, and Beethoven, composers had become experts at crafting harmonious pieces that fully exploited these central regions of musical orbifolds. Later composers then began to branch out, exploring new regions of musical space. By the 20th century, minimalist composers like Steve Reich were relying on burbling arpeggios of closely spaced notes right on the edges of orbifolds, while jazz composers like Miles Davis and Bill Evans were further exploring the chromatic spaces between the centers and edges of orbifolds pioneered by Wagner and Debussy.
This way of visualizing music provides new insights into how composers wrote some of the world’s most beautiful music. In his living room, Tymoczko plays Chopin’s “Prelude in E” through the speakers of his laptop as the computer mon itor displays a three-dimensional projection of a four-dimensional orbifold. With each new chord, a ball moves through a latticework of points on the screen. Repeatedly the prelude returns to a particular point in the lattice — representing a diminished seventh chord — from which it branches first to a chord on the immediate right and then to a chord on the immediate left. But at other points in the prelude, the ball moves freely along the lattice, a kind of improvisation more commonly associated with modern music. “Composers in the 19th century had an intuitive understanding of the bizarre geometry of musical chord space,” Tymoczko says. “In fact, they had a better feel for non-Euclidean, higher-dimensional spaces than did their mathematical contemporaries.”
Tymoczko’s approach also makes it possible to compare very different kinds of music. At various times in his new book he compares Schumann’s “Chopin” movement in Carna val to Nirvana’s “Heart-Shaped Box,” Shostakovich’s “G-Minor Piano Quintet” to the Black Sabbath song “Sabbath, Bloody Sabbath,” and Philip Glass’ opera Einstein on the Beach to the opening of the TV show Battlestar Galactica. “Music that superficially seems quite different, like Renaissance music and jazz, make remarkably similar use of musical space,” he says.
In fact, Tymoczko argues, the music of the past millennium in the West, and much of non-Western music as well, constitutes an “extended common practice” characterized by the continued broadening of usable musical space. That history has, of course, been shaped by the creativity and idiosyncrasies of individual composers. But it has followed certain broad paths because of how musical orbifolds are configured. It’s like a mountaineer ascending a rock face, he says. In principle, the climber is free to move in any direction. But the structure of the rock provides certain natural routes, offering handholds and footholds along the way. “The trick for the historian is to make room for historical contingency while also capturing the way in which music history sometimes follows the path of least resistance, like a climber ascending a cliff by way of a particularly inviting chute.”
Tymoczko’s work falls into the category of basic research — it doesn’t have a guaranteed payoff. Yet already it’s finding lots of applications. Among the first enthusiasts were composers who use computers to produce musical ideas — and in some cases, complete compositions. Composer Michael Gogins, for example, uses orbifolds to add harmonic and melodic structure to notes generated randomly by the computer. “It gives you more power,” he says.
Some music theorists have begun using orbifolds to teach composition and theory to undergraduates, a trend Tymoczko hopes his new book will encourage. The value of the idea, says Quinn at Yale, is that a geometric approach demonstrates both the possibilities and constraints of music. “Once composers said, ‘Let’s get out of the box,’ that’s when they began wrapping themselves around the singularities and other weird parts of the space.”
Other applications of orbifolds are still on drawing boards. Tymoczko envisions orbifold-inspired children’s toys that could teach chord structure through play. A dancer could move through a space wired to generate the chords associated with each location. Composers could write music by moving through orbifolds rather than writing notes on staves.
Has the discovery of music’s geometry achieved Tymoczko’s original purpose — to make his music sound better? It helps, he says, but it can’t replace inspiration. Composition has both a mechanical phase and a mysterious phase. The use of geometric ideas can suggest melodies and harmonies — it frees a composer “from repeating the formulas of the past.” But when he’s writing music, says Tymoczko, who is working on an album of jazz, funk, and classical fusion pieces, the mechanical part must be followed by the mysterious part, when a composer “learns what a piece is really about.”
The saxophone genius Charlie Parker put it a bit differently: Learn all the theory you can, he said, and then forget it when you play.
Steve Olson, based in Seattle, has written about genetics, race, evolution, climate change, talent, and punk-rock music, among other things. His most recent book, Anarchy Evolution: Faith, Science, and Bad Religion in a World Without God, co-written by Greg Graffin, was published in September by itbooks.
How is the Beatles’ “Help!” similar to Stravinsky’s “Dance of the Adolescents?” How does Radiohead’s “Just” relate to the improvisations of Bill Evans? And how do Chopin’s works exploit the non-Euclidean geometry of musical chords?
In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an age-old question: what makes music sound good?
A Geometry of Music provides an accessible introduction to Tymoczko’s revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing relationships among familiar chords and scales, giving readers the tools to translate between the musical and visual realms and revealing surprising degrees of structure in otherwise hard-to-understand pieces.
Tymoczko uses this theoretical foundation to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the “common practice” period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. He discusses a host of familiar pieces by a wide range of composers, from Bach to the Beatles, Mozart to Miles Davis, and many in between.
A Geometry of Music is accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers’ understanding, along with a series of appendices that explore the technical details of this exciting new theory.
Table of Contents
PREFACE
PART I. Theory
CHAPTER 1. Five Components of Tonality 1.1 The five features. 1.2. Perception and the five features. 1.3 Four Claims. A. Harmony and counterpoint constrain each other. B. Scale, macroharmony, and centricity are independent. C. Modulation involves voice leading. D. Music can be understood geometrically. 1.4 Music, magic, and language. 1.5 Outline of the book, and a suggestion for impatient readers.
CHAPTER 2. Harmony and Voice Leading 2.1 Linear pitch space. 2.2 Circular pitch-class space. 2.3 Transposition and inversion as distance-preserving functions. 2.4 Musical objects. 2.5 Voice leadings and chord progressions. 2.6 Comparing voice leadings. 2.7Voice-leading size. 2.8 Near identity. 2.9 Harmony and counterpoint revisited. 2.10 Acoustic consonance and near-evenness
CHAPTER 3. The Geometry of Chords 3.1 Ordered pitch space. 3.2 The Parable of the Ant. 3.3 Two-note chord space. 3.4 Chord progressions and voice leadings in two-note chord space. 3.5 Geometry in analysis. 3.6 Harmonic consistency and efficient voice leading. 3.7 Pure parallel and pure contrary motion. 3.8 Three-dimensional chord space. 3.9 Higher-dimensional chord spaces. 3.10 Voice leading lattices. 3.11 Triads are from Mars, seventh chords are from Venus. 3.12 Two musical geometries. 3.13 Study guide.
CHAPTER 4. Scales 4.1 A scale is a ruler. 4.2 Scale degrees, scalartransposition, scalar inversion. 4.3 Evenness and scalar transposition. 4.4 Constructing common scales. 4.5 Modulation and voice leading. 4.6 Voice leading between common scales . 4.7 Two examples. 4.8 Scalar and interscalar transposition. 4.9 Interscalar transposition and voice leading. 4.10 Combining interscalar and chromatic transpositions.
CHAPTER 5. Macroharmony and Centricity 5.1 Macroharmony. 5.2 Small-gap macroharmony. 5.3 Pitch-class circulation. 5.4 Modulating the rate of pitch-class circulation. 5.5 Macroharmonic consistency. 5.6 Centricity. 5.7 Where does centricity come from? 5.8 Beyond “tonal” and “atonal.”
PART II. History and Analysis
CHAPTER 6. The Extended Common Practice 6.1Disclaimers. 6.2 Two-voice medieval counterpoint. 6.3 Triads and the Renaissance. 6.4 Functional harmony. 6.5 Schumann’s Chopin. 6.6 Chromaticism. 6.7 Twentieth-century scalar music. 6.8 The extended common practice.
CHAPTER 7. Functional Harmony 7.1 The thirds-based grammar of elementary tonal harmony. 7.2 Voice leading in functional harmony. 7.3 Sequences. 7.4 Modulation and key distance. 7.5 The two lattices. 7.6 A challenge from Schenker. CHAPTER 8. Chromaticism 8.1 Decorative chromaticism. 8.2 Generalized augmented sixths. 8.3 Brahms and Schoenberg. 8.4 Schubert and the major-third system. 8.5 Chopin’s tesseract. 8.6 The Tristan Prelude. 8.7 Alternative approaches. 8.8Conclusion
CHAPTER 9. Scales in Twentieth-Century Music 9.1 Three scalar techniques. 9.2 Chord-first composition. A. Grieg’s “Drömmesyn,” (Vision), Op. 62 no. 5 (1895). B. Debussy’s “Fetes” (1899). C. Michael Nyman’s “The Mood That Passes Through You” (1993). 9.3 Scale-first composition. A. Debussy’s “Des pas sur la neige” (1910). B. Janácek’s “On an Overgrown Path,” Series II, no. 1 (1908). C. Shostakovich’s Fs minor Prelude and Fugue, Op. 87 (1950). D. Reich’s “New York Counterpoint” (1985). E. Reich’s “The Desert Music,” movement 1 (1984). F. The Who’s “Can’t Explain” (1965) and Bob Seger’s “Turn the Page” (1973). 9.4 The Subset Technique. A. Grieg’s “Klokkeklang,” (Bell Ringing), Op. 54 no. 6 (1891). B. “PetitAirs,” from Stravinsky’s Histoire du Soldat (1918). C. Reich’s “City Life” (1995). D. Stravinsky’s “Dance of the Adolescents” (1913). E. The Miles Davis Group’s “Freedom Jazz Dance” (1966). 9.5 Conclusion.
CHAPTER 10. Jazz. 10.1 Basic jazz voicings. 10.2 From thirds to fourths. 10.3 Tritone substitution. 10.4 Altered chords and scales. 10.5 Bass and upper-voice tritone substitutions. 10.6 Polytonality, sidestepping, and “playing out.” 10.7 Bill Evans’s “Oleo.” 10.8 Jazz as modernist synthesis.
CONCLUSION
APPENDIX A. Measuring voice-leading size APPENDIX B.Chord geometry: a technical look. APPENDIX C.Discrete voice leading lattices. APPENDIX D. The interscalar interval matrix. APPENDIX E.Scale, macroharmony, and Lerdahl’s “basic space.” APPENDIX F. Some study questions, problems, and activities.
Prelude: transposition along a collection 2. Rock logic 1. A melodic principle 2. A harmonic principle 3. A first chord-loop family 4. Two more families 5. Shepard-tone passacaglias 6. Minor triads and other trichords 7. A fourth family 8. Other modalities 9. Function and retrofunction 10. Continuity or reinvention?
Prelude: the Tinctoris transform 3. Line and configuration 1. The imperfect system 2. Voice exchanges 3. Other intervals 4. The circle of diatonic triads 5. Voice exchanges and multiple chord types 6. Four-voice triadic counterpoint 7. Counterpoint within the chord 8. Seventh chords 9. Harmony and counterpoint
Prelude: sequence and function 4. Repetition 1. Repetition reimagined 2. Repeating contrapuntal patterns 3. The geometry of two-voice sequences 4. Three voices and the circle of triads 5. Three voices arranged 2+1 6. Four voices 7. Contrary-motion sequences 8. Melodic sequences and near sequences 9. Near sequences 10. Sequences as reductional targets
Prelude: three varieties of analytical reduction 5. Nonharmonic tones 1. The first practice and the SNAP system 2. Schoenberg’s critique 3. Monteverdi’s “Ohimè” 4. The standardized second practice 5. A loophole 6. After nonharmonicity
Prelude: functional and scale-degree analysis 6. The origins of functional harmony 1. The logical structure of protofunctionality 2. Similarities and differences 3. Origin and meaning 4. Harmony and polyphony 5. The Pope Marcellus Kyrie 6. A broader perspective 7. “I Cannot Follow”
Prelude: could the Martians understand our music? 7. Functional progressions 1. A theory of harmonic cycles 2. A more principled view 3. Rameau and Bach 4. Functional melody, functional harmony 5. Fauxbourdon and linear idioms 6. Sequences 7. Bach the dualist
Prelude: chromatic or diatonic? 8. Modulation 1. Two models of key distance 2. Enharmonicism and loops in scale space 3. Minor keys 4. Modulatory schemas 5. Up and down the ladder 6. Modal homogenization and scalar voice leading 7. Generalized set theory
Prelude: hearing and hearing-as 9. Melodic strategies 1. Strategy and reduction 2. Two models of the phrase 3. Chopin and the Prime Directive 4. An expanded vocabulary of melodic templates 5. Simple harmonic hierarchy 6. The four-part phrase 7. Grouping, melody, harmony 8. Beyond the phrase: hierarchy at the level of the piece
Prelude: why Beethoven? 10. Beethoven theorist 1. Meet the Ludwig 2. From schema to flow 3. The Tempest 4. The Fifth Symphony 5. The “Pastorale” sonata, op. 28 6. Schubert’s Quartettsatz 7. The prelude to Lohengrin
From musical chords to twin primes / Jack Douthett, David Clampitt & Norman Carey
Hypercubes and the generalized Cohn cycle / Jack Douthett, Peter Steinbach & Rick Hermann
Associahedra, combinatorial block designs and related structures / Franck Jedrezejewski
Rhythmic and melodic l-canons / Jeremy Kastine
The Fibonacci sequence es metric suspension in Luigi Nono’s Il canto sospeso / Jon Kochavi
Note samba : navigating notes and their meanings within modes and exo-modes / Thomas Noll
Difference sets and all-directed-interval chords / Robert W. Peck
Harmonious opposition / Richard Plotkin
Orbifold path models for voice leading : dealing with doubling / James R. Hughes
Remarks on the geometry of chords / Thomas A. Ivey
Theoretical physics and category theory as tools for analysis of musical performance and composition / Maria Mannone
Intuitive musical homotopy / Aditya Sivakumar & Dmitri Tymoczko
Geometric generalizations of the Tonnetz and their relation to Fourier phases space / Jason Yust
Deterministic geometries : a technique for the systematic generation of musical elements in composition / Brent A. Milam
Flamenco music and its computational study / Francisco Gómez
Examining fixed and relative similarity metrics through jazz melodies / David J. Baker & Daniel Shanahan
In search of arcs of prototypicality / Daniel Shanahan.
Publisher’s summary Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music. remove. (source: Nielsen Book Data)
Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.
The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music.
remove Contents:
Section I:
From Musical Chords to Twin Primes (Jack Douthett, David Clampitt and Norman Carey)
Hypercubes and the Generalized Cohn Cycle (Jack Douthett, Peter Steinbach and Richard Hermann)
Associahedra, Combinatorial Block Designs and Related Structures (Franck Jedrzejewski)
Rhythmic and Melodic L-canons (Jeremy Kastine)
The Fibonacci Sequence as Metric Suspension in Luigi Nono’s II Canto Sospeso (Jon Kochavi)
One Note Samba: Navigating Notes and Their Meanings Within Modes and Exo-modes (Thomas Noll)
Difference Sets and All-Directed-Interval Chords (Robert W Peck)
Harmonious Opposition (Richard Plotkin)
Section II:
Orbifold Path Models for Voice Leading: Dealing with Doubling (James R Hughes)
Reflections on the Geometry of Chords (Thomas A Ivey)
Theoretical Physics and Category Theory as Tools for Analysis of Musical Performance and Composition (Maria Mannone)
Intuitive Musical Homotopy (Aditya Sivakumar and Dmitri Tymoczko)
Geometric Generalizations of the Tonnetz and Their Relation to Fourier Phases Spaces (Jason Yust)
Deterministic Geometries: A Technique for the Systematic Generation of Musical Elements in Composition (Brent A Milam)
Section III:
Flamenco Music and Its Computational Study (Francisco Gómez)
Examining Fixed and Relative Similarity Metrics Through Jazz Melodies (David J Baker and Daniel Shanahan)
In Search of Arcs of Prototypicality (Daniel Shanahan)
Readership: Students and researchers in Mathematical Music Theory.Mathematics and Music;Algebra;Geometry;Topology;Graph Theory;Combinatorics;Distance and Similarity Measures;Discrete Fourier Transform0 Key Features:
It includes the most prominent authors in the field
It gathers a gamut of the most recent work in the field, which is something very difficult to find in one volume
It will appeal to mathematicians, music theorists, and computer scientists. Within mathematics, it offers a variety of areas and techniques related to musical phenomena that cannot be found together in other volumes
1 Department of Mathematics and Computer Science, Saint Joseph’s University, Philadelphia, PA 19131, rhall@sju.edu
Music theorists have frequently invoked geometry in modeling musical objects such as chords, rhythms, and scales; however, no unified geometric perspective has hitherto emerged. On page XXX of this issue, Callender, Quinn, and Tymoczko (1) demonstrate that many musical terms can be understood as expressing symmetries of n -dimensional space. Identifying— “gluing together”—points related by these symmetries produces a family of non-Euclidean quotient spaces that subsume a large number of geometric models proposed in the literature. The use of mathematics to describe, analyze, and create music goes back millennia. Questions have previously emerged in music theory that are appealing, nontrivial, and, in several cases, connected to other scientific fields (2,3,4). Math inspired composers such as Schoenberg, Messaien, and Xenakis; musical investigations have even motivated mathematical discoveries (5). However, the “geometrical music theory” proposed by Callender, Quinn, and Tymoczko stands out both for the breadth of its musical implications and the depth of its mathematical content. It suggests a wealth of new techniques for studying music theoretical topics, including chord similarity and melodic contour, and provides attractive visualizations of harmonic relationships. Although the authors’ use of contemporary geometry departs from the discrete mathematics normally used to describe music, they make a convincing argument for the validity of their models. Musicians think in degrees of abstraction: While “middle C” is a particular pitch (frequency measured on a logarithmic scale), the letter name (or pitch class) “C” refers to any pitch that is a whole number of octaves away from it. Any collection of C, E, and G notes is a “C major chord.” A “major chord” is any transposition of a C major chord; a “consonant triad” is any three-note chord containing a major third, a minor third, and a perfect fifth. Music theorists invoke the mathematical concept of equivalence class —a set of objects that are “the same” if we ignore certain information—to describe these musical structures. Callender, Quinn, and Tymoczko show that many musical terms describe equivalence classes under combinations of five basic “OPTIC” relations: Octave shifts, permutation (reordering), transposition (the relation between pitches sharing the same succession of intervals, regardless of their starting note), inversion (turning a sequence “upside down”), and cardinality equivalence (ignoring repetitions). The authors go beyond traditional music theory by showing that there are two distinct ways in which these symmetries can apply to progressions of musical objects: “uniformly,” where the same symmetry applies to each object in a progression, and “individually,” where distinct symmetry operations apply to the harmonies in a progression. This distinction allows them to formalize the relationships among a large number of musical terms, as in table S1 of their paper. Following Tymoczko (6), they develop geometrical models of these relationships. Pitches correspond to real numbers; each point in the Euclidean n -dimensional “configuration” space represents a sequence of n pitches. Points near each other differ by microtones (a C major chord
The Geometry of Music
A composer has taken equations from string theory to explain why Bach and bebop aren’t so different
When you first hear them, a Gregorian chant, a Debussy prelude and a John Coltrane improvisation might seem to have almost nothing in common–except that they all include chord progressions and something you could plausibly call a melody. But music theorists have long known that there’s something else that ties these disparate musical forms together. The composers of these and virtually every other style of Western music over the past millennium tend to draw from a tiny fraction of the set of all possible chords. And their chord progressions tend to be efficient, changing as few notes, by as little as possible, from one chord to the next.
Exactly how one style relates to another, however, has remained a mystery–except over one brief stretch of musical history. That, says Princeton University composer Dmitri Tymoczko, “is why, no matter where you go to school, you learn almost exclusively about classical music from about 1700 to 1900. It’s kind of ridiculous.”
But Tymoczko may have changed all that. Borrowing some of the mathematics that string theorists invented to plumb the secrets of the physical universe, he has found a way to represent the universe of all possible musical chords in graphic form. “He’s not the first to try,” says Yale music theorist Richard Cohn. “But he’s the first to come up with a compelling answer.”
Tymoczko’s answer, which led last summer to the first paper on music theory ever published in the journal Science, is that the cosmos of chords consists of weird, multidimensional spaces, known as orbifolds, that turn back on themselves with a twist, like the Möbius strips math teachers love to trot out to prove to students that a two-dimensional figure can have only one side. Indeed, the simplest chords, which consist of just two notes, live on an actual Möbius strip. Three-note chords reside in spaces that look like prisms–except that opposing faces connect to each other. And more complex chords inhabit spaces that are as hard to visualize as the multidimensional universes of string theory.
But if you go to Tymoczko’s website music.princeton.edu/~dmitri) you can see exactly what he’s getting at by looking at movies he has created to represent tunes by Chopin and, of all things, Deep Purple. In both cases, as the music progresses, one chord after another lights up in patterns that occupy a surprisingly small stretch of musical real estate. According to Tymoczko, most pieces of chord-based music tend to do the same, although they may live in a different part of the orbifold space. Indeed, any conceivable chord lies somewhere in that space, although most of them would sound screechingly harsh to human ears.
The discovery is useful for at least a couple of reasons, says Tymoczko. “One is that composers have been exploring the geometrical structure of these maps since the beginning of Western music without really knowing what they were doing.” It’s as though you figured out your way around a city like Boston, for example, without realizing that some of your routes intersect. “If someone then showed you a map,” he says, “you might say, ‘Wow, I didn’t realize the Safeway was close to the disco.’ We can now go back and look at hundreds of years of this intuitive musical pathmaking and realize that there are some very simple principles that describe the process.”
Computational geometric aspects of rhythm, melody, and voice-leading
Godfried Toussaint 1 School of Computer Science and Center for Interdisciplinary Research in Music Media and Technology, McGill University, Montréal, Québec, Canada
Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.
Exploring Musical Spaces: A Synthesis of Mathematical Approaches
By Julian Hook
Oxford, 2022
Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models―the first examples of the musical spaces of the book’s title―depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, diatonic theory, and scale theory.
Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques.
A guided tour of the mathematical principles inherent in music.
Taking a “music first” approach, Gareth E. Roberts’s From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics.
Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational.
Topics discussed in the book include
• Rhythm • Introductory music theory • The science of sound • Tuning and temperament • Symmetry in music • The Bartók controversy • Change ringing • Twelve-tone music • Mathematical modern music • The Hemachandra–Fibonacci numbers and the golden ratio • Magic squares • Phase shifting
Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students’ analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.
physics and philosophy, but the most important is music, for in the patterns of the arts are the keys to all learning.
Plato, c. 428-347 BC
The first concrete argument for a fundamental link between mathematics and music was perhaps made by the early philosopher and mathematician Pythagoras (569-475 BC), often referred to as the “father of numbers.” He can also be considered the “father of harmony,” given that his discovery of the overtone series and analyses of the acoustics and ratios involved in music have served as the foundation of harmony in western-hemisphere music composition ever since. The Pythagorean, Quadrivium and Platonic classifications of mathematics were based on hierarchical dimensions, starting with arithmetic, then geometry, astronomy and finally music.
Reportedly, Pythagoras experimented with the tones produced when plucking strings of different lengths. He found that some specific ratios of string lengths created pleasing combinations (“harmonies”) and others did not. Based on his careful observations, Pythagoras identified the physics of intervals, or distances between notes, that form the primary harmonic system which is still used today (Parker, 2009, pp. 3-5).
Music is based on proportional relationships. The mathematical structure of harmonic sound begins with a single naturally occurring tone, which contains within it a series of additional frequencies above its fundamental frequency (“overtones”), of which we are normally unaware on a conscious level. Within this harmonic or overtone series, there is a mathematical relationship between the frequencies – they are specific integer multiples of each other. For example, if the slowest frequency (the “fundamental”) were 100 Hz, then the overtones would be 2 x 100 (200 Hz), 3 x 100 (300 Hz) and so forth. (The overtone series is often referred to as harmonics.)
Pythagoras observed several ratios of sound wave frequencies and the corresponding intervals between them, including 4:3 (known to musicians as the interval of a perfect fourth, or two pitches that are five semitones apart from each other) and 3:2 (a perfect fifth, seven semitones apart). Note that pitch is the frequency or rate of vibration of a physical source such as a plucked string.
The most prominent interval that Pythagoras observed highlights the universality of his findings. The ratio of 2:1 is known as the octave (8 tones apart within a musical scale).
When the frequency of one tone is twice the rate of another, the first tone is said to be an octave higher than the second tone, yet interestingly the tones are often perceived as being almost identical. For example, a woman’s voice may fluctuate around 220 Hertz while a man’s voice is around 110 Hertz, approximately half the frequency of the woman’s. However, if they sing together, it may sound as though they are singing the same melody together in unison, even though they are actually an octave apart. This 2:1 ratio is so elemental to what humans consider to be music, that the octave is the basis of all musical systems that have been documented – despite the diversity of musical cultures around the world. Moreover, this physical phenomenon is so fundamental that even non-human species such as monkeys and cats recognize it (Levitin, 2008, p. 31).
The inherent properties of physics and mathematics within music, perceived so long ago by Pythagoras, may help to explain why many physicists and mathematicians are also musicians. This point is illustrated by a quotation from Einstein: “The theory of relativity occurred to me by intuition, and music was the driving force behind that intuition….My new discovery was the result of musical perception” (Suzuki, 1969, 90).
*Portions of this article are adapted from Edel Sander’s chapter in Musik i forskola och tidiga skolar (2015), a Swedish textbook for music educators.
10 – Music theory and mathematics
from PART II – SPECULATIVE TRADITIONS
Published online by Cambridge University Press: 28 March 2008
In Chapter 6 of The Manual of Harmonics (early second century CE), Nicomachus of Gerasa narrates the legendary story of Pythagoras passing by the blacksmith’s shop, during which in an epiphany of sonorous revelation, he discovered the correlation of sounding intervals and their numerical ratios. According to Nicomachus, Pythagoras perceived from the striking of the hammers on the anvils the consonant intervals of the octave, fifth, and fourth, and the dissonant interval of the whole tone separating the fifth and fourth. Experimenting in the smithy with various factors that might have influenced the interval differences he heard (force of the hammer blows, shape of the hammer, material being cast), he concluded that it was the relative weight of the hammers that engendered the differences in the sounding intervals, and he attempted to verify his conclusion by comparing the sounds of plucked strings of equal tension and lengths, proportionally weighted according to the ratios of the intervals.
Physical and logical incongruities or misrepresentations in Nicomachus’s narrative aside, the parable became a fixture of neo-Pythagorean discourse because of its metaphoric resonance: it encapsulated the essence of Pythagorean understanding of number as material or corporeal, and it venerated Pythagoras as the discoverer of the mathematical ratios underlying the science of harmonics. The parable also established a frame of reference in music-theoretical thought in the association between music and number, or more accurately, music theory and mathematical models, since it is not through number alone but through the more fundamental notions of universality and truth embedded in Pythagorean and Platonic mathematics and philosophy that one can best begin to apprehend the broad range of interrelationships between music theory and mathematics.
A new way of visualizing Western music theory could inspire innovations in everything from musical instrument design to music composition techniques—even to new kinds of toys.
Western Music Moves in Three and Even Four (!) Dimensional Spaces: How the Pioneering Research of Princeton Theorist Dmitri Tymoczko Helps Us Visualize Music in Radical, New Ways
Supplementary Sets and regular complementary unending canons,
Dan Tudor Vuza,
Perspectives of New Music, Numbers 29(2), 30(1), 30(2), and 31(1). Journal of Mathematics and Music, Special Issue on Tiling Problems in Music, Vol. 3, No. 2
Computational Geometric Aspects of Rhythm, Melody, and Voice-Leading
Godfried Toussaint∗
School of Computer Science
and Center for Interdiisciplinary Research in Music Media and Technology McGill University Montr ́eal, Qu ́ebec, Canada
To appear in: Computational Geometry: Theory and Applications, 2009. doi:10.1016/j.comgeo.2007.01.003
Khajuraho temples use the 8×8 (64) Vastupurusamandala Manduka grid layout plan (left) found in Hindu temples. Above the temple’s brahma padas is a Shikhara (Vimana or Spire) that rises symmetrically above the central core, typically in a circles and turning-squares concentric layering design (right) that flows from one to the other as it rises towards the sky.[31][46]
Key Terms
Temple in Man
Panch Kosha Philosphy
Triguna Philosophy
Yoga Philosophy
Seven Chakras
Great Chain of Being
Higher/Lower Levels
Hierarchy Theory
Ascent of Men
Temple of Men
Beasts – Men – Devas – Gods
Ratha in Architecture
Triratha
Panchratha
Saptaratha
Navratha
Ayatan Plan Architecture
Ekayatan
Triayatan
Panchayatan
Saptayatan
Chhadyashikhar
Shikharanvit
Valabhichhandaj
Sandhar
Prasad
Nagara Architecture
Odisha
Chandella
Solanki Maru Gurjara
Mandapa in Architecture
Ardh Mandapa
Mandapa
Maha Mandapa
Vastu Purush Mandala
Ceilings in Architecture
Shikhar (Vimana, Spire)
Temple and the Tank
Temple and the Pond
Temple and the River
Temple and the Lake
Square and Circle
Earth and Heaven
As Above, So Below
As Below, So Above
Amalaka
Kalash
Garbh Graha
Outer to Inner
Lower to Higher
Purush Sukta
Shri Sukta
Antarala
Pada Devta
Lok Pals
Dwar Pals
Dik Pals
Ayadi Calculations
Toranas
Vitana (Ceiling)
Prakash and Vimarsh
Terrestial to Celestial
Nagara (North Indian) Hindu Temple Architecture
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar
Source: TEMPLE ARCHITECTURE AND SCULPTURE
Source: Symbolism in Hindu Temple Architecture and Fractal Geometry – ‘Thought Behind Form’
Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India
Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India
Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India
Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India
Source: Lakshmana Temple, Khajuraho, Madya Pradesh, India
Source: Lakshmana Temple, Khajuraho, Madya Pradesh, India
The layout plan of Kandariya Mahadeva Temple. It uses the 64 pada grid design. Smaller Khajuraho temples use the 9, 16, 36 or 49 grid mandala plan.[42]
Khajuraho temples, almost all Hindu temple designs, follow a grid geometrical design called vastu-purusha-mandala.[43] This design plan has three important components – Mandala means circle, Purusha is universal essence at the core of Hindu tradition, while Vastu means the dwelling structure.[44]
The design lays out a Hindu temple in a symmetrical, concentrically layered, self-repeating structure around the core of the temple called garbhagriya, where the abstract principle Purusha and the primary deity of the temple dwell. The shikhara, or spire, of the temple rises above the garbhagriya. This symmetry and structure in design is derived from central beliefs, myths, cardinality and mathematical principles.[45]
The circle of mandala circumscribe the square. The square is considered divine for its perfection and as a symbolic product of knowledge and human thought, while circle is considered earthly, human and observed in everyday life (moon, sun, horizon, water drop, rainbow). Each supports the other.[31] The square is divided into perfect 64 sub-squares called padas.[43]
Most Khajuraho temples deploy the 8×8 (64) padas grid Manduka Vastupurushamandala, with pitha mandala the square grid incorporated in the design of the spires.[42] The primary deity or lingas are located in the grid’s Brahma padas.
Khajuraho temples use the 8×8 (64) Vastupurusamandala Manduka grid layout plan (left) found in Hindu temples. Above the temple’s brahma padas is a Shikhara (Vimana or Spire) that rises symmetrically above the central core, typically in a circles and turning-squares concentric layering design (right) that flows from one to the other as it rises towards the sky.[31][46]
The architecture is symbolic and reflects the central Hindu beliefs through its form, structure, and arrangement of its parts.[47] The mandapas, as well as the arts, are arranged in the Khajuraho temples in a symmetric repeating patterns, even though each image or sculpture is distinctive in its own way. The relative placement of the images are not random but together they express ideas, just like connected words form sentences and paragraphs to compose ideas.[48] This fractal pattern that is common in Hindu temples.[49] Various statues and panels have inscriptions. Many of the inscriptions on the temple walls are poems with double meanings, something that the complex structure of Sanskrit allows in creative compositions.[29]
All Khajuraho temples, except one, face sunrise, and the entrance for the devotee is this east side.
An illustration of Khajuraho temple Spires (Shikhara, Vimana) built using concentric circle and rotating-squares principle. Four spires (left) are shown above, while the inside view of one Shikara ceiling (right) shows the symmetric layout.
Above the vastu-purusha-mandala of each temple is a superstructure with a dome called Shikhara (or Vimana, Spire).[44] Variations in spire design come from variation in degrees turned for the squares. The temple Shikhara, in some literature, is linked to mount Kailash or Meru, the mythical abode of the gods.[31]
In each temple, the central space typically is surrounded by an ambulatory for the pilgrim to walk around and ritually circumambulate the Purusa and the main deity.[31] The pillars, walls, and ceilings around the space, as well as outside have highly ornate carvings or images of the four just and necessary pursuits of life – kama, artha, dharma, and moksa. This clockwise walk around is called pradakshina.[44]
Larger Khajuraho temples also have pillared halls called mandapa. One near the entrance, on the east side, serves as the waiting room for pilgrims and devotees. The mandapas are also arranged by principles of symmetry, grids, and mathematical precision. This use of same underlying architectural principle is common in Hindu temples found all over India.[50] Each Khajuraho temple is distinctly carved yet also repeating the central common principles in almost all Hindu temples, one which Susan Lewandowski refers to as “an organism of repeating cells”.[51]
Source: PARAMETRIZING INDIAN KARNATA-DRAVIDA TEMPLE USING GEOMETRY
Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE
Layout of a Hindu temple pursues a geometrical design known as vastu-purusha-mandala, the name of which is derived from the three vital components of the design namely Vastu meaning Vaas or a place of dwelling; Purusha, meaning the Universal principle; and Mandala meaning circle. Vastupurushamandala is a mystical diagram referred in Sanskrit as a Yantra. The symmetrical and self-repeating model of a Hindu temple demonstrated in the design is derived from the primary convictions, traditions, myths, fundamentality and mathematical standards. According to Vastupurushamandala, the most sacred and typical template for a Hindu temple is the 8×8 (64) grid Manduka Hindu Temple Floor Plan also referred as Bhekapada and Ajira. The layout displays a vivid saffron centre with intersecting diagonals which according to Hindu philosophy symbolises the Purusha. The axis of the Mandir is created with the aid of the four fundamentally significant directions and thus, a perfect square is created around the axis within the available space. This square which is circumscribed by the Mandala circle and divided into perfect square grids is held sacred. On the other hand, the circle is regarded as human and worldly that can be perceived or noticed in daily life such as the Sun, Moon, rainbow, horizon or water drops. Both the square and the circle support each other. The model is usually seen in large temples while an 81 sub-square grid is observed in ceremonial temple superstructures. Each square within the main square referred as „Pada‟ symbolise a specific element that can be in the form of a deity, an apsara or a spirit. The primary or the innermost square/s of the 64 grid model called Brahma Padas is dedicated to Brahman. The Garbhagruha or centre of the house situated in the Brahma Padas houses the main deity. The outer concentric layer to Brahma Padas is the Devika Padas signifying facets of Devas or Gods which is again surrounded by the next layer, the Manusha Padas, with the ambulatory. The devotees circumambulate clockwise to perform Parikrama in the Manusha Padas with Devika Padas in the inner side and the Paishachika Padas, symbolising facets of Asuras and evils, on the outer side forming the last concentric square. The three outer Padas in larger temples generally adorn inspirational paintings, carvings and images with the wall reliefs and images of different temples depicting legends from different Hindu Epics and Vedic stories. Illustrations of artha, kama, dharma and moksha can be found in the embellished carvings and images adorning the walls, ceiling and pillars of the temples.
Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE
Source: VASTU PURUSHA MANDALA- A HUMAN ECOLOGICAL FRAMEWORK FOR DESIGNING LIVING ENVIRONMENTS
Source: Space and Cosmology in the Hindu Temple
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
According to Hindu philosophy, the main goal of man’s life is to achieve the ultimate liberation from the illusionary world where he suffers from his endless rebirth. Krishna says in Bhagavad Gita,9 if man worships, devotes and meditates to the manifested form of the wholeness of infinity, and sees the truth of its manifestation; he will surpass the confusion of the never-ending cycles of rebirth in this physical world, and finally will assimilate with the God. 10 Hindu calls it moksha.11 It brings the ultimate peace and harmony in man’s life. But how one, from the physical world, can practice to realize this ultimate truth of the creation? Brihatsamhita12 and Sthapatyaveda13 give the solution as the temple which should act as the microcosm of the cosmos [9]. It should be the bridge for the man of physical world to the God of divine world [10].
To connect the physical world with the divine world and to reflect the truth of creation, the layout of cosmos was copied graphically in the foundation of temple. Here, a mythical incident was interwoven where a formless being covered the sky and was, immediately, arrested down to the earth by the creator Brahma and other gods. This supernatural fact was depicted graphically by vastu purusha mandala, where vastu refers to physical environment, purusha refers to energy, power or cosmic being, and mandala is the diagram or chart. Central portion of the mandala represents the place of Brahma and other portions symbolize the other gods according to their capability. By laying down this metaphysical diagram in the foundation, various supernatural forces are captured beneath the temple whereas its centre is the source of cosmic energies. The basic shape of the vastu purusha mandala is a square which represents the earth [11]. Its four sides depict the four cardinal directions. It also symbolizes the order, the completeness of endless life and the perfectness of life and death [10]. According to Hindu philosophy, our mundane life is controlled by the number four—four castes, four stages of life, four great epochs or mahayugas, four heads of Brahma, the four Vedas etc. [12].
There are various types of vastu purusha mandala, which are nothing but the squares grids, produced from the basic shape; namely, a square which is known as sakala mandala. Each smaller square within the grid is called one pada. The number of pada may vary from 1, 4, 9, 16, 25 and so on 1024, where it follows the geometric progression of 1, 2, 3, 4, 5,…..,32 of common ratio 2. The mandala having even numbers of pada in its grid known as yugma squares mandala whereas the mandala, having odd numbers of pada known as ayugma squares mandala. Vastu purusha mandala is also known as different distinctive names according to the numbers of pada within the grid. The mandala having 1,4,9,16,25 and 36 numbers of pada within the grid are known as sakala mandala, pechaka mandala, pitah mandala, mahapitah mandala, manduka chandita mandala and para- masayika mandala, respectively14.
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Maṇḍala in Architecture: Symbolism and Significance for Contemporary Design Education in India
Source: Maṇḍala in Architecture: Symbolism and Significance for Contemporary Design Education in India
Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings
Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings
Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings
Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings
Source: Role of Fractal Geometry in Indian Hindu Temple Architecture
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.
Types of Vastu Purush Mandala
Source: Indian Architectural Theory: Contemporary Uses of Vastu Vidya
Sakala (1 x 1) = 1 Square
Pechaka (2 x 2) = 4 Squares
Pitha (3 x 3) = 9 Squares
Mahapitha (4 x 4) = 16 Squares
Upapitha (5 x 5) = 25 Squares
Ugrapitha (6 x 6) = 36 Squares
Sthandila (7 x 7) = 49 Squares
Manduka (Chandita) (8 x 8) = 64 Squares
Paramashayika (9 x 9) = 81 Squares
Types of Ceilings (Vitana) of Temples
Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India
Samatala vitāna decorated with padmaśila
Ksipta vitāna of Nābhicchanda order decorated with kola courses
Ksipta vitāna of Nābhicchanda order decorated with gajatalu courses
Ksipta vitāna of Nābhicchanda order in a set of diagonally arranged squares decorated with kola courses
Ksipta vitāna of Nābhicchanda order in a set of nine arranged squares decorated with kola courses
Ksipta vitāna of Nābhicchanda order in a set of eight circles intersected by a big circle in the center
Sama-ksipta vitāna decorated with figural groups in the boxes
Karotaka of Sabhmarga order
Ksipta vitāna of Nābhicchanda order made of three diminishing squares decorated with kola courses
Questions on Relationships
(Kshetra)
What is a relationship between the Temple and the Tank (Kunda)?
What is a relationship between the Temple and the Pond (Talab)?
What is a relationship between the Temple and the River (Nadi)?
What is a relationship between the Temple and the Lake (Jheel)?
Square and Circle in a Group
Shiva temples (Squares) and Shakti Temples (Circular)
(Group of Squares and Group of Circles)
Bateswer Group of Temples + Mitawali Chausath Yogini Temple (Circular)
Khajuraho Group of Temples + Chausath Yogini Temple (Rectangular)
Bhuvneswar Group of Temples + Hirapur Chausath Yogini Temple (Circular)
Temple Architecture Canonical Texts
Source: Rediscovering the Hindu Temple: The Sacred Architecture and Urbanism of India
A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India
Chelsea Gill
May 2008 A Senior Thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Archaeological Studies University of Wisconsin-La Crosse
TEMPLES, RITUAL, AND COSMIC SYMBOLISM IN THE ANCIENT WORLD
edited by DEENA RAGAVAN with contributions by Claus Ambos, John Baines, Gary Beckman, Matthew Canepa, Davíd Carrasco, Elizabeth Frood, Uri Gabbay, Susanne Görke, Ömür Harmanşah, Julia A. B. Hegewald, Clemente Marconi, Michael W. Meister, Tracy Miller, Richard Neer, Deena Ragavan, Betsey A. Robinson, Yorke M. Rowan, and Karl Taube
Papers from the Oriental Institute Seminar Heaven on Earth Held at the Oriental Institute of the University of Chicago 2–3 March 2012
“Naturalizing Buddhist Cosmology in the Temple Architecture of China: The Case of the Yicihui Pillar.”
Tracy Miller
In Heaven on Earth: Temples, Ritual, and Cosmic Symbolism in the Ancient World
(The University of Chicago Oriental Institute Seminars, No. 9), edited by Deena Ragavan, 17-39. Chicago: University of Chicago Oriental Institute, 2013.
Vastu Purusha Mandala – A human ecological framework for designing living environments.
Venugopal, Jayadevi
In Jetty, C, Chandra, B, Bhashyam, A, & Prabhakara, R (Eds.) Proceedings of International Conference on Advances in Architecture and Civil Engineering (AARCV 2012), Volume 2. Bonfring, India, pp. 870-877.
PANCHA KOSHA VIVEKA “Differentiation of the Five Sheaths”
PANCHADASHEE – 03
Composed by Sri Swami Vidyaranyaji
Reflections by SWAMI GURUBHAKTANANDA on the 32 Lectures delivered by Swami Advayanandaji, Acharya at the Chinmaya International Foundation, Veliyanad, Kerala. from April 28th 2017 – November 7th, 2017
Unit-5 Concept of Pancha Koshas, Concept of Panchmahabhutas
Issue Date: 2020 Publisher: Indira Gandhi National Open University, New Delhi URI: http://egyankosh.ac.in//handle/123456789/59786 Appears in Collections: Block-2 Yogic Concepts of Health
John McKim Malville University of Colorado Boulder
August 2014 DOI:10.1007/978-94-007-3934-5_9717-2 In book: Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (pp.1-16)
This site is best viewed on Microsoft Internet Explorer 4.0 or higher
with screen set to 1024 X 768 pixels, 24 bit …16 million colors.
Set … View/Text Size … to Meduim
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For more than twenty years, I have been studying the image generating properties of reflective spheres stacked in 52 degree angle pyramids. The 52 (51.827) degree angle slope of the sides of The Great Pyramid in Cairo, Egypt embodies the Golden Mean which is the ratio that is used in Nature to generate growth patterns in space. Sacred Geometry studies such primal systems which reveal the unity of the cosmos by representing the relationships between numbers geometrically. The Vesica Piscis is one of the most fundamental geometrical forms of this ancient discipline and it reveals the relationship between the The Great Pyramid and the 2 dimensional expansion of a circle of one unit radius R as shown in Figure 1. This relationship is more completely described in The New View Over Atlantis by John Michell published by Thames and Hudson.
Figure 1 Vesica Piscis in 2 Dimensions
In the early 1970s, I became very interested in the three dimensional representation of this geometry and I visualized this as a three dimensional pyramid inside two intersecting spheres shown in Figure 2.
Figure 2 Vesica Piscis in 3 Dimensions
In an effort to visualize these 3D relationships in yet another way, I stacked reflective Christmas Tree balls in an inverted pyramid shell. I discovered that the patterns of multiple reflections created on the interior surfaces of reflective spheres stacked at this angle produce images that relate to the human form as photographed and shown in Figure 3 and Figure 4.
I made many pyramidal configurations of reflective spheres with different light sources and I photographed the patterns on the interior surfaces from many points of view. Another is shown here in Figure 5.
Then in 1977, I discovered one stacking structure and viewpoint that produced a very convincing image of an archetypal human face. This structure of 10 spheres (2 5-ball pyramids) forming a cluster is shown in Figure 6.
The face image is shown in the lower third of the pattern on one sphere inside this 10 sphere cluster and is shown in Figure 7.
Figure 7 When I made this photograph, the structure was enclosed in a mirrored pyramid.
Later, I realized that the most natural structure for enclosing would be another 10 sphere, 2 pyramid structure that would totally enclose a smaller but similar cluster. I worked out the math to find that by multiplying the inner sphere’s diameters by Pi gives the dimension for the outer sphere’s diameters as shown in Figure 8.
Figure 8 One sphere is removed from the outer cluster to reveal the inner cluster. However, the inner cluster must be upside down with respect to the outer cluster to fit inside.
The expansion by Pi reinforced my suspicion that this 10 sphere cluster is a fundamental unit that is linked to the properties of three dimensional space.
Close-packed reflective spheres clustered in this concentric shell structure produce an optical distribution network that links the Golden Mean and Pi. The Golden Mean is expressed in the 52 degree angle pyramid structure and Pi is expressed in the ratio of the diameter to the circumference of each sphere of course; but it is also expressed in the ratio of the sizes of spheres in the 10 spheres within 10 spheres concentric shell structure that I discovered. This concentric shell structure could continue to expand with many shells and still retain the same ratio between shells. It was not until 1991 that I was able to build and photograph the images inside a ten-within-ten (2 shell) structure. I used 10 – 6″ diameter spheres and 10 – 19″ diameter hemispheres. The structure is shown here in Figure 9.
The photographs that I made from this 10-within-10 sphere structure revealed the inherent limitations of photography for this work. The final images were not what I was seeing with my eyes while directly viewing the interior of the structure. But this approach did reveal a more complete face form and I also realized that I would need at least 10 more spheres (about 60″ in diameter at $1500 each) to complete the enclosure and remove the remaining gaps in the images. Also, the lighting system was limited to the exterior and it was very difficult to control the positioning, color and brightness. So, in 1992, I purchased a computer to model these structures with ray-tracing software which enabled me to investigate more thoroughly the relationship between this cluster geometry and the archetypal images generated therein. During this tour into cyberspace, I could take a camera into the sacred chamber central to the concentric shells of reflective spheres which are simulated by a computer program (Real 3D by Realsoft Oy, Finland) that most accurately renders the effects of real-world light sources and records the patterns generated by multiple reflections on metal surfaces. With this method of investigation, I am able to more conveniently control the many variables which led to these discoveries and conclusions:
1.) At least 40 spheres (4-10 ball shells) are necessary to enclose the central area and fill in the gaps in the images.
2.) Most of the lights should be point sources placed at the points of contact between the innermost spheres of the structure; although additional symmetrically-paired, point-lights are necessary in the central area.
3.) Some of the innermost spheres are reduced in size and they can float within certain areas in the central space. Figure 12 shows typical positions and sizes.
4.) The camera position and field of view as shown in Figure 12 produces the most convincing image of the human form.
5.) The image of this artificial anthropoid that is produced in these structures can be animated when the positions and sizes of certain spheres are modulated as shown in the three animations listed below.
animation 1 .8 MB … estimated download time at 56k … 3.5 min.
animation 2 2.4 MB … estimated download time at 56k … 10 min.
Figure 13 Human Form From Sacred Geometry
The computer image in Figure 13 was made from the camera position and field of view shown in Figure 12.
Figure 14 is from the same camera position (shown in Figure 12) but the lens set at a very wide angle.
Figure 16 is a stereo image and it shows the interior of the cluster in 3D. Stare through the images with your eyes focused at a distant point and the two images will turn into three images and the center image will appear in 3D.
In 1996, I produced an animation Sacred Spaces (6 minutes) which has been screened in many national and international film festivals and it has won some awards (see resume). I have also produced Flesh Tones (5.5 Minutes) another animation completed in 1997 and I have produced many high resolution images for prints/slides (some examples are shown in Figures 13 through 36) which I have presented with the video animations at lecture presentations. These images are stills from the animations and they demonstrate the variety of image generating that is inherent in this system. I have concentrated on exhibiting the videotape documentation of my work thus far because it is the most portable presentation format that describes this research most completely.
This interdisciplinary research has taken me into many related areas of study. Geometry, Optics, Ancient History of Art and Religion, Computer Imaging, Photography, Animation, Graphic Art and of course Sculpture are the major connecting disciplines that have contributed to this work.
I feel that I have rediscovered some of what was a highly developed understanding of Mankind’s relationship to the Universe and this knowledge was utilized and documented in the geometry of ancient structures. Sacred Geometry, the study of the unity of the cosmos, demonstrates relationships between Number and Space and the Human Form. It was used in the construction of ancient glyphs and monuments thereby preserving the knowledge of these principles of Natural Law for future generations. This construction of reflective spheres may embody the technology that produced the animated images of the deities in the temples of antiquity. The Tree of Life which is a graphic representation of the interaction between cosmic forces is shown in Figure 22. It is found in many ancient texts of the Kabbalah.
The construction of this structure of clustered metal reflective spheres (offering bowls) is well within the capability of many ancient cultures and with the addition of a few glass lenses, these images could be projected onto walls or into smoke. Perhaps there is some Truth behind the smoke and mirrors of Ancient Religion … perhaps it is geometry … Sacred Geometry.
This research which is documented in four computer animations Sacred Spaces, Flesh Tones, Our Mothers and Sacred Spaces 2, in color computer prints, and in color slides has given me new insight into the motives that may have inspired the construction of The Great Pyramid.
1998 – 2000 update
The images shown in Figures 24 – 35 were made with 50 spheres and 144 point-light sources. Each of these images was made with unique brightness, color and value settings for various groupings of lights.
InFigure32, the camera is aimed at the sphere opposite the face shown inFigure 31.Figure 12 shows the camera position for Figure 31. The image in Figure 32 was formed when the camera position was rotated 180 degrees around the vertical axis shown in Figurre 12 and zoomed in. The face image (a child?) in Figure 32 is much smaller than the face (mother?) in Figure 31.
Figure35This is a stereo 3D image that requires shutter glasses to view and your monitor must be set to interlace mode.
Figure 36 Stereo Image for Cross-Eyed Viewing
With your monitor at arm’s length away, focus on a point 6 inches in front of your nose (put your index finger 6 inches in front of your nose and focus on it). You will see a third image in 3D between the two images on the monitor (at the tip of your finger). Shift your attention from your finger to this third/middle image which will appear in 3D.
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The cross-eyed viewing method is perhaps the most effective way to put the viewer inside the cluster to see the human image as it would exist in 3D from the cameras position as seen in Figure 12. There are many more identifiable images in this clustering geometry viewed from this position and from other camera positions and even more images with other color settings for the point lights.
Figure 37 Rods connecting centers of nearest neighbors in 3 shells
I’m now very curious about the relationship between Sacred Geometry and Sacred Music and the Human Form. Number relates to all that science measures by virtue of the way that 3D Space is defined. Number is also used to measure Time … and, as Pythagoras observed, Music is a manifestation of Number in Time. The distribution of sounds i.e. amplitude and frequency, may well find an idealized model in the 10 within 10 sphere, space-filling, close-packing geometric system. The representation of this geometry with sticks or strings or rods as shown in Figure 37 (in which the centers of nearest neighbor spheres in three shells are joined) may represent the ideal space-filling matrix of linear oscillating elements. It may also be used to define spatially distributed, hierarchical, cellular arrays.
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The camera positions on the symmetric planes within the cluster produce a bilateral symmetry that we identify with animal and human form. Naturally, we would expect this symmetry in the idealized images of higher life forms. The multiplicity of idealized beings in this cluster of reflective spheres suggests the presence of The Company of the Gods as described in many ancient Egyptian texts. It seems that the face images occur on a vertical plane linking the centers of spheres. There seem to be faces facing faces and faces within faces throughout this reflective environment. My guess is that the sacred part of what I have discovered is a result of the way the deity put the higher life forms in three dimensions.
This cluster geometry may have other properties that would be useful for spatial organization. The nesting of 10 reflective spheres within 10 reflective spheres geometry produces a distribution system that could be used for processing of optical information between the interior to the exterior of the structure. Because of the spatial distribution of the reflective spheres in two concentric shells of ten each, optical information must be reflected and diverged in order to enter or exit the system with the exception of a few radially arrayed directions. The system becomes a more selective filter of optical information as more shells are added to the structure. This inside-to-outside transformation/translation should have many practical applications in pattern recognition tasks. For example, any point source of coherent light (laser light) anywhere outside the two-shell cluster will produce a unique light distribution pattern on the inside as viewed from the center area of the cluster. This pattern could be recorded in a holographic medium and the exterior point source could then be reconstructed using conventional holographic means.
Is Number (Geometry and Time) the link between Art, Science and Religious experience? The language of number is perhaps the most convincing form of expression between humans and between humans and the Gods. We think and imagine in visual forms. Einstein constructed his mathematics based on mental images. He said that he would first try to visualize a space/time image and then mathematize it. We use mental images to construct possible scenarios of the future so that we don’t have to live out each one in “reality”. Words and pictures and mathematical formulas are ways to document, test, realize, and communicate these visions. Although there is seldom a need to mathematize images, we sense that it would be possible. We know that we could count and number the grains of sand on the beach. The geometry of our visions is what makes them real to us and it allows us to mentally work on them and to integrate them convincingly into our life here in 3D.
I’m not quite sure how the physical human form fits into the grand scheme of things but it does seem to be a result of the space-filling, spherical, close-packing geometric system that I’ve discovered and it is indeed “Sacred Geometry” by virtue of the definition God gave to three dimensions. I don’t think this geometric system is the matrix for all life systems but I think it can serve as a model for the interactions between the various dimensional realities in which we are immersed. It may also guide us in our attempt to develop new sciences and technologies that utilize the forces that operate in the regions that we now call consider paranormal.
According to contemporary Superstring Theory as described by Dr. Michio Kaku in his many recent writings, the mathematics that most appropriately describes the forces of nature requires an expression in ten dimensions. Einstein tried to describe the forces of nature in the mathematics of 3D and Time and found that the formulas were not broad enough to include all of the forces. We can perceive 3D and Time. The other 6D in Superstrings are hidden from our normal senses due to their incredibly small size according to Dr. Kaku. Our instincts inform us that there are more than 3 Dimensions and Time in the universe and the possiblity that something else exists mathematically beyond our perceptual horizon drives my curiosities about our possible links to these worlds. The new science of parapsychology has discovered many ways that humans can perceive by means beyond the physical senses and it has found that there are some people that are more capable of extrasensory perception than others. It may well be that our only contacts with this duality of nature are through numbers and mathematics on one hand or through dreams, remote viewing, telepathy, Ouija boards and tarot cards on the other. This would certainly confirm the existence of a God with a sense of humor
I am interested in any information relating this technology to ancient religious traditions. Any references that you send will certainly be appreciated and I will certainly credit any references in future publications.
A more detailed version of this research is now available on CD-ROM and it includes the 10 minute animation Sacred Spaces 2 in streaming format.
I am also making available unlimited editions of selected images on this website which will be printed in very high resolution on archival paper with archival inks.
For more specific information about the availability, formats, sizes and pricing of the prints and CD relating to this work you may contact me at:
Source: The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions
Source: The History of Mathematics From the Egyptians to Archimedes
Source: Platonic Solids, or, the power of counting
Key Terms
Prapanch (Five Fold)
Panch (Five)
5 Platonic Solids
14 Archimedean Solids
Catalan Solids
Regular Convex Polyhedra
Semi Regular Convex Polyhedra
Kepler-Poinsot Polyhedra
4D Polytopes
Five Elements
5 Kosh (Sheaths)
14 Lok (Levels, Realms)
7 Upper Worlds
7 Under Worlds
7 Chakras
5 Continents
7 Seas
Hierarchy Theory
Mount Meru
Nested Platonic Solids
Soccer Ball Geometry
Uniform Polyhedra
Johnson Solids
Goldberg Polyhedra
Albrechet Durer
Leonardo da Vinci
Johannes Kepler
Fra Luca Bartolomeo de Pacioli (ca.1447–1517)
Buckminster Fuller
Fullerenes
Virus Geometry
Symmetry
Polygons
Max Brückner
H.S.M. Coxeter
George W. Hart
Five Platonic Solids
Five Platonic Solids
Tetrahedron
Octahedron
Cube
Icosahedron
Dodecahedron
Five Elements
Fire – Tetrahedron
Air – Octahedron
Earth – Cube
Space – Dodecahedron
Water – Icosahedron
Five Sense Organs
Tongue -Taste – Water
Eyes – Form – Fire
Ears – Sound – Space
Nose – Smell – Earth
Skin – Touch – Air
Five Senses
Hearing
Sight
Touch
Taste
Smell
Source: Polyhedra: Plato, Archimedes, Euler
Source: Polyhedra: Plato, Archimedes, Euler
Source: Polyhedra: Plato, Archimedes, Euler
Source: Sacred Geometry and the Platonic Solids
Source: Sacred Geometry and the Platonic Solids
Source: The Platonic solids and fundamental tests of quantum mechanics
Source: The Platonic solids and fundamental tests of quantum mechanics
Source: The Platonic solids and fundamental tests of quantum mechanics
Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids
Source: Platonic Solids (Regular polytopes in 3D)
Source: Platonic Solids (Regular polytopes in 3D)
Source: Platonic Solids (Regular polytopes in 3D)
Source: Platonic Solids (Regular polytopes in 3D)
14 Archimedean Solids
Rhombicuboctahedron
Rhombicosidodecahedron
Cuboctahedron
Icosidodecahedron
Truncated Tetrahedron
Truncated Cube
Truncated Octahedron
Truncated Dodecahedron
Truncated Icosahedron
Truncated Cuboctahedron
Truncated Icosidodecahedron
Snub Cube
Snub Dodecahedron
Pseudorhombicuboctahedron ?
Source: Polyhedra: Plato, Archimedes, Euler
Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids
Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids
Source: The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions
Catalan Solids
Source: Catalan Solids
Kepler-Poinsot Solid
Source: Kepler Poinsot Solids
Source: The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions
Johnson Solids
Source: Johnson Solid
Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids
Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids
Source: Max Brücknerʼs Wunderkammer of Paper Polyhedra
Source: Max Brücknerʼs Wunderkammer of Paper Polyhedra
Platonic Solids and Plato’s Theory of Everything
Source: Platonic Solids and Plato’s Theory of Everything
The Socratic tradition was not particularly congenial to mathematics, as may be gathered from Socrates’ inability to convince himself that 1 plus 1 equals 2, but it seems that his student Plato gained an appreciation for mathematics after a series of conversations with his friend Archytas in 388 BC. One of the things that most caught Plato’s imagination was the existence and uniqueness of what are now called the five “Platonic solids”. It’s uncertain who first described all five of these shapes – it may have been the early Pythagoreans – but some sources (including Euclid) indicate that Theaetetus (another friend of Plato’s) wrote the first complete account of the five regular solids. Presumably this formed the basis of the constructions of the Platonic solids that constitute the concluding Book XIII of Euclid’s Elements. In any case, Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of (non-planar) points in space, and late in life he expounded a complete “theory of everything”, in the treatise called Timaeus, based explicitly on these five solids. Interestingly, almost 2000 years later, Johannes Kepler was similarly fascinated by these five shapes, and developed his own cosmology from them.
To achieve perfect symmetry between the vertices, it’s clear that each face of a regular polyhedron must be a regular polygon, and all the faces must be identical. So, Theaetetus first considered what solids could be constructed with only equilateral triangle faces. If only two triangles meet at a vertex, they must obviously be co-planar, so to make a solid we must have at least three triangles meeting at each vertex. Obviously when we have arranged three equilateral triangles in this way, their bases form another equilateral triangle, so we have a completely symmetrical solid figure with four faces, called the tetrahedron, illustrated below.
On the other hand, if we make four triangles meet at a vertex, we produce a square-bottomed pyramid, and we can obviously put two of these together, base to base, to give a completely symmetrical arrangement of eight triangular faces, called the octahedron, shown below.
Next, we can make five equilateral triangles meet at a point. It’s less obvious in this case, but if we continue this pattern, adding equilateral triangles so that five meet at each vertex, we arrive at a complete solid with 20 triangular faces. This is called the icosahedron, shown below.
Now, we might try putting six equilateral triangles together at a point, but the result is a planar arrangement of triangles, so it doesn’t give a finite solid. I suppose we could regard this as a Platonic solid with an infinite radius, which might have been useful in Plato’s cosmology, but it doesn’t seem to have been viewed this way. Perhaps this is not surprising, considering the well-known aversion of the ancient Greek mathematicians to the complete infinity. In any case, we clearly can’t construct any more perfectly symmetrical solids with equilateral triangle faces, so we must turn to other possible face shapes.
The next regular polygon shape is the square, and again we find that putting just two squares together does not yield a solid angle, so we need at least three squares to meet at each vertex. Putting three squares together we see that we can add three more to give the perfect solid with six faces, called the hexahedron (also known as the cube). This is shown below.
If we try to make four square faces meet at each vertex, we have another plane surface (giving another “infinite Platonic solid”), so clearly this is the only finite perfectly symmetrical solid with square faces.
Proceeding to pentagonal (five-sided) faces, we find that if we put together 12 pentagons so that three meet at each vertex, we arrive at the fifth Platonic solid, called the dodecahedron, illustrated below.
It isn’t self-evident that 12 identical regular pentagons would come together perfectly like this to form a closed solid, but it works, as Theaetetus proved and as Euclid demonstrates at the conclusion of The Elements. Of course, if we accept that the icosahedron works, then the dodecahedron automatically follows, because these two shapes are “duals” of each other. This means that the icosahedron has 20 faces and 12 vertices, whereas the dodecahedron has 12 faces and 20 vertices, and the angular positions of the face centers of one match up with the positions of the vertices of the other. Thus, once we have the icosahedron, we can just put a dot in the center of each face, connect the dots, and viola!, we have a dodecahedron. Similarly, the cube and the octahedron are duals of each other. Also, the tetrahedron is the dual of itself (so to speak).
Clearly it’s impossible for four (or more) pentagonal faces to meet at a vertex, because they subtend more than 360 degrees. For hexagonal (six-sided) faces, three hexagons meeting at a point constitute another “infinite solid”, i.e., a planar surface. It’s also obvious that no higher-order polygon can yield a solid, so the five solids already mentioned – tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron – are the only regular polyhedrons. Theaetetus not only proved that these solids exist, and that they are the only perfectly symmetrical solids, he also gave the actual ratios of the edge lengths E to the diameters D of the circumscribing spheres for each of these solids. This is summarized in Propositions 13 through 17 of Euclid’s Elements.
In Timaeus, Plato actually chose to constitute each of these solids from right triangles, which played the role of the “sub-atomic particles” in his theory of everything. In turn, these triangular particles consisted of the three legs (which we might liken to quarks), but these legs were ordinarily never separated. The right triangles that he chose as his basis particles were of two types. One is the “1,1,” isosceles triangle formed by cutting a square in half, and the other is the “1,2,” triangle formed by cutting an equilateral triangle in half. He used these to construct the faces of the first four solids, but oddly enough he didn’t just put two together, he used six “1,2, triangles to make a triangular face, and four “1,1,” triangles to make a square face, as shown below.
Of course, it’s not possible to build a pentagon from these two basic kinds of right triangles, and Plato doesn’t actually elaborate on how the faces of the dodecahedron are to be constructed, but from other sources we know that he thought each face should be composed of 30 right triangles, probably as shown on the right-hand figure above, so that the dodecahedron consisted of 360 triangles. The tetrahedron, octahedron, and icosahedron consisted of 24, 48, and 120 triangles (of the type 1,2,), respectively, and the hexahedron consisted of 24 triangles (of the type 1,1,).
Now, if the basic triangles were the subatomic particles, Plato regarded the solids as the “atoms” or corpuscles of the various forms of substance. In particular, he made the following identifications
The idea that all the constituents of nature consist of mixtures of a small number of “elements”, and in particular the selection of the four elements of earth, water, air, and fire, is attributed to an earlier Greek philosopher Empedocles of Agrigentum (495-435 BC). Empedocles believed that although these elements (which he called “the roots of all things”) could be mixed together in various proportions, the elements themselves were inviolable, and could never be changed. In contrast, one of the intriguing aspects of Plato’s theory was that he believed it was possible for the subatomic particles to split up and re-combine into other kinds of atoms. For example, he believed that a corpuscle of liquid, consisting of 120 “type 1” triangles, could be broken up into five corpuscles of plasma, or into two corpuscles of gas and one of plasma. Also, he believed that the “smaller” corpuscles could merge into larger corpuscles, so that (for example) two atoms of plasma could merge and form a single atom of gas. However, since the basic triangles making up “earth” (cubes) are dissimilar to those of the other forms of substance, he held that the triangles comprising cubes cannot be combined into any of the other shapes. If a particle of earth happened to be broken up into its constituent triangles, they will “drift about – whether the breaking up within fire itself, or within a mass of air or water – until its parts meet again somewhere, refit themselves together and become earth again”.
When Plato asserts that the [1,1,] triangles cannot combine into anything other than a cube, it’s conceivable that he was basing this on something more that just the geometric dissimilarity between this triangle and the [1,2,] triangle. He might also have had in mind some notion of the incommensurability of the magnitudes and , not only with the unit 1, but with each other. Indeed the same Theaetetus who gave the first complete account of the five “Platonic” solids is also remembered for recognizing the general fact that the square root of any non-square integer is irrational, which is to say, incommensurable with the unit 1. It isn’t clear whether Theaetetus (or Plato) knew that two square roots such as and are also incommensurable with each other, but Karl Popper (in his anti-Plato polemic “The Free Society and its Enemies”) speculated that this might have been known, and that Plato’s choice of these two triangles as the basic components of his theory was an attempt to provide a basis (in the mathematical sense) for all possible numbers. In other words, Popper’s idea is that Plato tentatively thought the numbers 1, , and are all mutually incommensurable, but that it might be possible to construct all other numbers, including , π, etc., as rational functions of these.
Of course, Book X of Euclid’s Elements (cf. Prop 42) dashes this hope, but it’s possible that the propositions recorded there were developed subsequent to Plato’s time. Popper also makes much of the numerical coincidence that + is approximately equal to π, and speculates that Plato might have thought these numbers were exactly equal, but this doesn’t seem credible to me. For one thing, it would give a means of squaring the circle, which would certainly have been mentioned if anyone had believed it. More importantly, the basic insight of Theaetetus was in recognizing the symmetry of all the infinitely many irrational square roots, and it just doesn’t seem likely that he (or Plato) would have been misled into supposing that just two of them (along with the unit 1) could form the basis for all the others. It’s a very unnatural idea, one that would not be likely to occur to a mathematician. (Still, an imaginative interpreter could probably discern correspondences between the four basis vectors of “The Platonic Field“, i.e., numbers of the form A + B+ C + D and Plato’s four elements, not to mention the components of Hamilton’s quaternions.)
It’s also interesting that Plato describes the “1,1,” triangle as the most “stable”, and the most likely to hold its shape, thus accounting for the inert and unchanging quality of the solid elements. He didn’t elaborate on his criterion for “stability”, although we can imagine that he had in mind the more nearly equal lengths of the edges, being closer to equilibrium. On the other hand, this would suggest that the equilateral triangle (which is the face of Plato’s “less stable” elements) was highly stable. Plato made no mention of the fact that the cube is actually the only unstable Platonic solid, in the sense of rigidity of its edge structure. In addition, the cube is the only Platonic solid that is not an equilibrium configuration for its vertices on the surface of a sphere with respect to an inverse-square repulsion. Nevertheless, the idea of stability of the sub-atomic structure of solid is somewhat akin to modern accounts of the stability of inert elements.
We can also discern echoes of Plato’s descriptions in Isaac Newton’s corpuscular theory. Newton’s comments about the “sides” of light particles are very reminiscent of Plato’s language in Timaeus. It’s also interesting to compare some passages in Timaeus, such as
And so all these things were taken in hand, their natures being determined by necessity in the way we’ve described, by the craftsman of the most perfect and excellent among things that come to be…
with phrases in Newton’s Principia, such as
…All the diversity of created things, each in its place and time, could only have arisen from the ideas and the will of a necessarily existing being…
…all phenomena may depend on certain forces by which the particles of bodies…either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede…
…if anyone could work with perfect exactness, he would be the most perfect mechanic of all…
Plato explicitly addressed the role of necessity in the design of the universe (so well exemplified by the five and only five Platonic solids), much as Einstein always said that what really interested him was whether God had any choice in the creation of the world. But Plato was not naive. He wrote
Although [God] did make use of the relevant auxiliary causes, it was he himself who gave their fair design to all that comes to be. That is why we must distinguish two forms of cause, the divine and the necessary. First, the divine, for which we must search in all things if we are to gain a life of happiness to the extent that our nature allows, and second, the necessary, for which we must search for the sake of the divine. Our reason is that without the necessary, those other objects, about which we are serious, cannot on their own be discerned, and hence cannot be comprehended or partaken of in any other way.
The fifth element, i.e., the quintessence, according to Plato was identified with the dodecahedron. He says simply “God used this solid for the whole universe, embroidering figures on it”. So, I suppose it’s a good thing that the right triangles comprising this quintessence are incommensurate with those of the other four elements, since we certainly wouldn’t want the quintessence of the universe to start transmuting into the baser substances contained within itself!
Timaeus contains a very detailed discussion of virtually all aspects of physical existence, including biology, cosmology, geography, chemistry, physics, psychological perceptions, etc., all expressed in terms of these four basic elements and their transmutations from one into another by means of the constituent triangles being broken apart and re-assembled into other forms. Overall it’s a very interesting and impressive theory, and strikingly similar in its combinatorial (and numerological) aspects to some modern speculative “theories of everything”, as well as expressing ideas that have obvious counterparts in the modern theory of chemistry and the period table of elements, and so on.
Timaeus concludes
And so now we may say that our account of the universe has reached its conclusion. This world of ours has received and teems with living things, mortal and immortal. A visible living thing containing visible things, and a perceptible God, the image of the intelligible Living Thing. Its grandness, goodness, beauty and perfection are unexcelled. Our one universe, indeed, the only one of its kind, has come to be.
The speculative details of Plato’s “account of the universe” are not very satisfactory from the modern point of view, but there’s no doubt that – at least in its scope and ambition as an attempt to represent “all that is” in terms of a small number of simple mathematical operations – Plato’s “theory of everything” left a lasting impression on Western science.
Source: Platonic Solids, or, the power of counting
Six (6) 4D Polytopes
Source: 4d-polytopes described by Coxeter diagrams and quaternions
Discovery of the Platonic solids; tetrahedron, cube, octahedron, icosahedron and dodecahedron dates back to the people of Scotland lived 1000 years earlier than the ancient Greeks and the models curved on the stones are now kept in the Ashmolean Museum at Oxford [1]. Plato associated tetrahedron with fire, cube with earth, air with octahedron, and water with icosahedron. Archimedes discovered the semi-regular convex solids and several centuries later they were rediscovered by the renaissance mathematicians. By introducing prisms and anti-prisms as well as four regular non-convex polyhedra, Kepler completed the work in 1620. Nearly two centuries later, in 1865, Catalan constructed the dual solids of the Archimedean solids now known as Catalan solids [2]. Extensions of the platonic solids to 4D dimensions have been made in 1855 by L. Schlaffli [3] and their generalizations to higher dimensions in 1900 by T.Gosset [4]. Further important contributions are made by W. A. Wythoff [5] among many others and in particular by the contemporary mathematicians H.S.M. Coxeter [6] and J.H. Conway [7].
The 3D and 4D convex polytopes single out as compared to the polytopes in higher dimensions. The number of Platonic solids is five in 3D and there exist six regular polytopes in 4D contrary to the higher dimensional cases where there exist only three platonic polytopes which are the generalizations of tetrahedron, octahedron and cube to higher dimensions. The Platonic and Archimedean solids [8] as well as the Catalan solids [9] can be described with the rank-3 Coxeter groupsW(A3),W(B3) and W(H3).
The 4D polytopes are described by the rank-4 Coxeter groups W(A4 ), W(B4 ), W(H4 ) and the group W (F4 ).
Source: The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions
Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces
Source: Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces
Source: Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces
Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses
Stan Schein stan.schein@gmail.com and James Maurice Gayed
Edited by Patrick Fowler, The University of Sheffield, Sheffield, United Kingdom, and accepted by the Editorial Board January 7, 2014 (received for review June 10, 2013) February 10, 2014 111 (8) 2920-2925 https://doi.org/10.1073/pnas.1310939111
*Budapest University of Technology and Economics Budapest, Műegyetem rkp. 3., H-1521 Hungary tarnai@ep-mech.me.bme.hu
Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures 28 September – 2 October 2009, Universidad Politecnica de Valencia, Spain Alberto DOMINGO and Carlos LAZARO (eds.)
Interaction / Intersection of Menger Sponge and Apollonian Sphere in cosmic geometry produces pythagorean triples / triads.
Seven plus one spheres to triads of three.
Apollonian Gasket, Circle and Sphere Packing and Cosmic Geometry
Source: PRECISE CALCULATION OF HAUSDORFF DIMENSION OF APOLLONIAN GASKET
A transfer operator method is proposed to calculate 𝑑𝐻, the Hausdorff dimension of the Apollonian gasket. Compared with previous operator-based methods, we make two improvements in this paper. We adopt an infinite set of contractive Möbius transformations (rather than a finite set of parabolic ones) to generate the Apollonian gasket. We also apply an efficient finite matrix approximation of an infinite sum of infinite-dimensional operators. By using this method, a high precision estimate of 𝑑𝐻 is obtained:
𝑑𝐻=1.305 686 728 049 877 184 645 986 206 851 0….
Source: THE FRACTAL DIMENSION OF THE APOLLONIAN SPHERE PACKING
The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2−19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both algorithms are given.
Source: THE FRACTAL DIMENSION OF THE APOLLONIAN SPHERE PACKING
Source: What Type of Apollonian Circle Packing Will Appear?
Source: Self-similar space-filling sphere packings in three and four dimensions
Source: Self-similar space-filling sphere packings in three and four dimensions
Sapta Matrikas
Brahmi
Maheswari
Kumari
Vaishnavi
Varahi
Indrani
Chamunda
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
In Varahapurana the Devi, Vaishnavi in account of the creation of the matrika, is doing asceticism on mount Mandara. At one point she losses her concentration. From her disgraced mind, several beautiful attendants were created. They later became Devi’s helpmates on the battlefield when she fights the demon. Although the Matrikas are described as lovely in this account, it is important to note that they are born when Devi losses control of her concentration. This suggests that the matrikas are essentially of uncontrolled natures. Born from lack of mental control, they lack control themselves. Varahapurana relates them to vices or inauspicious emotions; Brahmi of Mada, Maheswari of Krodha, Kumari of Moha, Vaishnavi of Lobha, Indrani of Matsarya, Varahi of Asuya and Chamunda of Paisunya.
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
The follower of Tantrasara has an esoteric interpretation of the seven matrikas. According to them, Brahmi represents the primordial Nada, the energy in which even the first throb has not yet appeared. This is the manifest sound, the origin of all creation. It is the same substance or energy represented by the pranava. When Brahmi creates the universe, the power of Vaishnavi gives definite shape. The symmetry, beauty, organization and order in the universe are the work of Vaishnavi. Maheswari stands for the power that gives individuality to the created beings. She resides in the hearts of all and makes them play, as dolls mounted on a machine. Kumari represents the ever present force of aspiration of the evolving soul. She is ‘Guruguha’, the Guru in guha (the cave of the heart, the intellect). Varahi is the all-consuming power of assimilation and enjoyment. Because of her, all living beings get their food and enjoyments. Indrani symbolizes the terrible power that destroys all that opposes the cosmic law. Chamunda is the force of concentrated awareness, the spiritual awakening in the heart that devours that ceaseless activity of the immature mind and uplifts it to the highest level (Harshananda 1981.95-99).
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
Source: Matrikas/Siddha Pedia
Source: The Seven Ancient Mothers
The Pythagoreans considered the figure seven as the image and model of the divine order and harmony in nature. As the harmony of cosmic sound takes place on the space between the seven planets, the harmony of audible sound takes place on a smaller plane within the musical scale of the seven tones. Therefore, the syrinx of the nature god Pan consists of seven pipes, and the lyre of Apollo (the god of music) consists of seven strings. As the number seven is a union between the number three (the symbol of the divine triad) and of four (the symbol of the cosmic forces or elements), the number seven points out symbolically to the union of the divine with the universe.
Pierre Auclair∗ Cosmology, Universe and Relativity at Louvain (CURL), Institute of Mathematics and Physics, University of Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium (Dated: November 15, 2022)
Estimate for the fractal dimension of the Apollonian gasket in d dimensions
R. S. Farr* Unilever R&D, Olivier van Noortlaan 120, AT3133 Vlaardingen, The Netherlands and The London Institute for Mathematical Sciences, 22 S. Audley Street, Mayfair, London, United Kingdom
E. Griffiths
297 Sandy Bay Road, Sandy Bay, Tasmania, Australia
A Study of the Sapta Matrikas’ Origins and Evolution: From the Perspectives of the Art and Literature of Western India Dating from B.C. 1400 to 500 A.D.
Rebecca Sholes 1982
Mother Goddess in Central India
Om Prakash Misra Agam Kala Prakashan, 1985
The Little Goddesses (mātrikās)
K. C. Aryan Rekhā, 1980
The Iconography of the Saptamatrikas: Seven Hindu Goddesses of Spiritual Transformation
Katherine Anne Harper E. Mellen Press, 1989
ISBNS 9780889460614, 0889460612 OCLC ocm19921123
Saptamātr̥kā Worship and Sculptures: An Iconological Interpretation of Conflicts and Resolutions in the Storied Brāhmanical Icons
Self-similar space-filling sphere packings in three and four dimensions *
D. V. St ̈ager 1 , ∗ and H. J. Herrmann 1, 2 , † 1 Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland 2 Departamento de F ́ısica, Universidade Federal do Cear ́a, 60451-970 Fortaleza, Ceara ́, Brazil
Indira’s Pearls: Apollonian Gasket, Circle and Sphere Packing
Source: Matrikas
Key Terms
Circle Packing
Disk Packing
Sphere Packing
Indira’s Net
Indira’s Pearls
Apollonian Gasket
Fractals
Square and Circle
Circular Temple
Earth and Heaven
64 Yogini Hindu Shakti Temples
64 Bhairav
64 Yogini
7 Matrikas
Mandala
Curvilinear Sierpinski Gasket
Poincare Disc
Jagannath Puri, Odisha
Bhuvneswar, Odisha
Konark, Odisha
Non Euclidian Geometry
Euclidian Geometry
Fractal Geometry
Sacred Geometry
Hyperbolic Geometry
1, 2, 3 where is the 4th
1 2 3 4
Fifth and Fourth in Music Theory
Fractals, Circle, and Sphere Packing
Source: Circle Packing Explorations
Source: Circle Packing Explorations
Source: On a Diophantine Equation That Generates AllIntegral Apollonian Gaskets
Source:
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: On the cover: Apollonian packing
What can you do with this space? So asks Andrew Stacey. ‘Fill it’ is the prompt reply, but fill it with what? Maybe like Andrew you want to use a single curve, but I want to use circles. If you do this in the way shown above in blue, the result is called an Apollonian packing, a variant of which can be seen on the cover of this issue.
Here we shall explore the history of this entrancing object, which spans over 2000 years, and percolates into a surprising variety of mathematical disciplines. Starting in the familiar world of Euclidean geometry, Apollonian packings extend into fractal geometry and measure theory; Möbius transformations and the hyperbolic plane; and then on into the distant reaches of geometric group theory, number theory, orbital mechanics, and even ship navigation.
At times we may wonder off into thickets of more obscure mathematics, so those readers who get lost should feel free to skip ahead to the next section.
Apollonius of Perga
Apollonius (c 230 BC) was a Hellenistic mathematician, considered one of the greatest after Euclid and Archimedes. Perhaps his most important work was his eight book treaties Κωνικα on conic sections—once lost to European civilisation, but fortuitously preserved by the more enlightened Middle Eastern scholars and later reintroduced by Edmund Halley in 1710. The same unfortunately cannot be said of Έπαφαι (De Tractionibus or Tangencies). Although now lost, we have accounts of the work from other ancient authors, particularly in the writings of Pappus of Alexandria. In it, Apollonius posed and solved the following problem.
Problem: Given three geometric objects in the plane (points, lines, and/or circles), find all circles which meet all three simultaneously (ie which pass through any points, and are tangent to any lines or circles).
So for example, given three points which don’t lie on the same line, there is exactly one circle which passes through all three. The case which interests us at present is when we are given three circles, each of which is tangent to the other two. In the very special case that all three are tangent at the same point there are infinitely many circles tangent to all three. Usually, however, the circles will be pairwise tangent at three distinct points, in which case there are exactly two other circles tangent to all three simultaneously.
Given three mutually tangent circles (black) there are usually exactly two others (blue) tangent to all three.
This is as far as Apollonius went; the next step would not be taken until 1643, when René Descartes discovered a formula for the size of the two tangent circles, which he wrote in a letter to Princess Elizabeth of the Palatinate. The same formula was later rediscovered by Frederick Soddy and published as a poem in Nature in 1936.
The size of a circle is determined by its radius rr. If rr is small, the circle will be small, but it will also be very curved. We can define the curvature of the circle to be k=1/rk=1/r. Descartes showed that if three given circles are mutually tangent at three distinct points, and have curvatures k1k1, k2k2, and k3k3, then a fourth circle which is tangent to all three has curvature k4k4 satisfying
For technical algebraic reasons, sometimes this equation gives negative values for the curvature k4k4, which we can interpret as corresponding to a circle with curvature |k||k|which contains the other circles in its interior. Notice that this equation is quadratic in the variable k4k4, so there are two solutions; these will correspond to the two possibilities for the fourth circle found by Apollonius.
Apollonian packings
So far we have constructed at most 5 mutually tangent circles. The step to infinity may seem obvious, but took another 63 years and some 1900 years after Apollonius. The earliest description seems to appear in a letter from Leibniz to des Bosses (11 March 1706):
Imagine a circle; in it draw three other circles that are the same size and as large as possible, and in any new circle and in the space between circles again draw the three largest circles of the same size that are possible. Imagine proceeding to infinity in this way…
A finite iteration of a nested Apollonian packing similar to the one described by Leibniz. Image: adapted from Todd Stedl, CC BY-SA 4.0
What Leibniz is describing is in fact a nested Apollonian packing, since at each step he fills in every circle as well as the gaps between circles. This early description makes the nested Apollonian packing one of the first fractals, although it wasn’t studied properly until mathematicians like Cantor, Weierstrass, von Koch, and Sierpinski started discovering other fractals in the late nineteenth and early twentieth centuries. This may be because Leibniz was not interested in the mathematical construction, but rather was trying to draw an analogy to argue against the existence in infinitesimals in nature.
Henceforth we shall only consider the un-nested Apollonian packing. As a fractal, it has a number of interesting properties: it is a set of measure 0, which means that if you tried to make it by starting with a disc of metal, and then drilled out infinitely many ever smaller holes (and if you ignore that metal is made out of atoms), then you would finish up with a single piece of metal (you haven’t removed everything), but nevertheless with exactly 0 mass. It has fractal dimension approximately 1.30568, which means that mathematically it lives somewhere between a 1D curve and a 2D area. Finally, if you look at just the portion of an Apollonian packing which lies in the triangular region between three tangent circles, this is homeomorphic to the Sierpinski triangle, which means that one can be bent and stretched to look like the other.
A portion of an Apollonian packing is homeomorphic to the Sierpinski triangle—just squash all of the circles to make them triangular. Images: adapted from Todd Stedl, CC BY-SA 4.0 and Beojan Stanislaus, CC BY-SA 3.0.
There is a curious combinatorial consequence of Descartes’ formula for Apollonian packings. If we start with three mutually tangent circles with curvatures k1k1, k2k2, and k3k3, we can solve (1)(1) to find that the curvatures k+4k4+ and k−4k4− of the other two circles are
The integral Apollonian packing starting with curvatures -10, 18, 23, and 27. Image: adapted from Todd Stedl, CC BY-SA 4.0.
Now suppose we start constructing an Apollonian packing by drawing four mutually tangent circles whose curvatures k1k1, k2k2, k3k3, and k+4k4+ are all integers. From equation (2)(2) it follows that 2√k1k2+k2k3+k3k12k1k2+k2k3+k3k1 must be an integer since k+4k4+ is an integer, and so k−4k4− is also an integer. Now we can build the packing by filling in a fifth circle wherever we see four mutually tangent circles. By the observation above, if the four circles have integer curvatures, the fifth circle will also have integer curvature. Inductively therefore we will end up with an Apollonian packing consisting of infinitely many tangent circles, all of which have integer curvatures.
Hyperbolic geometry
All these scorpions have the same hyperbolic size.
If you have some familiarity with non-Euclidean geometry, Apollonian packings may remind you of the Poincaré model of the hyperbolic plane. The hyperbolic plane H2H2 is a 2D surface on which we can do geometry just like we can on the flat Euclidean plane. Whereas a sphere has constant positive curvature (it curves the same way in all directions), and the Euclidean plane has constant zero curvature (it’s flat), H2H2 is an infinite surface which has constant negative curvature, which means that at every point it curves in the same way as a Pringle. This negative curvature makes the surface crinkle up on itself more and more as you move out towards infinity, which is inconvenient when we try to work with it. Usually then we represent it on a flat surface so we can draw pictures of it in magazines and the like. One way to do this is with the Poincaré model. This views the hyperbolic plane as a disc. In order to fit the whole infinity of H2H2 into a finite disc, we have to shrink distances as we move out towards the edge of the disc. Using this skewed way of measuring distances, the circular edge of the disc is infinitely far away from its centre.
We can think of an Apollonian packing as living in the Poincaré disc, with the outermost circle of the packing as the boundary circle of H2H2. Then the circles in the packing which are not tangent to this boundary are also circles in the strange hyperbolic way of measuring distance, that is, all points are equidistant from some other point in the plane—the circle’s hyperbolic centre. Circles in the packing which are tangent to the boundary are called horocycles (in Greek this literally means border circle), which are circles with infinite radius in the hyperbolic metric. Horocycles have no analogue in the Euclidean plane.
Something interesting happens when we see what an Apollonian packing looks like in the upper half-plane (UHP) model for H2H2. This model is similar to the Poincaré model, but instead of using a disc, we use the half-plane above the xx-axis {(x,y)∈R2:y>0}{(x,y)∈R2:y>0}, where the xx-axis behaves like the boundary circle and should be thought of as at infinity. There is a problem, in that in the Poincaré disc, the boundary of H2H2 was a circle, and so it closed up on itself. In the UHP, the boundary is a line which doesn’t close up on itself, but these are supposed to be models for the same thing. To fix this, we imagine there is a point at infinity ∞∞ which joins up the two ends of the boundary to form an infinite diameter circle.
If we start with any Apollonian packing living in the Poincaré disc, there is a map from the disc to the UHP preserving hyperbolic distances, under which the outer circle of the packing becomes the xx-axis (together with the point at infinity), and exactly one of the horocycles (one of the circles tangent to the outer circle in the packing) becomes the horizontal line y=1y=1. All other circles and horocycles in the packing are sent to circles which are tangent to each other as before, but are now sandwiched between the lines y=0y=0 and y=1y=1.
If we focus on just those circles which meet the xx-axis we get what are called Ford circles. Remarkably each of these circles is tangent to the xx-axis at a rational number p/qp/q, and has radius 1/2q21/2q2. Moreover every rational number is the point of tangency of one of the circles (see below). Now some magic happens: suppose the Ford circles at a/ba/b and c/dc/d are tangent to each other, then there is a unique circle sandwiched between these two circles and the xx-axis. The rational point at which this circle meets the xx-axis is given by the Farey sum of a/ba/b and c/dc/d
ab⊕cd=a+bc+dab⊕cd=a+bc+d
Note that for this to be well-defined, a/ba/b and c/dc/d must be written in their simplest form. This Farey sum, and the associated Farey sequences FnFn you get by looking at all rational numbers between 0 and 1 which can be written as a fraction with denominator at most nn, turn up in several places across number theory. These include rational approximation of irrational numbers and the Riemann Hypothesis.
Möbius transformations
If you haven’t seen hyperbolic geometry before, you may wonder how we can map the Poincaré disc model to the UHP model, and in such a way that the strange distance measure in the two models is preserved—for a start one is a finite region while the other is an infinite half-plane. The answer is to view both models as living inside the complex plane CC (or more accurately the extended complex plane ˆC=C∪{∞}C^=C∪{∞}): the Poincaré disc is the unit disc {z∈C:|z|<1}{z∈C:|z|<1}, and the UHP is the region above the real axis {z∈C:{z∈C: Im(z)>0}(z)>0}. Then a function like
z↦−iz+1z−1=−iz−iz−1(3)(3)z↦−iz+1z−1=−iz−iz−1
will do the trick. This function is an example of a Möbius transformation, which in general is a complex function of the form
z↦az+bbz+dz↦az+bbz+d
were we require ad−bc≠0ad−bc≠0 so that this function is invertible. The function (3)(3) sends the unit disc to the UHP, but it is not the only Möbius transformation which does this. In fact there are infinitely many such functions, all of which preserve the hyperbolic metric. In the previous section I claimed that starting with any Apollonian packing, we could choose one of these Möbius transformations such that the image had a very specific form, sandwiched between the lines Im(z)=0(z)=0 and Im(z)=1(z)=1.
An exercise: If you have seen Möbius transformations before, you may wish to try and prove that the purported mapping exists yourself. (Hint: remember that Möbius transformations send circles and lines to circles and lines, and are completely determined by their image on 3 distinct points.)
The upshot of this is that all Apollonian packings are the same in the hyperbolic plane, because they can all be mapped to the same packing by (invertible) functions which preserve hyperbolic distance. Once we have started thinking about the Apollonian packing living in the complex plane, the whole world of complex functions is open to us, and we can start to do crazy things. If we don’t restrict ourselves to just Möbius transformations, but see what happens when we apply holomorphic or anti-holomorphic functions to the packing (these are complex functions with a good notion of derivative in the sense of calculus, which in particular have the property that they preserve angles between intersecting curves), we can get some very pretty designs. We need not even require (anti-)holomorphicity. The patterns featured on the front and back covers were drawn in this way.
Beyond the packing
Let us return to Apollonius of Perga. Remember that his treaties Έπαφαι, where he stated and solved the problem of finding tangent circles, is lost to history—how then do we know what he proved and how? The answer is that we don’t. The only record we have appears in the writings of Pappus of Alexandria, who lived some 400 years after Apollonius, but who references many of Apollonius’ works, including six which are no longer extant. All he says of Tangencies is the general problem which Apollonius was interested in, and that he solved it by solving many simple special cases and working up from there.
The first person to reprove Apollonius’ results in ‘modern’ times was Adriaan van Roomen in 1596. His solution, however, does not use ruler and compass constructions, so cannot have been the one Apollonius used. The result was later proved using methods available to Apollonius, and in the way described by Pappus, by van Roomen’s friend François Viéte.
A ship’s location determined by its distance from three points.
The method of Viéte was later reworked and simplified by several mathematicians, including Isaac Newton in his Principia. Newton related the position of the centre of the fourth circle to its distance from the centres of the three circles to which it is supposed to be tangent. This is called hyperbolic positioning or trilateration. Newton used this viewpoint to describe the orbits of planets in the solar system, but it can also be used to help navigate ships, and to locate the source of a signal based on the different times the signal is received at three different locations. In the first world war this was used to locate artillery based on when shots were heard. This is also how modern GPS works (not by triangulation as is commonly believed).
So this 2000-year-old problem in abstract geometry turned out to have extremely useful applications in the real world. The Apollonian packing also shows up in lots of different areas of mathematics. For example, Ford circles inspired the Hardy–Littlewood circle method, an important tool in analytic number theory which was used to solve Waring’s Problem: for an integer kk, can every integer be written as a sum of at most nn kkth powers for some value of nn? This is true: for example, every integer is the sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 2013, Harald Helfgott used the circle method to prove the weak Goldbach conjecture: every odd number greater than 5 is the sum of 3 primes.
To infinity
As a final application, I am a geometric group theorist, and I cannot help but talk about one place the Apollonian packing shows up in my field. Be warned: there is definitely some advanced maths coming up, but if you don’t mind skipping over some of the details, there are some very pretty pictures to make it worthwhile.
It turns out that the extended complex plane ˆCC^ can be thought of as the boundary of 3 dimensional hyperbolic space H3H3. If we model H3H3 as the upper half-space {(x,y,z)∈R3∣z≥0}∪{∞}{(x,y,z)∈R3∣z≥0}∪{∞} then ˆCC^ is identified with the plane {(x,y,z)∈R3∣z=0}∪{∞}{(x,y,z)∈R3∣z=0}∪{∞}. When Möbius transformations act on ˆCC^, they also act on the whole of H3H3, and preserve hyperbolic distance. If we start by choosing just a few Möbius transformations, these generate a group which acts on H3H3. In doing so, the group creates a pattern on the complex plane called its limit set. This is a picture of how the group acts ‘at infinity’. Choosing the Möbius transformations carefully gives a group whose limit set is precisely the Apollonian packing.
A valid arrangement of circles, with their interiors shaded. Note that the straight line is an infinite radius circle through ∞∞, so it still counts.
Let’s be a bit more precise; pick a point p∈ˆCp∈C^ and choose gg pairs of circles (C+i,C−i)gi=1(Ci+,Ci−)i=1g, each of which doesn’t intersect pp. Each circle cuts ˆCC^ into two regions, call the region containing pp the exterior of that circle, and the complementary region the circle’s interior. We also want to arrange things so that no two circles have overlapping interiors (although two circles are allowed to be tangent). Next, for each pair of circles (C+i,C−i)(Ci+,Ci−) choose a Möbius transformation mimiwhich maps C+iCi+ to C−iCi− and which sends the interior of C+iCi+ to the exterior of C−iCi−. The group G=⟨m1,…,mg⟩G=⟨m1,…,mg⟩ generated by these transformations is called a (classical) Schottky group and it acts as a subgroup of the group of isometries of H3H3. Since we chose the circles to have non-overlapping interiors, we can use the delightfully named ‘Ping-Pong Lemma’ to prove that GG is abstractly isomorphic to the free group on gggenerators.
A packing with the four starting circles emphasised in black, and the Schottky group generators shown in pink and blue.
So how do we get a Schottky group whose limit set is the Apollonian packing? We can cheat slightly by working backwards; starting off with the picture we want to create, then we will choose the pairs of circles in the right way. Remember that one way we thought about constructing the Apollonian packing was to start off with four mutually tangent circles and then inductively draw the fifth circle wherever we can. Our strategy will be to choose Möbius transformations which do the same thing. We are helped by the following curious fact which you may want to try and prove yourself (again using Möbius transformations): given any three mutually tangent circles, there is a unique circle (possibly through ∞∞) which passes through all three circles at right angles. Given the four initial circles, there are (43)=4(43)=4 triples of mutually tangent circles, so we let C±1C1± and C±2C2± be the four circles orthogonal to each of these triples, as shown on the left. The corresponding Möbius transformations are :
The limit set of G=⟨m1,m2⟩G=⟨m1,m2⟩ is indeed the Apollonian packing we started with. If we perturb the starting Möbius transformations just slightly by varying the matrix entries (while being careful to ensure that the resulting group acts nicely on H3H3), we get a group whose limit set is a twisted Apollonian packing.
The limit sets of a one parameter family of groups which contains the Schottky group GG. Click the image to view a higher quality version.
Even though some of these perturbed limit sets look like they are still made up more or less of circles, they are in fact made up of a single continuous closed curve which is fractal, and does not intersect itself anywhere. They are examples of Jordan curves and illustrate why the Jordan Curve Theorem is so difficult to prove despite being ‘obvious’. Playing around more with different choices of Möbius generators we can produce even more beautiful examples of fractal limit sets; below are just a few to finish off. If you want to learn more about Schottky groups, their limit sets, and how to draw these pictures, I highly recommend the book Indra’s pearls: the vision of Felix Klein. It is the basis of this final section of this article, and gives details on exactly how you can draw these and many other pictures yourself.
David Sheard
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Chausath Jogini Mandir (64 Joginis Temple) is situated in a hamlet called Hirapur, 20 km outside Bhubaneswar, the capital of Odisha state of Eastern India.
The temple is believed to be built by the Queen Hiradevi of Bramha dynasty during 9th century.
It’s built in a circular fashion, completely put together with blocks of sand stone. The inside of the circular wall has cavities, each housing the statue of a Goddess. There are around 56 such idols, made of black granite, inscribed within the wall cavities, centring on the main idol which is the Goddess Kali, who stands on a human head representing the triumph of the heart over the mind. The temple houses a central altar (Chandi Mandapa) which has the remaining 8 Goddess idols on all 4 sides. Some historians believe that an idol of Maha Bhairava was worshiped in the Chandi Mandapa.
64 Joginis Temple is a tantric temple, with hypaethral architecture as tantric prayer rituals involve worshiping the bhumandala (environment consisting all the 5 elements of nature – fire, water, earth, sky and ether).
The legend behind the temple according to local priests is of the Goddess Durga taking the form of 64 demi-goddesses in order to defeat a demon. After the fight the 64 goddesses (Joginis) asked Durga to commemorate them in the form of a temple structure.
The Jogini idols are generally representing a female figurine standing on an animal, a demon or a human head depicting the victory of Shakti (Feminine power). The idols express everything from rage, sadness, pleasure, joy, desire and happiness.
Such temples are also seen at Ranipur-Jharial site of the Balangir district in Odisha and seven other places in India.
The number 64 finds its reference in Hindu mythology in various forms viz Kala for time, Kala for performing arts etc. The temple complex is maintained by Archaeological Survey of India.
Source: 64 Yogini Temple in Hirapur, Odisha
Source: Chausathi Yogini Temple
Source: Chausathi Yogini Temple
Chaunsath Yogini ( 64 योगिनी )
Jul 26 2019
In very ancient days, eight great Female Goddesses Shaktis emerged from the cosmic soul of the Principle Dieties and formed Kali Durga, the Universal Shakti Power, These were the grand Mothers (Ashta Matrikas) of all subsequent Yoginis. According to Kaula Tantra, these eight manifested each in turn into eight Divine Shaktis, thus resulting in the 64 Tantric Yoginis. These 64 powerful Goddesses have unique personas and powers to fulfill your desires, drive away negativity and fear, prevent misfortunes, and give you knowledge, peace, all-around prosperity, good progeny, and auspiciousness of all kinds.
An examination of the ancient Tantric tradition reveals a particular sanctity assigned to the number eight. The eight mother faculties (tatvas) of the manifested universe, the eight directions with four cardinal and four intermediate points (digbandahs), the eight miraculous yogic powers (siddhis), eight “limbs” of Yoga (astanga) eight forms of the Divine Mother (matrikas) and eight primary mystic symbols (mudras) are just a few examples. The square of eight, or sixty-four, occupies an even more profound position in the field of Tantra which, from the point of view of the practitioner, first and foremost identifies the sixty-four Tantric Yoginis.
Legends about the 64 Yoginis
The Lalitha Sahasranama and Vishnu Bhagavatha Purana reverently call the Divine Mother Shakti as Maha Yogini (Great Ascetic) and Kula Yogini. The Yogini Sahasranamaand Kaula Kularnava Tantra refers to the Supreme Devi as Maha Yogini. In ancient days, 8 great Female Shaktis emerged from the Universal Shakthi Energy called Parashakthi – these were the Divine Grand Mothers (called Ashta Matrikas) of all subsequent Yoginis. These 8 Matrikas manifested each in turn into Eight Sacred Shaktis, thus resulting in the 64 Tantric Yoginis.
The Yoginis are also popularly known and worshipped in Hinduism, Jainism and Buddhism in several countries as Bhairavis, Dhakinis, Shakinis, Sakinis, Sakthis, etc. The 64 Yoginis are known to be capable to manifest in physical form to give dharshan even in this Kali Yuga.
References about the 64 Yoginis and their powers can be found in the ancient scriptures like : Brahmananda Purana, Agni Purana, Skanda Purana, Kalika Purana, Jnanarnava Tantra, Brihad Nandikeswara Purana, Chandi Purana of Sarala Das, Brihndla Tantra, Bata Avakasa of Balaram Das, etc. Historical romances and semi-historical literature like Somadevasuri’s Yasastilaka of AD 959, Kalhana’s Rajatarangini of c.1150 and Somadeva’s Kathasarit Sagara of c.1070 contain legendary stories about the all-powerful Yoginis.
Each form of the Divine Mother Kali is a Mahavidya. Dasa Mahavidyas are:
Kālī
Tārā
Ṣodaśī
Bhuvaneśvarī
Chinnamastā
Bhairavī
Dhūmāvatī
Bagalāmukhī
Mātangī
Kamala
Shodashi (Sixteen) Nityas of Lalita
Kamesvari
Bhagamalini
Nityaklinna
Bherunda
Vahnivasini
Mahavajresvari
Sivaduti
Tvarita
Kulasundari
Nitya
Nilapataka
Vijaya
Sarvamangala
Jvalamalini
Citra
Mahanitya
Shodashi (Sixteen) names of Devi and 16 days worship Rituals
Source: The Millennium old 16-day Durga Puja in Odisha
Source: The sixteen names of Durgā and their explanations
Chausath Yogini Temples in India
Source: When was Chousath Yogini Temple built?
There are 17 Chausath Yogini temples found across India out of which two are in Odisha, five in Madhya Pradesh, three in Uttar Pradesh, and one in Tamil Nadu. The most prominent ones are Hirapur, Ranipur Jharial, Khajuraho, Bhedaghat, Mitauli, Dudhai, and badoh. Several yogini temples have been lost, whether through deliberate destruction or plundering for building materials. Yogini images have been discovered from Shahdol, Hinglajgadh, Lokhari, Nareshwar, Rikhiyan, shahdol, Kanchipuram, greater Bengal, Varanasi, and Delhi.
Chausath Yogini Temple of Hirapur, also called Mahamaya Temple, is 20 km outside Bhubaneswar, the capital of Odisha state of Eastern India. It is devoted to the worship of the yoginis, auspicious goddess-like figures. The temple is believed to have been built by Queen Hiradevi of the Bramha dynasty during the 9th century. The temple is small and circular, only 25 feet in diameter. It is hypaethral and built of blocks of sandstone. The inside of the circular wall has niches, each housing the statue of a Goddess. 56 of the 64 idols, made of black stone, survive. They surround the main image at the center of the temple, the Goddess Kali
Chausath Yogini Temple of Ranipur Jharial in Balangir District, Odisha, The Chausath Yogini Temple, built in the 9th or 10th century in Ranipur-Jharial, in an isolated position some miles from the towns of Titilagarh and Kantabanjhi in Balangir district, Odisha, is a circular, hypaethral, 64-yogini temple made of sandstone, some 50 feet in diameter. 62 of the yogini images survive. The site, on an outcrop of rock, must have been important, given the presence of a large temple built of brick and several small temples of stone. The primary entrance is an opening in the circular wall towards the east; unlike at the Hirapur yogini temple, there was once a further opening towards the south, now filled in.
Chausath Yogini temple in Khajuraho town of Madhya Pradesh, India. Dated to the late 9th century, it is the oldest surviving temple at Khajuraho. According to an inscription dated to 1323 CE (Vikram Samvat 1383), the temple was built by the Kachchhapaghata king Devapala (1055 – 1075CE). It is said that the temple was the venue for providing education in astrology and mathematics based on the transit of the Sun. It is the earliest extant temple in the Chandela capital, Khajuraho. The temple is among the Western group of temples on a 5.4 m high platform. It has a rectangular plan measuring 31.4 m x 18.3 m. It is one of the historic Yogini temples across India; many of the others have a circular plan, though those at Rikhiyan and Badoh are also rectangular, so there was at least a local tradition of building them in this shape. Like all Yogini temples, the Khajuraho temple is hypaethral, open to the air. The temple is made of large, coarse granite blocks, with an open courtyard at the center. The centeryard was originally surrounded by 65 shrine cells: 10 on the front (north) wall, 11 on the back wall, and 22 on each side. Only 35 of these 65 cells now survive; each has a small doorway made of two squared granite pillars, a lintel stone, and a curvilinear tower roof. Above the lintel of the best-surviving cells is a triangular pediment. There is no surviving trace of a central shrine, whether to Shiva or the Goddess, as found in other Yogini temples.
Chausath Yogini Temple, Bhedaghat, also called the Golaki Math, is one of India’s yogini temples, but exceptionally it has shrines for 81 rather than the usual 64 yoginis. The temple is the largest of the circular yogini temples, some 125 feet in diameter. The scholar Shaman Hatley calls it the “most imposing and perhaps best known of the yogini temples”. It has a covered walkway with 81 cells for yoginis around the inside of its circular wall; three niches, two to the west, and one to the southeast remain open as entrances. There is a later shrine in the center of the courtyard; the temple was adapted as a Gauri-Shankar temple with the construction of the building in the south-centre of the circle in 1155 AD, at which time the central deities were moved. The temple was built early in the 11th Century AD by King Yuvaraja II, of the dynasty of the Kalachuris of Tripuri; he lived around 975-1025 AD
Chausath Yogini Temple, Mitaoli, also known as Ekattarso Mahadeva Temple, is an 11th-century temple in the Morena district in the Indian state of Madhya Pradesh. It is one of the few well-preserved Yogini temples in India. The temple is formed by a circular wall with 65 chambers, apparently for 64 yoginis and the goddess Devi, and an open mandapa in the center of a circular courtyard, sacred to Shiva. According to an inscription dated to 1323 CE (Vikram Samvat 1383), the temple was built by the Kachchhapaghata king Devapala (1055 – 1075CE). It is said that the temple was the venue for providing education in astrology and mathematics based on the transit of the Sun. The temple is on a hill about 100 feet (30 m) in height; there are 100 steps to climb up to the entrance. It is circular with a radius of 170 feet (52 m), while inside it has 65 small chambers, each with a mandapa which is open and a facia of pilasters and pillars. The roof of the ring of shrines is flat, as is that of the central shrine to Shiva; the circular courtyard is hypaethral, open to the sky, with an open porch as its entrance. The parliament building of India is said to have been based on this temple. The temple is in the Seismic Zone III region and has survived several earthquakes, seemingly without any serious damage. This fact was cited when the issue of safety from the earthquake effect of the circular Parliament House, its design supposedly based on the Mitaoli temple, was debated in the Indian Parliament.
Chausath Yogini Temple, Dudhai, Lalitpur, Uttarpradesh, details of temples are not available. The temple locally called as Akhada /Akhara, is situated in a forest of Buri Dudhai. Temple is believed to be built in the 10th century CE by Chandela kings. The temple is on a circular plan and has a hypaethral elevation. The temple is about 50 feet in diameter. The temple has 12 flat-roofed cells on the north, and five flat-roofed cells on the south, and the eastern and western portions are completely lost.
Chausath Yogini Temple, Badoh, Some 30 miles from Dudahi, at Badoh in Vidisha district, Madhya Pradesh is the Gadarmal temple of the Mothers, another 42-niche yogini temple, and one of the few that are rectangular. 18 broken images of the goddesses that once fitted into grooves in the temple platform are preserved from the waist down. It is composed of a rectangular shrine and a tall and massive Shikhara, adjacent to some Jain temples. Vidya Dehejia writes that the yogini temple must once have been hypaethral. Gadarmal Devi temple dates back to the 9th century. The architecture of this yogini temple is a fusion of Pratihara and Parmara styles.
LokhariThere appears to have been an early 10th-century yogini temple on a hilltop at Lokhari, Banda District, and Uttar Pradesh. A set of twenty images, nearly all theriomorphic, the figures having the heads of animals such as horse, cow, rabbit, snake, buffalo, goat, bear, and deer, has been recorded. Dehejia describes these as striking rather than specially artistic.
Nareshwar Another set of twenty 10th-century images, with careless later inscriptions from the 12th century, was rescued from Nareshwar (also called Naleshvar and Naresar) in Madhya Pradesh, a site which still has some twenty small Shaivite temples, to the Gwalior Museum, some fifteen miles away.
Hinglajgarh The site of Hinglajgarh, on the border of Madhya Pradesh and Rajasthan, was cleared of statuary for the building of the Gandhi Sagar Dam. The rescued statues contain enough fragments of yogini images for Dehejia to state that there was once a yogini temple at Hinglajgarh
Rikhiyan, Some 150 miles north of Khajuraho on the south bank of the River Yamuna, in the Banda District, Uttar Pradesh, are the fragmentary remains of what seems to have been a rectangular 64-Yogini temple in the Rikhiyan valley. This is part of a complex of other temples, unlike the solitary Yogini temples such as Mitaoli. When the site was photographed in 1909, ten four-Yogini slabs were present. Dehejia states that the multiples of 4 suggest a 64-Yogini total, while the straightness of the slabs implies a rectangular plan (as at Khajuraho). Seven were stolen on various occasions, and the last 3 of the slabs were moved to Gadhwa fort nearby for their safety. The slabs portray the Yoginis on a plain background without the usual attendant figures. They sit in the ceremonial pose of Lalitasana, one leg resting on their animal vehicle. They have “heavy breasts, broad waist[s] and large stomach[s]”. One has the head of a horse and holds a corpse, a severed head, a club, and a bell, and so maybe Hayanana, “The Horse-headed”. This and other Yoginis shown with corpses link the temple to a corpse ritual. Also photographed in 1909 were three three-Matrika slabs; Dehejia suggests that these formed part of a rectangular shrine to the Eight Matrikas accompanied by Ganesh. A well-preserved four-Yogini slab from Rikhiyan is held in the Denver Art Museum
Shahdol, Yogini images from Shahdol district (anciently Sahasa-dollaka) in Madhya Pradesh have been taken to the Dhubela Museum near Khajuraho, the Indian Museum at Calcutta, and the village temples of Antara and Panchgaon in Shahdol district. The yoginis are seated in the ceremonial Lalitasana pose, and they have haloes flanked by flying figures behind their heads
Kanchipuram or Kaveripakkam Dehejia publishes and discusses yogini images of the Chola period, around 900 AD, recovered from northern Tamil Nadu. These include one now in the British Museum, others in the Madras Museum, the Brooklyn Museum, the Minneapolis Institute of Arts, the Detroit Institute of Arts, and the Royal Ontario Museum. The British Museum yogini is ascribed to Kanchipuram; the collection site is not known, but many sculptures of the same style were recovered from a large “tank” (artificial lake) at Kaveripakkam, seemingly derived from nearby temples. The image formed part of a large set of yoginis.
Varanasi, 12th-century texts including the Varanasimahatmya of the Bhairavapradurbhava suggest that there was a circular hypaethral yogini temple in Varanasi (also called Benares and Kashi) in the 11th century. Several yogini-related sites have been identified in the city. Just above the Chaumsathi Ghat cremation ground is Chaumsathi Devi temple; it is not mentioned in the scriptures but is where modern-day devotees gather, especially at Holi, as prescribed in the Kashikhanda.
Delhi, Legend has it that a yogini temple was built in the south Delhi district of Mehrauli; tradition places this as the Yogmaya Temple there, without reliable evidence. The region outside the imperial city of Indraprastha, described in the Mahabharata, was called Yoginipura, the yogini city. Indraprastha has been identified with Delhi.
Source of Picture: Google
Chausath Yogini Temple, Bhedha Ghat, Jabalpur, MP, India
Source: Chausath Yogini Temple – Complete Inventory of Goddesses and Gods
Source: Chausath Yogini Temple – Complete Inventory of Goddesses and Gods
Dupuis, Stella (2008). The Yoginī Temples of India : in the pursuit of a mystery, travel notes. Varanasi: Pilgrims Publishing. ISBN978-81-7769-665-3. OCLC298129207.
Dehejia, V. 1986. Yoginī Cult and Temples: A Tantric Tradition. New Delhi: National Museum.
Donaldson, Thomas E. 1985. Hindu Temple Art of Orissa 3 vols. Leiden: Brill.
Gadon, Elinor. ‘Probing the Mysteries of the Hirapur Yoginis’. In ReVision Vol. 25, no. 1 (2002): 33-41.
Shaw, Miranda. 1994. Passionate Enlightenment: Women in Tantric Buddhism. New Jersey: Princeton University Press.
Mahapatra, K. N. 1953. ‘A Note on the Hypaethral Temple of Sixty-four Yoginis at Hirapur,’ Orissa Historical Research Journal II: 23–40; reprinted in H. K. Mahtab, ed., Orissa Historical Research Journal, Special Volume, 1982.
Hatley, Shaman. 2007. ‘The Brahmayamalatantra and Early Saiva Cult of Yoginis’. Unpublished PhD. dissertation, University of Pennsylvania.
Stietencron, Henrich von. 2013. ‘Cosmographic buildings of India: The circles of the yogini,’ Yogini in South Asia: Interdisciplinary Approaches(Routledge Studies in Asian Religion and Philosophy), ed. István Keul, pp. 70-83. London: Routledge.
Further readings
Hatley, Shaman. ‘Matr to Yogini: Continuity and Transformations in the cult of the Mother Goddesses,’ in Transformations and Transfer of Tantra in Asia and Beyond, edited by István Keul, pp. 99–129, Berlin: Walter de Gruyter, 2012.
Mishra, P.B. Orissa under the Bhauma Kings. Calcutta: Vishwamitra Press, 1934.
Panigrahi, K. C. Archaeological Remains at Bhubaneswar. Bombay: Orient Longmans, 1961.
Sharma, Rajkumar. The Temple of Chaunsatha-yogini at Bheraghat. Delhi: Agam Kala Prakashan, 1978.
Source: From theatrics to metatheatre: The Enron Drama
We review briefly dramatism and theatrics and attempt to synthesize several views of metatheatre to suggest that metatheatre has the potential to govern theatrics. We posit metatheatre can influence theatrics by dis-integrating dominant theatrics to reveal differences, and provide those competing theatrics a stage for enactment. On the other hand, metatheatre may provide a means for competing theatrics to achieve dialectic resolution through antenarrative synthesis. We think metatheatre serves both dialectic purposes, sometimes by engendering a dialectic of disintegration as well as resolving competing theatrics when metatheatre offers an antenarrative that either synthesizes theatrical dialectics or enables them to co-exist on the same stage. These two metatheatrical governed events occur simultaneously and continuously, continuing without closure in rhizomatic fashion. The only criteria metatheatrics must met is that it allows theatrics to continue to find a way to move (forward, backward, sideways, up, down, etc.), so that players, in our case, people in organizations, can continue to coordinate ongoing action.
Aligning ourselves with dramatism and the narrative paradigm in general, we begin by reviewing theatrics and metatheatre. We then explore Enron’s context and the related theatrics. We demonstrate how metatheatre mediates all these theatrics and then discuss how metatheatre operates and the implications of metatheatre for future research on the topic of organizational theatre. We hope to offer a view of metatheatre that can be applied widely in the analysis of social life.
Dramaturgy, Theatrics, and Antenarrative
The two main influences of dramaturgy in organizational studies are Kenneth Burke and Erving Goffman. Both of them encourage us to look at social interaction through a theatrical lens, seeing how people dramatize and mystify everything they do in everyday life. Focusing this lens on organizational life allows us to analyze organizations as context made up of actors playing roles on a particular stages. Dramaturgy assumes a close relationship between language and action (Burke, 1966, 1969). Dramas are literary compositions that tell a story, often of human conflict, by means of dialogue and action, and are performed by actors’ in front of an audience (Turner, 1986).
In organizational studies, drama is used as an umbrella metaphor (Czarniawska, 1997) encompassing a wide range of events where meaning emerges from social interaction: story-as-performed, enacted stories, scripts, cultural performance, plays, etc. Mangham and Overington (1987) present a dramaturgical model for understanding organizational interactions and the staging of organizational reality, looking at the various ways organizational members project appearances in various organizational settings. Continuing that work, Clark and Mangham (2004) look at dramatic events that are explicitly scripted and produced. They examine the power corporate theatre has in suggesting new values and challenging everyday assumptions as the authors, audience and actors collude in turning scripts into an imaginative life made present. The difference with the examination we present is that the audience was an unwitting one.
Clark and Mangham (2004) review a piece of corporate theatre where the audience was invited to what they knew was a corporate show, a Hollywood-style production. In the performance we analyze here, the audience was an unwitting one. They were not aware they were watching scripted theatrics. And while the producers knew the show was make-believe, the audience thought they were witnessing everyday life, and not a special show created in a liminal space set aside for the scripted performance. Our example provides a backstage peek at some Goffmanesque trickery and mystification.
Dramaturgy’s bent on social construction is that social reality is co-constructed through a community of players and witnesses to the play (Barry, 1997). Dramatrugists assert that human behavior is constituted by rhetoric; individuals use discourse to persuade and influence the behavior of others (Brissett & Edgley, 1990).
“The synthetic nature of stories goes beyond saying what did happen, to imply what should happen or what can happen. In this way, they not only influence thought, feeling, and will, but the construction of social reality itself” (Feldman, 1990, italics in original).
In addition to discourse, dramaturgy recognizes that part of the meaning of peoples’ social interaction is to be found in the manner which they express themselves during interaction (Brissett & Edgley, 1990). Thus, how people express themselves to, and in conjunction with, others to create meaning is the central focus of dramaturgy (Gardner & Avolio, 1998).
Organizational theatrics remind us that a drama is never really complete until it is performed; acted on some kind of a stage before an audience.
‘Even the most trivial forms of day-to-day talk involve immense skill and presume a great deal of learning. Talk can become art in the sense that particular forms of convention or contrivance may be employed to secure certain expressive and communicative ends. Storytelling, displays of wit, rhetoric and drama exist in all types of society. The ‘success’ of these verbal forms, however, is directly involved with their performance…’ (Giddens, 1987).
Those particular theatrics, their production and for whom and what purpose, by what means, become organizational episodes, along with their scripts, that we can explore using narrative and ethnographic methods. For the organization itself, organizational theatre provides a model, in one concentrated image, of the ways in which members can give meaning and direction to their work lives. While organizational narratives furnish an ideal, theatrics enact particular stories, specifying roles suited for particular actors and contexts. By engaging in theatrics in various settings for particular audiences, actors attempt to shape their definition of the situation Goffman (1959).
But this shaping can be met with resistance. People have their own projects and there are represented by competing discourses and expressed by people performing those discourses in competing theatrics (Boje, 2002a; Boje and Rosile, 2003b). They are enacting new and alternative narratives and fighting for stage time.
This competition often begins with antenarrative. Antenarrative theory (Boje, 2001) is closely tied to Kristeva (1980: 36) and Bakhtin (1981), who suggest that each text has an intertextual “trajectory” that is historical and social (Boje, 2001, O’Connor, 2002). Used as an adverb, “ante” combined with “narrative” or “antenarrative” means earlier than narrative. Antenarrative shifts the focus of narrative analysis from “what’s the story here” to questions of “why and how did this particular story emerge to dominate the stage?” Used as a noun, ‘ante’ indicates a bet, a pre-narrative speculation that offers new possibility. As opposed to an existing or dominant narrative, antenarrative is nothing more than a wager. Antenarrative signifies discourse that is fragmented, nonlinear, incoherent, collective, yet unplotted. The bet is that such a pre-story can be told and theatrically performed to thereby enroll stakeholders in intertextual ways that transform the world of action into theatrics. A narrative of dominant discourse usually tries to retain its elite status by silencing alternative discourse or swashing antenarrative newcomers. People in power re-double their efforts at theatrics that enact the narratives that serve their own purposes and projects. They do not want to share stage time with competing theatrics or antenarrative rookies, much less rebels.
Antenarrative is threatening to narrative because its theatrics are potentially liberating. Clark and Mangham(2004) explains how in theatrics, imaginative events take on for a moment the presentness of physical events; in theatre, ‘physical events take on for a moment the perfection of imaginative form’ (Cole, 1975, cited in Clark and Mangham). These moments involve a power to reveal the seen but unnoticed by juxtaposing the lived and experienced world with the presentation of the narrative ideal (to contrast the profane with the sacred), and the role that theatrics play depends on the manifestation of imaginative life as physical presence (Clark and Mangham, 2004).
But theatrics are tenuous, and liberation and change is far from guaranteed. While organizations themselves are becoming more fragmented, polyphonic (many voiced) and collectively produced, and we would like to think that as modernist organizations make way for postmodern organizations, dominant narratives must make way for multiple stories. The issues of power and politics, along with the marginalization and silence of alternative discourses and antenarrative is ever present.
The focus of our analysis are these currently-under-construction and fragmented theatrics, the discourses just beginning to compete through enactment. These organizational theatrics are the theatrics of the antenarrative versus the theatrics of narrative. Looming over this staged battle is metatheatre.
Metatheatre
In this paper, we synthesize two metatheatre perspectives. Victor Turner (1985: 181) uses the term to indicate all the communication regarding a particular communication process, the metacommentary that spectators and actors engage in as they reflect upon their own roles and actions on various stages. We all engage in this metacommentary, reflecting and talking about what we have said and done or even what we are saying and doing. “Thus, if daily living is a kind of theater, social drama is a kind of meta-theater, that is, a dramaturgical language about the language of ordinary roleplaying and status-maintenance which constitutes communication in the quotidian social process” (Turner, 1985:181). This metatheatre involves reflexivity by everyday actors about their performances where they reveal to spectators what they are doing.
For Boje and Rosile (2003a), the term metatheatre highlights the simultaneity of multiple theatres. For them, metatheatre indicates the multiple and contending theatres that constitute organizations. Boje (1995) uses the metaphor of a multi-play performance called Tamara, where the audience follows different threads of stories, following actors from room to room. Doing so, the audience only ever sees portions of the larger script. They get peaks and pieces depending on the theatrics they witness. The metatheatre here is represented by the grand written script, the meta-script, by Tamara playwright John Krizanc (1981, 1989). For actors and audience alike, there is no ‘larger’ script. The only thing the audience can experiences in chasing actors from room to room are fragmented, emerging, incomplete theatrics. As with a grand narrative (Lyotard, 1979/1984), no one is ever in the position to witness metatheatre, but only local narratives and theatrics. The same is true for all of us, whether it be the metatheatrics of ‘capitalism’ or our organizations. We never see all the theatre performed; it is occurring simultaneously on different stages; some you see and perform, but other acts you hear about from colleagues, vendors, and customers (Boje & Rosile, 2002b). We know them, and change them, only through the theatrics that involve us.
The relation between theatrics and metatheatre might be compared to Gidden’s (1987) deep and surface structure which entails a duality of some construct simultaneously influencing and being influenced by, social interaction. The metatheatre provides an abstract frame for our theatrical actions, and those actions in turn influence the frame. Our actions also change our conception of metatheatre. We see similar dualities in organizational studies, as with the perspective of a culture that resides ‘above’ individual behavior or the simultaneous and recursive emergence between structure and strategy.
We synthesize these two metatheatre approaches so we can reflect on theatrics and their purposes and agenda, as well as attend to Tamara-esque fragmentation, liminality and indeterminacy of processual aspects of theatrics. Both perspectives take postmodern turns in dramaturgy, with Turner’s focusing on reflexivity and metacommentary while Boje and Rosile’s focus on fragmentation and multiple theatrics.
As we will see, both of these processual theatrical qualities are endemic to Enron (Boje, 2002b), and can be reveal through Enron actors reflecting on their actions and theatrics. At Enron, each integrating attempt by presidents, corporate leaders and boards of directors to evoke spectacular theatre, to control the center stage, and to enroll a cast of characters that will influence spectators, soon disintegrates as the pull of multiple scripts, plots, and characters spin out of control.
On one hand, metatheatre reveals a dialectic of disintegration that opposes integrative attempts by executive players. On the other hand, metatheatre might also offer a resolution by offering an antenarrative that allows multiple theatrics to co-exist on the same stage (that will become a Tamaraland), or perhaps the new antenarrative will turn out to be a good bet, bringing together or silencing the previous narratives, carving out enough space and stage to become the new dominant theatrics by subsuming or displacing competing theatrics. And eventually, these evolutions and revolutions in dialectic cycles of theatric-integration and disintegration, the networking of simultaneous stage-crafted performances seeking to instruct and control spectators and actors, also erupt into more fragmentation (Boje, 2002b: 10).
Our synthesis also allows us to make reflexive hermeneutic shifts. We make shifts between the metatheatric perspectives, between actor commentary about their theatrics and the theatrics that emerge from those reflections, as well as shift between the multiple competing theatrics. By making these shifts, we are able to explore and reveal fragmented, emerging theatrics that might plant a seed for change through dialectic confrontation or synthesis, as well as the actor and spectator reflections on their own performances and participation in theatrics.
Source: VICTOR TURNER’S POSTMODERN THEORY OF SOCIAL DRAMA: Implications for Organization Studies
Metatheatre – Turner (1985: 181) invents the term ‘meta-theater.’ Where for Burke and Goffman, all the world is a theatre stage, for Turner, ‘meta-theatre’ is the communication about the communication process, spectators and actors reflect upon how the actors do what they do on stage, ‘the ability to communicate about the communication process itself’ (p. 181). In contrasting his own dramaturgy work with Goffman’s, Turner (1985; 181) says that for him ‘dramaturgical analysis begins when crises arise in the daily flow of social interaction.’ Turner continues, ‘Thus, if daily living is a kind of theater, social drama is a kind of meta-theater, that is, a dramaturgical language about the language of ordinary role-playing and status-maintenance which constitutes communication in the quotidian social process’ (p. 181). Metatheatre then is for Turner, reflexivity by everyday actors about the communication system, where they consciously show spectators what they are doing. Turner studies reflexivity in crisis phase of social interaction, but also within the redressive phase. Turner theorizes four phases, breech, crisis, redressive action, and reintegration in what he calls ‘social drama.’
Metacommentary, is a term (Turner, 1982a: 104) borrows from Geertz, ‘a story a group tells itself about itself’ or ‘a play a society acts about itself.’ Metatheatre then builds upon the idea of metacommentary, ‘an interpretive reenactment of its experience’ (Turner, 1982a: 104). In the positive, metatheatre reenacts conflicts, giving them contextualization, so that with metacommentary, facets are illuminated and accessible for remedial action. Through multiple reflections, spectators are able to provoke transformations in everyday life. On the negative side, the metatheatre distorts event and context in ways that provoke conformity. For example, our weekly street theatre is a metacommentary on global, national, and local conflicts, a time for reflection and reflexivity. Our signs are commentary, and we resist conformity. We are opposed by metacommentary of our critics, what see our acts as traitorous, seditious, and rebellious. Both sides use drama to provoke and persuade.
Metatheatre is about the dialectic process of framing through theatre, in ways that appeal to the frame of mind of the spectator; resistance is about bringing counter-frames to bear on dominant frames.
UNDERSTANDING METATHEATRE
Introduction
“Meta” is a Greek word, meaning “after” and “with” or “alongside”. It becomes a very active prefix. Like other words that are coined with a prefix “meta”, such as metafiction, metahistory, and metanarrative, metatheatre1, also denoting the concept of self-reflectivity, is a scrutiny of and concern with theatricality and the making of theatre. The practice of metatheatre has a long history; however, the term, both seminal and much-debated, was initially used in 1963 by the critic and playwright Lionel Abel in his work Metatheatre: A New View of Dramatic Form, a loose collection of essays on metatheatre. He declared that a new form has succeeded tragedy as the dominant force of modern theatre, which he termed “metatheatre”. Abel’s discussion has aroused the interest of critics. Since his coinage of the term, other important works on metatheatre have been published, trying to demarcate the domain of metatheatre and attempting at the itematization of the features of metatheatre.
Definitions and Features of Metatheatre
Abel (1963) defined metatheatre as “theatre pieces about life seen as already theatricalized” (p. 60), a theatre not concerned with the world “outside” the theatre, but only with the theatre itself. His theory of metatheatre rested on two basic postulates: First, the world was a stage; second, life was a dream (Abel, 2003, p. 163).
Abel pointed out the nature of metatheatre—a subset of theatricality, or the glorification of the theatre itself, and he also presented the particulars of metatheatre in a miscellaneous and combined manner2, but his definition was loose and sometimes erratic. Later critics redefined the term and expanded the scope of metatheatre.
Calderwook (1971) offered his definition of metatheatre in Shakespearean Metadrama: Metatheatre “is a dramatic genre that goes beyond drama (at least drama of a traditional sort), becoming a kind of anti-form in which the boundaries between the play as a work of self-contained art and life are dissolved” (p. 4). Calderwook’s understanding of metatheatre does not go beyond the confines established by Abel.
June Schluter in her Metafictional Characters in Modern Drama (1979) understood metatheatre broadly as a concept focused on identity, a tool for the playwright to mark the distinctions between reality and illusion, art and life. Her study included postmodern dramatists, such as Peter Weiss, Edward Albee, and Tom Stoppard. She concluded that all plays under scrutiny in the book “reflect the modern artist’s ongoing awareness of the constantly changing dialectic of reality and illusion” (as cited in Gallagher, 2010, p. 2).
In 1982, Manfred Schmeling in his French monograph on metatheatre defined it as plays that refer to themselves as plays, primarily through the device of the play within the play. For instance, Shakespeare’s use of “The Murder of Gonzago” in Hamlet becomes a paradigm for such a device. Schmeling brought out the intertextuality in metatheatre and extended his study to certain Romantic plays. (Gallagher, 2010, p. 3) Schmeling was not the first to note the intertexuality in metatheatre. Gentili in Theatrical Performance in the Ancient World: Hellenistic and Roman Theatre (1979) observed the intertextuality when remarking “I used the term ‘metatheatre’ in the sense that played constructed from previously existing plays” (p. 15).
More critics emphasize interxtuality in metatheatre. Anderson in his Barbarian Play: Plautus’s Roman Comedy (1993) warned us of “this voguish term ‘metatheatre’” and remarked that we “must confine its usage” (p. 139). He reserved metatheatre to the changes one play rings upon another, or to its argument with it. Boyle (1997) referred to the process of play making as “The Palimpsestic Code” in Tragic Seneca. (ch. 5)
Slawomir Swiontek, a Polish drama theorist and semiotician, in his contribution to the theory of metatheatre in 1986 and 1993, called dramatic dialogue “meta-enunciative” in that it always contains the two theatrical axes of communication, among characters onstage and toward the audience (Gallagher, 2010, p. 2). William Fitzhenry in “The N-Town Plays and the Politics of Metatheatre” (2003) discussed two contrasting theatrical models: the monologic and the dialogic. He thinks:
The second model posits a more dynamic and interactive form of drama that initiates an open-ended dialogue between stage and audience. In this model, the boundaries between playwright, actor, and audience do not collapse to underwrite a single, overarching idea, but rather remain in tension with one another, multiplying interpretive possibilities rather than reducing them. (p. 23)
Most importantly in 1986, Richard Hornby (1986) published his seminal work Drama, Metadrama and Perception, giving a clear definition and taxonomy of metatheatre. Metatheatre, in his view, means “drama about drama”. It occurs “whenever the subject of a play turns out to be, in some sense, drama itself” (p. 31). It implies self-reflexivity, or a type of literature that is aware of itself as a literary object and concerned with the process of its own making. It is characterized with theatricality, self-awareness, self-reflexivity, and self-knowledge.
The various definitions and features of metatheatre examined here indicate that the understanding of metatheatre is deepened and the confines of it enlarged, as can be further demonstrated in the evolution of metatheatre.
The Evolution of Metatheatre
Baroque metatheatre and modernist metatheatre are generally regarded as the two great eras of metatheatricality. Abel and the critics following him believe that metatheatrical plays first appeared in the late sixteenth and early seventeenth centuries. For him, Shakespeare, Calderon, and other baroque playwrights wrote plays “about life seen as already theatricalized” with characters “unlike those in tragedy […] aware of their own theatricality” (as cited in Gallagher, 2010, p. 1). Abel later stresses the differentiation between tragedy and metatheatre in Tragedy and Metatheatre: Essays on Dramatic Form, a collection of essays published in 2003. For him, tragedy deals “with the real world” and metatheatre “with the world of the imagination” (Abel, 2003, p. v). He regarded Macbeth as Shakespeare’s only tragedy, while Hamlet is a typical metatheatrical play.
Richard Fly (1986) noted that a growing body of scholars is concerned with the self-reflexive themes and techniques in Shakespeare’s plays. They find in his plays a preoccupation with “the materials and processes of art-making itself” (p. 124), and intend to view his masterpieces as “‘mirrors’ reflecting the artist’s ongoing struggle to understand and master the expressive potential of his medium” (p. 124). They conclude that
the drama in the plays becomes dislodged from plot and character and situated instead in the playwright’s self-conscious interaction with himself, his medium, and his audience. With this redirection of the creative process, mimesis gives way to self-analysis, and drama is subsumed in “metadrama”. (p. 124)
Later playwrights in the 20th-century, such as Luigi Pirandello, Bertolt Brecht, Jean Genet, and Samuel Beckett inherited the legacy of the baroque playwrights and represented the illusory and theatrical qualities of both life and theatre. The Theatre of the Absurd is also considered as metatheatre.
In a remarkable introduction to Abel’s Tragedy and Metatheatre: Essays on Dramatic Form, Martin Puchner differentiates between baroque metatheatre and modernist metatheatre. Despite the similarity that both represent and mediate on theatre, baroque playwrights used the form to “celebrate the theatre” and theatricality, while modernist playwrights “view the theatre with mistrust and suspicion”, exposing its problematic nature (Abel, 2003, p. 17). Puchner pointed out that German dramatist and director Brecht admitted to being driven by a continual “mistrust of theatre” and attempted to make the theatre less theatrical in his experiment with Epic Theatre (Abel, 2003, p. 17).
Most of Pirandello’s plays explore his meditation on reality as a problematic concept, which can be apprehended neither objectively nor scientifically. He produced a group of plays exploring the art of theatre, knowns as “theater in the theatre”, which include Six Characters in Search of an Author (1921), Each in His Own Way (1924), Tonight We Improvise (1930), and The Mountain Giants (1934). He used ingeniously play-within-the play to tear down the fourth wall, to keep reminding the audience that he is facing not reality and real people but a created work, performed by actors. In this way, he achieved the effect of estrangement and explored the themes of artistic creation, of the relationship between art and the everyday reality. Such a technique influenced Brecht.
Six Characters in Search of an Author is a piece of metatheatre, a play-within-the-play. There are two interwoven plots. The first plot is a sentimental melodrama of a family, composed of six members: father, mother, son, stepdaughter, boy, and girl. These characters are conjured and then abandoned by their author, but have by now life in their own veins; therefore, they seek the means to exhibit their drama. The second plot centers on a company is about to do rehearsals for a new play, Pirandello’s The Game of Role-Playing. When the company starts the rehearsal of the second act, the family of the six characters comes on stage, claiming that they are really six most interesting characters and that their story should be acted out. The director agrees to actualize their drama.
The staging of a play and even the process of creation is put on stage. This play is what Pirandello called “the mirror theatre”, a play that turns a mirror onto the theatre itself, and exposes and renews the operating principles of the drama. In this way, Pirandello discussed the making of theatre and the nature of artistic creation, expressing his dissatisfaction with and rebellion against the bourgeois theatre of the turn of the 20th century. By attacking the actors’ style of performance in the play, Pirandello denunciated the commercial theatre at the turn of the century.
Different from Abel, Hornby holds the view that metatheatre is not a narrow phenomenon limited to certain playwrights or certain periods in the history of drama. In fact, it is always occurring. The evolvement of the theory of metatheatre has seen an extension of the application of its concept to classical Greek comedy and tragedy (Abel denied the theatricality of tragedy in his 1963 study) as well as to non-Western drama and world drama.
Critics observe metatheatre in the tragedies of Greek tragedian Sophocles. Batchelder in his The Seal of Orestes: Self-reference and Authority in Sophocles’s Electra (1995) read Sophocles’s Electra as a play about playwriting. He remarked “In its close association with speech and action, falsehood and truth, the [empty] urn also functions as a symbol of the deception of the theatrical situation per se. In this respect it is…a ‘metatragic’ symbol of tragedy…” (p. 35). Ringer’s Electra and the Empty Urn: Metatheatre and Role-Playing in Sophocles (1998) is another impressive study on metatheatrical devices in Sophocles’s plays.
F. J. Lelièvre sees the connection between Aristophanes’s use of parody in his old comedies and metafiction (as cited in Zen, 2016, p. 31). Of his 44 plays, critics note that The Frogs (405 BC) and The Thesmophoriazusae (410 BC) are metadrama, which are concerned with drama itself.
Brian Crow in “African Metatheatre: Criticizing Society, Celebrating the Stage” (2002) claimed that “anotable feature of contemporary African drama is the persistence with which its writers foreground the act of performance itself and seem concerned to investigate its status” (p. 133). He attributed it partly to “a deep-seated pleasure in many African cultures in playful theatricalizing and comic, often satirical observation and parody of different kinds of behavior at the everyday social level” (p. 133), and partly to the fact that it is “a natural offshoot of the intense ‘theatricality’ of so much African theatre” (p. 134). He then observed that African uses of the theatrical image are strikingly different from “the primarily individualistic, existential, and often introspective quality of many Western examples of metatheatre” (p. 134). For African dramatists, the exploration of the metatheatrical aims to “anatomize oppression and injustice and to celebrate the capacity of theatre and the theatrical to function as modes of survival, resistance, and even, in their more optimistic moments, change in contemporary African societies” (p. 134). Crow also remarked that the ways in which African metatheatricality has been realized are formally varied. Besides setting a play in a theatrical environment or the performance of a play within the play, the less direct ways include the allusion to the oral narrative instead of dramatic and/or the theatrical conventions, establishing intertexuality with well-known African plays, and the use of “scripted ‘improvisations’ involving role-play and play-acting,” etc. (p. 138).
Increasing attention has been given to metatheatricality in Asian drama and drama in other parts of the world as well. This is due to the comprehensive and profound understanding of metatheatre.
Five Types of Metatheatre
As to metatheatricality, Hornby identified five types (devices) of metatheatre in Drama, Metadrama and Perception: the play within the play, the ceremony within the play, role-playing within the role, literary and real-life reference, and self-reference.
Hornby (1986) held that “the play within the play is projected onto life itself, and becomes a means of gauging it” (p. 45). The play within the play creates two strikingly distinctive layers of performances for the audience to have the experience of seeing double and noting the multiple layers of action. It indirectly reminds the audience of the illusory nature of theatre.
It is a strategy for constructing play texts that contain, within the perimeter of their fictional reality, asecond or internal theatrical performance, in which actors appear as actors who play an additional role. It doubles an aesthetic experience of a double reality (Fischer & Greiner, 2007, p. xi).
Hornby (1986) thought that ceremonies within plays are ubiquitous (p. 49). Anthropologists and theatre historians have long posited a generic and historical relationship between ritual and theatre. Drama had an origin in rituals; tragedy originated in ancient Greece. The word tragedy, of Greek origin, means “goat song” and may possibly refer to archaic performances in which dancers either competed for a goat as a prize or were dressed up as goats. Drama competition was enacted at the Great Dionysia, an annual religious and cultural festival held in honor of the god Dionysus, the god of wine. As part of the festival, four plays of each playwright—three thematically connected tragedies and a satyr play were played and judged. When Hornby talked about the ceremony, he meant the play involves a formal spectacle of some kind that is separated from the surrounding action; for instance, the crowning of a king, a wedding or a ritual sacrifice. The ceremony offers a communal pleasure that derives from the understanding of things that would otherwise be confusing or ephemeral.
Margaret Croyden (1974) gave insightful comments on the validity of “ritual”:
Attempts at ritual seem less successful when actors and audience share no common ground. In actual fact, ritual has always had a moral, religious, practical, or psychological significance, and has never existed for its own sake. Rites were a need. (as cited in Graham-White, 1976, p. 319)
William Frost lists numerous rites and ceremonies common to Elizabethan drama, such as prayers, formal curses, funerals, marriages, dances, feasts, banquets, ceremonial arrivals and departure, formal oaths, trials, banishments, royal court scenes, etc. (Stroup, 1977, p. 139). Critics note how Chekhov’s plays are built around such ceremonies as arrivals, departures, anniversaries, and parties (Graham-White, 1976, p. 320). Dramatists believe in the immense symbolical, allegorical, and universalizing values of these essentially dramatic phenomena.
Role-playing within the role, in Hornby’s (1986) view, is “an excellent means for delineating characters, by showing not only who the character is, but he wants to be” (p. 67). It is an effective revelation of the psychological truth of the character. Hornby further classified it into three broad types: voluntary, involuntary, and allegorical.
Literary and real-life reference stresses the intertextuality of theatre and the importance of understanding aplay in its cultural code. Hornby defined literary reference as:
Direct, conscious allusion to specific works (except for parody, which may be more general) that are recent and popular. The work or works referred to must not yet be part of the drama culture complex, but should preferably be avant-garde, or at least somewhat controversial. (1986, p. 90)
There are four categories of literary reference: citation, allegory, parody, and adaptation. Real-life reference means allusions to real persons, places, events, objects, etc.
Self-reference is the most extreme, intense form of metatheatre, which calls attention to itself as a play, thus producing “alienation effect” and avoiding catharsis.
Though these five varieties of metatheatre have their distinctive effects, they often occur together or blend into one another. They have “no truth in and of themselves, but rather a means of discovering truth” (Hornby 1986, p. 32). Metatheatre criticizes the conventions of realist and naturalist dramas and breaks the illusion onstage by tearing down the “fourth wall”, which is an invisible, imagined wall that separates the actors from the audience. Metatheatre discomforts the audience and creates the aesthetic effect of alienation or estrangement, consequently prompting the audience to meditate on life.
METATHEATER AND SANSKRIT DRAMA
About the Book
In 1963, Lionel Abel’s book, Metatheatre: A New View of Dramatic Form, was published. The basic idea of metatheater is that of multiple ‘layers’ of illusion. The prefix ‘meta-‘, here, suggets ‘beyond’, ‘above’, or ‘within’. Metatheater, in one of its senses, can be viewed as one make-believe (dramatic) world superimposed upon another make-believe (dramatic) world. Or as one dramatic world framed within another dramatic world. The most easily relationship is the ‘play-within-the-play’.
The question might be asked what relevance such a recent topic of literary criticism in the West would have to a study of ancient Sanskrit drama. Each of the six essays in Part One of this book provides an effective answer. In the sixth essay, a translation is given of the passage in the Abhinavabharati, wherein Abhinavagupta comments on the term ‘natyayita’. Remarkably, this ancient Sanskrit term is most appropriately translated by the freshly minted English word, ‘metatheater’! And it is through an understanding of this 30-year-old English term (‘metatheater’) that one is able to obtain a revealing insight into what Abhinava was saying one thousand year ago about ‘natyayita’, term used in the Natya-Sastra, in the section on Sarira Abhinaya, and illustrated by Abhinava with a reference to Subandhu’s play, Vasavadatta Natyadhara
The first five essays illustrate how profoundly a knowledge of the metadramatic structure of Sanskrit plays will affect the way in which they are to be understood and translated.
Part two of this book presents the text and translation of, and commentary on, two Sanskrit farces which were written in the seventh century A.D. by the South Indian king, Mahendravarman. These two plays superbly illustrate the multi-dimensional splendor of ‘metatheater’ in Sanskrit drama.
About the Author
Michael Lockwood and Vishnu Bhat are both faculty members of Madras Christian College, Tambaram.
Dr. Lockwood (Dept. of Philosophy) has written a number of articles on the Pallavas, and has published two books which deal with the art, philosophy and history of the Pallavas: Mamallapuram and the Pallavas and Mahabalipuram Studies (co-authored by Dr. Gift Siromoney and Dr. P. Dayanandan).
Dr. Vishnu Bhat (Dept. of English) had much of his early education in a Sanskrit school before receiving his higher degrees in the field of English literature and language. He has collaborated previously with Dr. Lockwook in writing several articale on the inscriptions of the Pallavas.
Preface
In 1963, Lionel Abel’s book Metatheatre: A New View of Dramatic Form, was published. In this book Abel introduced a new term: ‘Metatheatre’. According to him, ‘metatheatre’ is the right term to describe the only form possible to the contemporary playwright who wishes to treat a subject gravely. He held that tragedy, invented by the Greeks to describe pain and yet give pleasure, is unrealizable today. In the late Renaissance, a revolution occurred in human consciousness which made tragedy impossible. But playwrights such as Calderon and Shakespeare wrote ‘serious’ plays which were self-reflexive: the illusion that sustains the play worlds also sustains the world outside the plays – the so-called ‘real world’.
Abel’s theory of metatheater is not a simple one, and it is, perhaps, better to look at a later analysis of this and related terms: ‘metadrama’, ‘metaplay’, etc.
The basic idea of metatheater is of multiple ‘layers’ of illusion. The prefix, ‘meta-‘, here, suggests ‘beyond’, ‘above’, or ‘within’. Metatheater, in one of its senses, can be viewed as one make-believe (dramatic) world superimposed upon another make- believe (dramatic) world. Or as one dramatic world framed within another dramatic world. The most easily understandable example of this relationship is the ‘play-within-the-play’. Of course, this idea did not come into being in the age of Calderon and Shakespeare. The idea of multiple layers of illusion is as old as theater itself. But it is only since Abel’s book was published in 1963 that a whole area of criticism and theory has sprung up in the West under the general heading of , meta theater’ or ‘metadrama’. Richard Hornby, in his book, Drama, Metadrama, and Perception (1986), has given a clear and concise analysis of different types of ‘metatheater/metadrama’:
1. The play within the play: i) The Inset type – the inner play is secondary ii) The Framed type – the inner play is primary 2. The ceremony within the play: In all cultures we find plays that contain feasts, balls, pageants, tournaments, games, rituals, trials, inquests, processions, funerals, coronations, etc. 3. Role playing within the role: i) Voluntary, ii) Involuntary, iii) Allegorical 4. Literary and real-life references: i) Citation, ii) Allegory, iii) Parody, and iv) Adaptation Self-reference: The play directly calls attention to itself as a play, an imaginative fiction
The question might be asked what relevance such a recent topic of literary criticism in the West would have to a study of ancient Sanskrit drama. Each of the six essays in Part One of this book provides, we hope, an effective answer. In our sixth essay, we translate the passage in the Abhinavabharati, where in Abhinavagupta comments upon the term ‘natyayita. Remarkably, this ancient Sanskrit term is most appropriately translated by the freshly minted English word, ‘metatheater’! And it is through an understanding of this 30-year-old English term (‘metatheater’) that we are able to obtain a revealing insight into what Abhinava was saying one thousand years ago about ‘natyayita’, a term used in the Natya-Sastra, in the section on Sarira Abhinaya, and illustrated by Abhinava with a reference to Subandhu’s play, Vasavadatta Natyadhara.
Part Two of this book presents the text and translation of, and our commentary on, two Sanskrit farces which were written in the seventh century A.D. by the South Indian king, Mahendravarman. These two plays superbly illustrate the multi- dimensional splendor of ‘metatheater’ in Sanskrit drama.
Contents
Preface
v
1.
Sanskrit Drama – Its Continuity of Structure
1
2.
Natya-Yajna (Drama as Sacrifice)
11
3.
The Victorianization of Sakuntala
19
4.
Bhasamana-Bhasah or the Case of the Chimerical Kavi
33
5.
You or Us?
37
6.
Abhinavagupta’s Discussion of Metadrama (c. 1000 A.D.)
In T. Bakken, & T. Hernes (Eds.), Autopoietic organization theory: drawing on Niklas Luhmann’s social systems perspective (pp. 151-182). Copenhagen Business School Press.
The dialogic imagination: Four essays
Bakhtin, M. M. (1981).
(C. Emerson & M.Holquist Trans; M. Holquist, Ed.). Austin, Texas: University of Texas Press.
Organisational theatre as polyphonic enterprise: ambiguity & process in health care transformation.
Matula, L. J., Badham, R., & Meisiek, S. (2013).
In 2013 Academy of Management Annual Meeting proceedings (pp. 1-40). Academy of Management (AoM).
Introduction: Polyphony and Organization Studies: Mikhail Bakhtin and Beyond
Source: Mandapa: Its Proportion as a tool in Understanding Indian Temple Architecture
Proportion and measurements were the guiding tools for Indian temple construction starting from the 5th century onwards and it continuous even now. Through out the history proportion dominated as a tool, which determined the monuments both spatial arrangements as well as form. The ancient texts, therefore, insist on a high degree of precision in their measurements.
The standard text Mayamata mentions- ”Only if the temple is constructed correctly according to a mathematical system can it be expected to function in harmony with the universe. Only if the measurement of the temple is in every way perfect, there will be perfection in the universe as well.”
Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE
Source: TEMPLE ARCHITECTURE AND SCULPTURE
Source: Symbolism in Hindu Temple Architecture and Fractal Geometry – ‘Thought Behind Form’
Source: TEMPLE ARCHITECTURE AND SCULPTURE
Source: TEMPLE ARCHITECTURE AND SCULPTURE
Key Terms
Fractals
Cosmology
Temple Architecture
Fractal Dimension
Recursive
Algorithmically
Hindu Temples
Vastu Purush Mandala
Vastu Shastra
Shilpa Shastra
Nagara Style
Dravidian Style
Vesara Style
Kalinga Style
64 Yogini Temple Style
Jain Temple Architecture
Buddhist Stupa Architecture
Cellular Automata
3D Fractals
2D Cellular Automata
Nine Cell Square
Nav Grah Yantra
Sierpinski Carpet
Box Counting Method
Biophilic Architecture
Symbolism
Square and Circle
Earth and Heaven
Squaring the Circle
Correspondence
Equivalence
Symmetry
As Above, So below
Cosmic Mirrors
Hindu Temples: Models of a Fractal Universe
Source: Hindu temples: Models of a fractal universe
Hindu philosophy views the cosmos to be holonomic and self-similar in nature. According to ancient architectural tradition, Hindu temples are symbols of models of the cosmos and their form represents the cosmos symbolically.
The procedures and methods used in the construction of Hindu temples bear a striking resemblance to the procedures of computer graphics, including discretization, fractalization and extensive use of recursive procedures, including self-similar iteration. The instructions given in ancient Vastu shastras (texts on architecture) work like general programmes to generate various types of temples.
The paper is an attempt to draw attention to the similarities between the procedures and resulting forms in computer graphics and Hindu temple architecture and to explain the relationship that exists between the form of the temple and the concepts of Hindu philosophy. It is proposed that Hindu temples may be viewed as three dimensional fractal models and that the use of fractal geometry procedures has a special symbolic meaning in the generation of the forms of Hindu temples.
Introduction to the Temple Architecture in the Indian Context
Source: Temples of Odisha- the Geometry of Plan Form
The evolution of temple architecture is marked by a strict adherence to the original ancient models, that were derived from sacred thought which persisted over many centuries. The commencement of the main style of Hindu temple architecture in India dates back to the Mauryan period i.e 3rd century BC, as evident from the archaeological excavation at Sanchi (Madhya Pradesh, temple no.40 and18) and Bairat (Rajastan), (DB Garnayak , 2007) . The Indian Silpasastras recognize three main types of temples known as the Nagara, Dravida and Vesara. Nagara temple belongs to the country from the Himalaya to the Vindhya, Vesara from the Vindhya to the Krishna and the Dravida from the Krishna to the Cape Comorin (DB Garnayak , 2007). An inscription in 1235 A.D in the mukhamandapa of the Amritesvara temple at Holal in Bellary district of Karnataka speaks of the fourth style i.e. Kalinga, in addition to the above three. The Kalinga style of Architecture is explained exclusively in the texts like Bhubana Pradip, Silpa Prakasa, Silpa Ratnakosha etc.
Source: Investigating Architectural Patterns of Indian Traditional Hindu Temples through Visual Analysis Framework
Source: Fractal Geometry as a source of innovative formations in interior design
Source: The fractal analysis of architecture: calibrating the box-counting method using scaling coefficient and grid disposition variables
Source: PARAMETRIZING INDIAN KARNATA-DRAVIDA TEMPLE USING GEOMETRY
Vastu purusha mandala
Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE
Layout of a Hindu temple pursues a geometrical design known as vastu-purusha-mandala, the name of which is derived from the three vital components of the design namely Vastu meaning Vaas or a place of dwelling; Purusha, meaning the Universal principle; and Mandala meaning circle. Vastupurushamandala is a mystical diagram referred in Sanskrit as a Yantra. The symmetrical and self-repeating model of a Hindu temple demonstrated in the design is derived from the primary convictions, traditions, myths, fundamentality and mathematical standards. According to Vastupurushamandala, the most sacred and typical template for a Hindu temple is the 8×8 (64) grid Manduka Hindu Temple Floor Plan also referred as Bhekapada and Ajira. The layout displays a vivid saffron centre with intersecting diagonals which according to Hindu philosophy symbolises the Purusha. The axis of the Mandir is created with the aid of the four fundamentally significant directions and thus, a perfect square is created around the axis within the available space. This square which is circumscribed by the Mandala circle and divided into perfect square grids is held sacred. On the other hand, the circle is regarded as human and worldly that can be perceived or noticed in daily life such as the Sun, Moon, rainbow, horizon or water drops. Both the square and the circle support each other. The model is usually seen in large temples while an 81 sub-square grid is observed in ceremonial temple superstructures. Each square within the main square referred as „Pada‟ symbolise a specific element that can be in the form of a deity, an apsara or a spirit. The primary or the innermost square/s of the 64 grid model called Brahma Padas is dedicated to Brahman. The Garbhagruha or centre of the house situated in the Brahma Padas houses the main deity. The outer concentric layer to Brahma Padas is the Devika Padas signifying facets of Devas or Gods which is again surrounded by the next layer, the Manusha Padas, with the ambulatory. The devotees circumambulate clockwise to perform Parikrama in the Manusha Padas with Devika Padas in the inner side and the Paishachika Padas, symbolising facets of Asuras and evils, on the outer side forming the last concentric square. The three outer Padas in larger temples generally adorn inspirational paintings, carvings and images with the wall reliefs and images of different temples depicting legends from different Hindu Epics and Vedic stories. Illustrations of artha, kama, dharma and moksha can be found in the embellished carvings and images adorning the walls, ceiling and pillars of the temples.
Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE
Source: VASTU PURUSHA MANDALA- A HUMAN ECOLOGICAL FRAMEWORK FOR DESIGNING LIVING ENVIRONMENTS
Source: Space and Cosmology in the Hindu Temple
Source: Exploring Ancient Architectural Designs with Cellular Automata
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
According to Hindu philosophy, the main goal of man’s life is to achieve the ultimate liberation from the illusionary world where he suffers from his endless rebirth. Krishna says in Bhagavad Gita,9 if man worships, devotes and meditates to the manifested form of the wholeness of infinity, and sees the truth of its manifestation; he will surpass the confusion of the never-ending cycles of rebirth in this physical world, and finally will assimilate with the God. 10 Hindu calls it moksha.11 It brings the ultimate peace and harmony in man’s life. But how one, from the physical world, can practice to realize this ultimate truth of the creation? Brihatsamhita12 and Sthapatyaveda13 give the solution as the temple which should act as the microcosm of the cosmos [9]. It should be the bridge for the man of physical world to the God of divine world [10].
To connect the physical world with the divine world and to reflect the truth of creation, the layout of cosmos was copied graphically in the foundation of temple. Here, a mythical incident was interwoven where a formless being covered the sky and was, immediately, arrested down to the earth by the creator Brahma and other gods. This supernatural fact was depicted graphically by vastu purusha mandala, where vastu refers to physical environment, purusha refers to energy, power or cosmic being, and mandala is the diagram or chart. Central portion of the mandala represents the place of Brahma and other portions symbolize the other gods according to their capability. By laying down this metaphysical diagram in the foundation, various supernatural forces are captured beneath the temple whereas its centre is the source of cosmic energies. The basic shape of the vastu purusha mandala is a square which represents the earth [11]. Its four sides depict the four cardinal directions. It also symbolizes the order, the completeness of endless life and the perfectness of life and death [10]. According to Hindu philosophy, our mundane life is controlled by the number four—four castes, four stages of life, four great epochs or mahayugas, four heads of Brahma, the four Vedas etc. [12].
There are various types of vastu purusha mandala, which are nothing but the squares grids, produced from the basic shape; namely, a square which is known as sakala mandala. Each smaller square within the grid is called one pada. The number of pada may vary from 1, 4, 9, 16, 25 and so on 1024, where it follows the geometric progression of 1, 2, 3, 4, 5,…..,32 of common ratio 2. The mandala having even numbers of pada in its grid known as yugma squares mandala whereas the mandala, having odd numbers of pada known as ayugma squares mandala. Vastu purusha mandala is also known as different distinctive names according to the numbers of pada within the grid. The mandala having 1,4,9,16,25 and 36 numbers of pada within the grid are known as sakala mandala, pechaka mandala, pitah mandala, mahapitah mandala, manduka chandita mandala and para- masayika mandala, respectively14.
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho
FRACTAL DESIGN, ARCHITECTURE AND ART IN HUMAN HISTORY
Source: Working with Fractals
Fractals have permeated cultures spanning across many centuries and continents, classical art and vernacular architecture from the column capitals of ancient Greece, Egyptian, Aztec, Incan civilisations, the art of Ancient Mayans, Islamic and Hindu temples, Angkor Wat in Cambodia, the Eifel Tower in Paris, and the structures of Santiago Calatrava. Fractals are also evident in such well known works as those of Botticelli, Vincent van Gogh, and Jackson Pollock. Their visual properties were also explored by mathematicians when Benoit Mandelbrot published The Fractal Geometry of Nature (1982) in which he catalogued nature’s statistical fractals and discussed them using mathematical methods for their replication.
Fractals constitute a central component of human daily experience of the environment (Taylor & Spehar, 2016). While extensive research has documented the negative effects of environments that do not have a complement of rich experiential aesthetic variety (Mehaffy & Salingaros, 2013), their proliferation in art and design has continued to grow and diversify, creating architecture, interiors and products designed for human needs (Taylor & Spehar, 2016). Over the past two decades, interdisciplinary teams have confirmed that the aesthetic qualities of nature’s fractal patterns can induce striking effects on health.1
PARAMETERS OF FRACTAL PREFERENCE
Source: Working with Fractals: For the Love of Fractals
The universal preference for ‘statistical’ fractals peaks at low to moderate degree of complexity, while universal preference for ‘exact’ fractals peaks at a higher complexity. The high level of symmetry in exact fractals enables greater tolerance for visual complexity compared to statistical fractals (Abboushi et al., 2019). Four factors influence complexity in exact fractals:
1. Fractal dimension (D)
2. Symmetry
3. Recursion
4. Number of elements introduced at each recursion
Fractal dimension.
The Euclidean simplicity and symmetry of exact fractals increases tolerance and peak preference for medium-high complexity exact fractals (D= 1.5–1.7) (Abboushi et al., 2019). Medium- high exact fractals can enhance visual preference and mood, particularly in less complex Euclidean interior spaces (Abboushi et al., 2019; Taylor et al., 2018).
When complex fractal patterns are experienced within a low-complexity interior space, the visual preference can shift to those available higher D values (1.5 to 1.7, medium-high range), suggesting that a low complexity environment enables a tolerance and preference for higher complexity statistical fractals such as found in artworks or casted light patterns unique within that space (Abboushi et al., 2019). A good example of this scenario is museums with an abundance of geometrical rooms and white walls adorned with highly complex artworks that captivate.
Symmetry.
Patterns with symmetry and geometry, such as common among exact fractals, can be visually appealing as they balance interest and comprehensibility. Mirror symmetry is generally considered one of the most predictive factors when judging whether a geometric pattern is ‘beautiful’. A lack of radial and mirror symmetry can be overcome by including more recursion and higher fractal dimensionality.
The orderliness of exact fractals allows a pattern to approach the maximum use of space at a particular dimension while retaining its elegance. Patterned tiles and carpet, wall coverings and textiles, artefacts and ornaments found in many cultures (Eglash, 2002) are evidence of this spatial orderliness and symmetry.
Recursion.
Fractals generated by a finite subdivision rule bear a striking resemblance to both nature and human ornament. In mathematics, the finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. In a sense, subdivision rules are generalisations of regular exact fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals (Cannon, et al., 2001).
Source: The application of complexity theory and Fractals in architecture, urban planning and design
Source: The application of complexity theory and Fractals in architecture, urban planning and design
Fractal Geometry And Self-Similarity In Architecture: An Overview Across The Centuries
Nicoletta Sala Academy o f Architecture o f Mendrisio, University o f Italian Switzerland Largo Bernasconi CH- 6850 Mendrisio Switzerland E-mail: nsala @ arch.unisLch
Fractal Geometry and Architecture Design: Case Study Review
Xiaoshu Lu1,2, Derek Clements-Croome3, Martti Viljanen1
1Department of Civil and Structural Engineering, School of Engineering, Aalto University, PO Box 12100, FIN-02150, Espoo, Finland E-mail: xiaoshu@cc.hut.fi 2Finnish Institute of Occupational Health, Finland 3School of Construction Management and Engineering, Whiteknights, University of Reading, PO Box 219, Reading RG6 6AW, UK
The influence of traditional Indian architecture in Balkrishna Doshi’s IIM Complex at Bangalore: A comparative analysis using fractal dimensions and lacunarity
PRINCIPLES OF FRACTAL GEOMETRY AND APPLICATIONS IN ARCHITECTURE AND CIVIL ENGINEERING
Anton Vrdoljak, M.Sc. Faculty of Civil Engineering, University of Mostar, anton.vrdoljak@gf.sum.ba Kristina Miletić, B.Sc.(Math.) Faculty of Civil Engineering, University of Mostar, kristina.miletic@gf.sum.ba
Vastu Purusha Mandala – A human ecological framework for designing living environments.
Venugopal, Jayadevi
In Jetty, C, Chandra, B, Bhashyam, A, & Prabhakara, R (Eds.) Proceedings of International Conference on Advances in Architecture and Civil Engineering (AARCV 2012), Volume 2. Bonfring, India, pp. 870-877.
While plots may be “archetypal” when they exhibit certain forms, in this post we are concerned with character archetypes.
In modern storytelling, to consider them as archetypes might suggest a bit of a corset, perhaps even a straightjacket for the characters. For today’s author, to present a character as an archetype does not seem conducive to achieving psychological verisimilitude.
But an archetype is not the same as a stereotype. An advisor or mentor does not need to be a wise old man like Obi-Wan Kenobi. And an antagonist does not need to be a baddy.
Consider archetypes as powers within a story. Like planets in a solar system, they have gravity and they therefore exert force as they move.
Archetypes denote certain general roles or functions for characters within the system of the story. There is ample room for variation within each role or function. Boundaries between one archetype and another may be fuzzy. And it is possible for one character to stand for more than one archetype.
Archetypes Through The Ages
Certain archetypes are ancient and have been around as long as stories have been told. Others may have a Christian background. Some are modern interpretations of ancient archetypes seen in the light of dramaturgical principles.
We may distinguish between three sorts of archetypes.
Ancient – archetypes that we find in the very oldest stories, and in very modern ones
Classical – archetypes that we find in works of literature of the past two thousand years
Role-based – modern variants that consider the dramaturgical function of characters
This categorisation has overlaps. The ancient, original archetypes, such as the Mentor, are of course also classical. And certain role-based archetypes, such as the Protagonist, may correspond to ancient ones, such as the Prince.
The Protagonist is sometimes called the Hero, a word which in terms of ancient archetypes might refer to a number of archetypes, for instance Warrior (Achilles) or Trickster (Odysseus). In the modern sense of role-based archetypes, this is the person (or rabbit, or robot, or whatever) the story is primarily about, the one whose travails the recipient, the audience or reader, follows through to the end of the story.
The Protagonist’s opposing power is the antagonism, which may be personified in an individual Antagonist. It helps to remember that in terms of function within the story, an antagonist does not necessarily have to be a villain, but is a counterforce to the protagonist (for an ancient example, consider Agamemnon and Achilles in the Iliad).
The antagonistic force is sometimes referred to as the “Shadow”. This can be misleading, since really almost every archetype has its own shadow side. A Patriarch may be presented as benevolent or “light”, or as tyrannical and “dark”. Indeed, in one story the character (or characters) representing such an archetype might show signs of both.
Characters Wearing Hats
Several of the roles or functions that you find in all sorts of stories – such as the Mentor, the Ally, the Patriarch – do not always have to be riveted to one specific character. For instance, it is quite possible that one character may have the Mentor hat on at one point in the story, and the Ally hat at another.
The point is that such forces or functions tend to be present in stories, and characters express these forces through their role or function within the story at each point in the narrative.
There is even an archetype for a character that explicitly changes roles in the story, where it becomes part of that character’s function to jump role at one or more points along the story. That is a Shapechanger.
Some archetypes are gender specific. The Patriarch/Father/King stands for different values from the Matriarch/Mother/Queen. For other archetypes, whether the character is male or female is not the point. A Shapechanger or a Trickster is defined by what the character does in the story.
So archetypes are really little more than signposts. Assigning a character an archetype is not to pressure that character into behaving in a certain way. Calling a character an archetype is merely to give us a pointer to that character’s role and function in the story. Characters that can be labelled as several archetypes tend to be multi-facetted. Hamlet, for instance, fulfils the criteria for several archetypes. So thinking about characters in terms of which archetypal roles they may play is actually a way of making the characters richer, giving them more depth, making them appear psychologically real and ultimately human.
System archetypes are the pattern which are recurrent.
Key Terms
Systems
System Archetypes
Feedback
Causal Loops
Delays
Leveraged Networks
The Systems Thinker
Daniel H Kim
Peter Senge
Barry Richmond
STELLA
VENSIM
ITHINK
Ventana Systems
Isee Systems
WHAT IS A SYSTEMS ARCHETYPE?
Source: Systems Archetype Basics : From Story to Structure
Without having to climb beanstalks or push anyone into an oven, children learn lessons from fairy tales about how to hide from powerful, cruel beings, build solid dwellings, and be respectful of old people. Literary themes also show us the hero’s journey, the trials of hard work, the outcomes of faithful love and misguided passion, and the ennui of a materialistic life. In these examples from literature, the term archetype signifies a recurring, generic character, symbol, or storyline. In systems thinking, the term has a very similar meaning. It refers to recurring, generic systemic structures that are found in many kinds of organizations, under many circumstances, and at different levels or scales, from internal personal dynamics to global international relations.
Captured in the stories, structures, and behavior over time of the archetypes are similar teachings about competition, addiction, the perils of quick fixes, and the high flyer’s downfall. And as we do with stories and fairy tales, we can use the archetypes to explore generic problems and hone our awareness of the organizational dramas unfolding around us. We can even use archetypes to sharpen our ability to anticipate difficulties, communicate about them with our colleagues, and find ways to address them together.
The systems archetypes, as a group, make up one of the 10 current categories of systems thinking tools. (See Appendix B for a complete list of these tools.) Each archetype features a storyline with a distinctive theme, a particular pattern of behavior over time that can be graphed, and a unique systemic structure that can be depicted in a causal loop diagram. The value of archetypes is that we can study them apart from a specific story, problem, or organizational situation and take away generic, transferable learnings that we can then apply to many situations in our own lives.
WHERE DID ARCHETYPES COME FROM?
In the 1960s and 1970s, Jay Forrester, Dennis Meadows, Donella Meadows, and other pioneers of systems thinking observed several recurring systemic structures. In the 1980s, Michael Goodman, Charles Kiefer, Jenny Kemeny, and Peter Senge built on that work, in part with the help of notes developed by John Sterman, by describing, diagramming, and cataloguing these generic systemic structures as systems templates. When Peter Senge authored The Fifth Discipline: The Art and Practice of the Learning Organization, he referred to those structures as systems archetypes. Since then, the notion of systems archetypes has become quite popularized, and systems thinking practitioners have continued to teach, apply, and write about these recurring generic structures as well as investigate and test the potential of identifying new ones.
List of Key System Archetypes
Drifting Goals
Escalation
Fixes that Fail
Growth and Underinvestment
Limits to Success
Shifting the Burden/Addiction
Success to the Successful
Tragedy of the Commons
Source: Systems Thinking Tools: A User’s Reference Guide
Source: SYSTEMS ARCHETYPES I
Source: Systems Thinking Tools: A User’s Reference Guide
Source: SYSTEMS ARCHETYPES I
System Archetypes and Their Storylines
Source: SYSTEMS ARCHETYPES AND THEIR APPLICATION
Archetypes and their Applications
Source: SYSTEMS ARCHETYPES AND THEIR APPLICATION
Source: Systems Archetype Basics : From Story to Structure
Source: Systems Archetype Basics : From Story to Structure
Source: Systems Archetype Basics : From Story to Structure
Source: Systems Archetype Basics : From Story to Structure
Growth Archetypes
Source: A theory of spatial system archetypes
➤ A Glossary of Systems Thinking Tools
Source: Systems Archetype Basics : From Story to Structure
Systems thinking can serve as a language for communicating about complexity and interdependencies. To be fully conversant in any language, you must gain some mastery of the vocabulary, especially the phrases and idioms unique to that language. This glossary lists many terms that may come in handy when you’re faced with a systems problem.
Accumulator
Anything that builds up or dwindles; for example, water in a bathtub, savings in a bank account, inventory in a warehouse. In modeling software, a stock is often used as a generic symbol for accumulators. Also known as Stock or Level.
Balancing Process/Loop
Combined with reinforcing loops, balancing processes form the building blocks of dynamic systems. Balancing processes seek equilibrium: They try to bring things to a desired state and keep them there. They also limit and constrain change generated by reinforcing processes. A balancing loop in a causal loop diagram depicts a balancing process.
Balancing Process with Delay
A commonly occurring structure. When a bal- ancing process has a long delay, the usual response is to overcorrect. Over- correction leads to wild swings in behavior. Example: real estate cycles.
Behavior Over Time (BOT) Graph
One of the 10 tools of systems thinking. BOT graphs capture the history or trend of one or more variables over time. By sketching several variables on one graph, you can gain an explicit understanding of how they interact over time. Also called Reference Mode.
Causal Loop Diagram (CLD)
One of the 10 tools of systems thinking. Causal loop diagrams capture how variables in a system are interrelated. A CLD takes the form of a closed loop that depicts cause-and-effect linkages.
Drifting Goals
A systems archetype. In a “Drifting Goals” scenario, a gradual downward slide in performance goals goes unnoticed, threatening the long- term future of the system or organization. Example: lengthening delivery delays.
Escalation
A systems archetype. In the “Escalation” archetype, two parties compete for superiority in an arena. As one party’s actions put it ahead, the other party “retaliates” by increasing its actions. The result is a continual ratcheting up of activity on both sides. Examples: price battles, the Cold War.
Feedback
The return of information about the status of a process. Example: annual performance reviews return information to an employee about the quality of his or her work.
Fixes That Fail
A systems archetype. In a “Fixes That Fail” situation, a fix is applied to a problem and has immediate positive results. However, the fix also has unforeseen long-term consequences that eventually worsen the problem. Also known as “Fixes That Backfire.”
Flow
The amount of change something undergoes during a particular unit of time. Example: the amount of water that flows out of a bathtub each minute, or the amount of interest earned in a savings account each month. Also called a Rate.
Generic Structures
Structures that can be generalized across many different settings because the underlying relationships are fundamentally the same. Systems archetypes are a class of generic structures.
Graphical Function Diagram (GFD)
One of the 10 tools of systems thinking. GFDs show how one variable, such as delivery delays, interacts with another, such as sales, by plotting the relationship between the two over the entire range of relevant values. The resulting diagram is a concise hypothesis of how the two variables interrelate. Also called Table Function.
Growth and Underinvestment
A systems archetype. In this situation, resource investments in a growing area are not made, owing to short-term pressures. As growth begins to stall because of lack of resources, there is less incentive for adding capacity, and growth slows even further.
Learning Laboratory
One of the 10 tools of systems thinking. A learning lab- oratory embeds a management flight simulator in a learning environment. Groups of managers use a combination of systems thinking tools to explore the dynamics of a particular system and inquire into their own understand- ing of that system. Learning labs serve as a manager’s practice field.
Level
See Accumulator.
Leverage Point
An area where small change can yield large improvements in a system.
Limits to Success
A systems archetype. In a “Limits to Success” scenario, a company or product line grows rapidly at first, but eventually begins to slow or even decline. The reason is that the system has hit some limit— capacity constraints, resource limits, market saturation, etc.—that is inhibiting further growth. Also called “Limits to Growth.”
Management Flight Simulator (MFS)
One of the 10 tools of systems thinking. Similar to a pilot’s flight simulator, an MFS allows managers to test the outcome of different policies and decisions without “crashing and burning” real companies. An MFS is based on a system dynamics computer model that has been changed into an interactive decision-making simulator through the use of a user interface.
Policy Structure Diagram
One of the 10 tools of systems thinking. Policy structure diagrams are used to create a conceptual “map” of the decision- making process that is embedded in an organization. It highlights the fac- tors that are weighed at each decision point.
Rate
See Flow.
Reference Mode
See Behavior Over Time Graph.
Reinforcing Process/Loop
Along with balancing loops, reinforcing loops form the building blocks of dynamic systems. Reinforcing processes com- pound change in one direction with even more change in that same direc- tion. As such, they generate both growth and collapse. A reinforcing loop in a causal loop diagram depicts a reinforcing process. Also known as vicious cycles or virtuous cycles.
Shifting the Burden
A systems archetype. In a “Shifting the Burden” situa- tion, a short-term solution is tried that successfully solves an ongoing prob- lem. As the solution is used over and over again, it takes attention away from more fundamental, enduring solutions. Over time, the ability to apply a fundamental solution may decrease, resulting in more and more reliance on the symptomatic solution. Examples: drug and alcohol dependency.
Shifting the Burden to the Intervener
A special case of the “Shifting the Burden” systems archetype that occurs when an intervener is brought in to help solve an ongoing problem. Over time, as the intervener successfully handles the problem, the people within the system become less capable of solving the problem themselves. They become even more dependent on the intervener. Example: ongoing use of outside consultants.
Simulation Model
One of the 10 tools of systems thinking. A computer model that lets you map the relationships that are important to a problem or an issue and then simulate the interaction of those variables over time.
Stock
See Accumulator.
Structural Diagram
Draws out the accumulators and flows in a system, giving an overview of the major structural elements that produce the system’s behavior. Also called flow diagram or accumulator/flow diagram.
Structure-Behavior Pair
One of the 10 tools of systems thinking. A structure- behavior pair consists of a structural representation of a business issue, using accumulators and flows, and the corresponding behavior over time (BOT) graph for the issue being studied.
Structure
The manner in which a system’s elements are organized or interre- lated. The structure of an organization, for example, could include not only the organizational chart but also incentive systems, information flows, and interpersonal interactions.
Success to the Successful
A systems archetype. In a “Success to the Success- ful” situation, two activities compete for a common but limited resource. The activity that is initially more successful is consistently given more resources, allowing it to succeed even more. At the same time, the activity that is initially less successful becomes starved for resources and eventually dies out. Example: the QWERTY layout of typewriter keyboards.
System Dynamics
A field of study that includes a methodology for constructing computer simulation models to achieve better understanding of social and corporate systems. It draws on organizational studies, behavioral decision theory, and engineering to provide a theoretical and empirical base for structuring the relationships in complex systems.
System
A group of interacting, interrelated, or interdependent elements form- ing a complex whole. Almost always defined with respect to a specific pur- pose within a larger system. Example: An R&D department is a system that has a purpose in the context of the larger organization.
Systems Archetypes
One of the 10 tools of systems thinking. Systems archetypes are the “classic stories” in systems thinking—common patterns and structures that occur repeatedly in different settings.
Systems Thinking
A school of thought that focuses on recognizing the inter- connections between the parts of a system and synthesizing them into a unified view of the whole.
Table Function
See Graphical Function Diagram.
Template
A tool used to identify systems archetypes. To use a template, you fill in the blank variables in causal loop diagrams.
Tragedy of the Commons
A systems archetype. In a “Tragedy of the Commons” scenario, a shared resource becomes overburdened as each person in the system uses more and more of the resource for individual gain. Eventually, the resource dwindles or is wiped out, resulting in lower gains for everyone involved. Example: the Greenhouse Effect.
The above glossary is a compilation of definitions from many sources, including:
Innovation Associates’ and GKA’s Introduction to Systems Thinking coursebooks
The Fifth Discipline: The Art and Practice of the Learning Organization, by Peter Senge
High Performance Systems’ Academic User’s Guide to STELLA
The American Heritage Dictionary and The Random House Dictionary.
Systems Thinking Tools
Source: THE “THINKING” IN SYSTEMS THINKING: HOW CAN WE MAKE IT EASIER TO MASTER?
Source: Systems Thinking Tools: A User’s Reference Guide
Source: Systems Thinking Tools: A User’s Reference Guide
Systems Thinking: A Review and Bibliometric Analysis
Niamat Ullah Ibne Hossain , Vidanelage L. Dayarathna, Morteza Nagahi and Raed Jaradat *
Department of Industrial and Systems Engineering, Mississippi State University, Mississippi State, MS 39762, USA; ni78@msstate.edu (N.U.I.H.); vld66@msstate.edu (V.L.D.); mn852@msstate.edu (M.N.) * Correspondence: jaradat@ise.msstate.edu
A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
An Approach for the Development of Complex Systems Archetypes
Walter Lee Akers Old Dominion University, akers.walt@gmail.com
(2015). Doctor of Philosophy (PhD), Dissertation, Engineering Management & Systems Engineering, Old Dominion University, DOI: 10.25777/6xmx-r674 https://digitalcommons.odu.edu/emse_etds/18
Use of System Archetypes to Accelerate, Advance, and Deepen Systems Thinking Skills of Nurses
Daniel J Pesut PhD RN FAAN Professor of Nursing Population Health and Systems Cooperative Unit Director of the Katharine Densford International Center for Nursing Leadership University of Minnesota School of Nursing 308 Harvard St. SE
MPLS MN 55455 USA
Judith Pechacek, DNP, RN, CENP University of Minnesota, School of Nursing
Clinical Associate Professor Director, Doctor of Nursing Practice (DNP) Program 308 Harvard St. SE
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Karl Dodd1, 
Viewpoints which Matter 
