A Unifying Model of Arts

A Unifying Model of Arts

Key Terms

  • Natya Shastra of Bharata Muni
  • Poetics of Aristotle
  • Narrative Arts
  • Narrative Psychology
  • Drama Therapy
  • Social Simulation
  • Learning and Reflection
  • Normative Choices
  • Social Psychology
  • Social Mirrors
  • Psychological Mirrors
  • Self as Other
  • Other as Self
  • Coordination Arts
  • Competition Vs Cooperation
  • Networks and Hierarchy
  • Dance
  • Music
  • Drama/Films/Theater
  • Visual Arts
  • Diegesis
  • Haple diegesis
  • Diegesis dia mimeseos
  • Diegesis di’ amphoteron
  • Mimesis

Source: A Unifying Model of the Arts: The Narration/ Coordination Model

The Narration/Coordination model is presented as a unifying model of the arts with regard to psychological processing and social functions. The model proposes a classification of the arts into the two broad categories of the narrative arts and the coordinative arts. The narrative arts function to tell stories, often to promote social learning through the modeling of prosocial behaviors. The coordinative arts function to stimulate group participation through synchronized action, thereby serving as a reinforcer of group affiliation and a promoter of social cooperation. These two categories vary with regard to a number of psychological and social features related to personal engagement, role playing, cognitive structure, and performance. The arts are evolutionarily adaptive because they promote social cooperation through two distinct routes: the simulation of prosocial behaviors via the narrative arts, and the stimulation of group synchronization and cohesion via the coordinative arts.

Narrative and Coordinative Arts


Narration/Coordination Model of the Arts


Features of Narrative and Coordinative Arts


Classification of Arts


Interaction among the Arts


Modular Aspects of Performance Arts


Connections Between the arts: an Indian Perspective


The view that the arts belong to the domain of the sacred and that there is a connection between them is given most clearly in a famous passage in the Vishnudharmottara Purana in which the sage Markandeya instructs the king Vajra in the art of sculpture, teaching that to learn it one must first learn painting, dance, and music:

Vajra: How should I make the forms of gods so that the image may always manifest the deity?

Markandeya: He who does not know the canon of painting (citrasutram) can never know the canon of image-making (pratima lakshanam).

Vajra: Explain to me the canon of painting as one who knows the canon of painting knows the canon of image-making.

Markandeya: It is very difficult to know the canon of painting without the canon of dance (nritta shastra), for in both the world is to be represented.

Vajra: Explain to me the canon of dance and then you will speak about the canon of painting, for one who knows the practice of the canon of dance knows painting.

Markandeya: Dance is difficult to understand by one who is not acquainted with instrumental music (atodya).

Vajra: Speak about instrumental music and then you will speak about the canon of dance, because when the instrumental music is properly understood, one understands dance.

Markandeya: Without vocal music (gita) it is not possible to know instrumental music.

Vajra: Explain to me the canon of vocal music, because he, who knows the canon of vocal music, is the best of men who knows everything.

Markandeya: Vocal music is to be understood as subject to recitation that may be done in two ways, prose (gadya) and verse (padya). Verse is in many meters.

My Related Posts:

The Social Significance of Drama and Narrative Arts

Narrative Psychology: Language, Meaning, and Self

Psychology of Happiness: Value of Storytelling and Narrative Plays

Aesthetics and Ethics

Arts and Moral Philosophy

Drama Therapy: Self in Performance

Aesthetics and Ethics: At the Intersection

Understanding Rasa: Yoga of Nine Emotions

Rituals | Recursion | Mantras | Meaning : Language and Recursion

Meta Integral Theories: Integral Theory, Critical Realism, and Complex Thought

Levels of Human Psychological Development in Integral Spiral Dynamics

Truth, Beauty, and Goodness: Integral Theory of Ken Wilber

Luminosity and Chromaticity: On Light and Color

Geometry of Consciousness

Drama Theory: Acting Strategically

Drama Theory: Choices, Conflicts and Dilemmas

Third and Higher Order Cybernetics

Key Sources of Research:

Toward a Unification of the Arts

Steven Brown*

Front. Psychol. 9:1938. 2018

doi: 10.3389/fpsyg.2018.01938



Psychology of Narrative Art

Keith Oatley and Maja Djikic


A Unifying Model of the Arts: The Narration/ Coordination Model

Steven Brown

Empirical Studies of the Arts 2019, Vol. 37(2) 172–196

Click to access NarrCoord.pdf

Interaction, narrative, and drama: Creating an adaptive interactive narrative using performance arts theories

Magy Seif El-Nasr


Art, dance, and music therapy


Healing Dramas and Clinical Plots: The Narrative Structure of Experience (Cambridge Studies in Medical Anthropology)1st Edition

Cheryl Mattingly

A hypothesis on the biological origins and social evolution of music and dance

Tianyan Wang




Narrative, Emotion, and Insight

Edited by Noël Carroll, and John Gibson


The narrative arc: Revealing core narrative structures through text analysis

  • Ryan L. Boyd1,*
  • Kate G. Blackburn2 and 
  • James W. Pennebaker2

 Science Advances   07 Aug 2020:
Vol. 6, no. 32, eaba2196
DOI: 10.1126/sciadv.aba2196

Historical Narratives and the Philosophy of Art

Noël Carroll

The Journal of Aesthetics and Art Criticism 

Vol. 51, No. 3, Philosophy and the Histories of the Arts (Summer, 1993),

pp. 313-326 (14 pages) Published By: Wiley 


Narratives and Narrators: A Philosophy of Stories

Gregory Currie

The Poetics, Aesthetics, and Philosophy of Narrative

Noël Carroll

Wiley-Blackwell (2009)


The Psychology of Narrative Thought: How the Stories We Tell Ourselves Shape our lives

By Lee Roy Beach

Narrative: State of the Art

Click to access Bamberg,%20%20%20%20%20%20Narrative-State%20of%20the%20Art,%20%20%20%20%20%20Georgakopoulou%20Thinking%20Big%20with%20small%20stories%20in%20narrative%20and%20%20%20%20%20%20identity%20analysis.pdf

Narrative Psychology, Trauma and the Study of Self/Identity

Michele L. Crossley

Theory and Psychology Vol 10, Issue 4, 2000

First Published August 1, 2000 



The “Who” System of the Human Brain: A System for Social Cognition About the Self and Others

Steven Brown*

  • Department of Psychology, Neuroscience & Behaviour, McMaster University, Hamilton, ON, Canada

Front. Hum. Neurosci., 19 June 2020 | https://doi.org/10.3389/fnhum.2020.00224



The Synthesis of the Arts: From Ceremonial Ritual to “Total Work of Art”

Steven Brown1* and Ellen Dissanayake2

  • 1Department of Psychology, Neuroscience & Behaviour, McMaster University, Hamilton, ON, Canada
  • 2School of Music, University of Washington, Seattle, WA, United States

Front. Sociol., 15 May 2018 | https://doi.org/10.3389/fsoc.2018.00009


Storytelling Is Intrinsically Mentalistic: A Functional Magnetic Resonance Imaging Study of Narrative Production across Modalities

Ye Yuan, Judy Major-Girardin, and Steven Brown


The neural basis of audiomotor entrainment: an ALE meta-analysis

Léa A. S. ChauvignéKevin M. Gitau and Steven Brown*

Front. Hum. Neurosci., 30 September 2014 | https://doi.org/10.3389/fnhum.2014.00776


The Evolution and Ontogeny of Ritual

Part VI. Culture and Coordination

Cristine H. LegareRachel E. Watson‐Jones

The Handbook of Evolutionary Psychology

First published: 18 November 2015 https://doi.org/10.1002/9781119125563.evpsych234


On the distinction of empathic and vicarious emotions

Frieder M. Paulus1,2*, Laura Müller-Pinzler1Stefan Westermann1 and Sören Krach1*

Front. Hum. Neurosci., 15 May 2013 | https://doi.org/10.3389/fnhum.2013.00196


The Narrative Construction of Reality

Jerome Bruner


Click to access bruner1991narrative.pdf

Cooperation and the evolution of hunter-gatherer storytelling

DOI: 10.1038/s41467-017-02036-8



Ancient or Modern? Alexander G. Baumgarten and the Coming of Age of Aesthetics

Alessandro Nannini

Click to access 0353-57381503629N.pdf


Stephen Davies, Philosophy, University of Auckland


Diegesis – Mimesis

Stephen Halliwell
Created: 17. October 2012 Revised: 12. September 2013

Published on the living handbook of narratology (http://www.lhn.uni-hamburg.de)


Art and Cosmology in India

Subhash Kak 2006

Cosmic Mirror Theory

Cosmic Mirror Theory

Source: Why some physicists really think there’s a ‘mirror universe’ hiding in space-time

Why does the Universe look like a Gemstone? A Jewel? An Opal maybe?

Why does the Universe look like an egg? Brahmanda?

Concept of Mirror is invoked in four ways in Cosmology

  • Shape of the Universe – Multi-connected manifolds – Dodecahedron topology – Reflecting Surfaces – Hall of Mirrors – Cosmic Crystals
  • Mirror Universe – a Parallel universe- Universe having a mirror twin – before the Big Bang – Symmetrical Opposite
  • Black Holes – Black Holes have mirror opposite in White Holes
  • Universe as a Hologram

Key Terms

  • Cosmic Hall of Mirrors
  • Parallel universe
  • Black holes
  • White Holes
  • Big Bang Theory
  • Shape of the Universe
  • Multiverse
  • Indira’s Net
  • Buckminster Fuller
  • Mirror Symmetry
  • Quantum Biology
  • Relational Science
  • Entanglements
  • Action at a distance
  • Holographic Universe
  • Fractal Universe
  • Recursive Universe
  • Universe as a Cow
  • Universe as a Human
  • Universe as Brahmanda
  • CBOE
  • WMAP
  • ACT
  • Curvature of Space
  • Topology of Space
  • Cosmic Topology
  • Cosmic Harmonics
  • Dodecahedral Space
  • Triloka (Three Universes)
  • Trikaal (Three Times)
  • MultiConnected Manifolds
  • Age of Universe – 13.77 Billion Years
  • 14 Lokas in Hinduism – Realms – Levels
  • Anthromorphic Universe
  • Maha Vishnu
  • 5 Sheaths (Kosh) in Humans
  • Tripartite Universe
  • Triguna
  • Interconnected Hypothesis
  • Cosmic Microwave Background CMB
  • Dark Energy
  • Dark Matter
  • Mirrorverse

Shape of the Universe


Cosmic Microwave Background from different probes

Source: Pintrest/478366791654117997/



Source: PLANCK Data 2018

Source: Decoding the cosmic microwave background

Decoding the cosmic microwave background

The Big Bang left behind a unique signature on the sky. Probes such as COBE, WMAP, and Planck taught us how to read it.

By Liz Kruesi | Published: Friday, July 27, 2018

This all-sky map, released in March 2013 and based on 15.5 months of observation, shows tiny fluctuations in the temperature of the CMB. These variations correspond to minute under- and over-densities of matter that ultimately led to the large-scale structure we see in the universe today. The redder areas represent above-average temperatures, and bluer areas show temperatures colder than average.

European Space Agency, Planck Collaboration

A glow undetectable to the human eye permeates the universe. This light is the remnant signature of the cosmic beginning — a dense, hot fireball that burst forth and created all mass, energy, and time. The primordial cosmic microwave background (CMB) radiation has since traveled some 13.8 billion years through the expanding cosmos to our telescopes on Earth and above it.

But the CMB isn’t just light. It holds within it an incredible wealth of knowledge that astronomers have been teasing out for the past few decades. “It’s the earliest view we have of the universe,” says Princeton University cosmologist Joanna Dunkley. “And it gives us so much information because all the things that we now see out in space — the galaxies, the clusters of galaxies — the very earliest seeds of those, we see in this CMB light.”

Extracting these clues from the CMB has taken multiple generations of telescopes on the ground, lofted into the atmosphere, and launched into space. In the mid-1960s, when Arno Penzias and Robert Wilson discovered the CMB’s pervasive microwave static across the sky, it appeared identical everywhere. It would take satellites launched above Earth’s obscuring atmosphere to map that microwave glow to precisions on the order of millionths of a degree. Specifically, three satellites — COBE, WMAP, and Planck — revealed that our current cosmos, which is complex and filled with clusters of galaxies, stars, planets, and black holes, evolved from a surprisingly simple early universe.

The Planck satellite produced the most detailed image of the cosmic microwave background (CMB) to date.

The universe began with the Big Bang 13.8 billion years ago as a fiery sea that expanded rapidly. A few minutes later, the universe’s constituent primordial subatomic particles glommed together into an elemental soup of atomic nuclei containing hydrogen, helium, and trace amounts of lithium. Electrons and light collided and scattered off of those atomic nuclei. Over the next thousands of years, the cosmos continued to expand, giving the particles more room to move and allowing the universe’s temperature to cool bit by bit. Around 380,000 years after the Big Bang, the temperature dropped to about 3,000 kelvins, cool enough for electrons to latch onto hydrogen nuclei. The universe became mostly neutral hydrogen, with some heavier elements swirled in.

With fewer individual particles zooming around, light could finally move about freely. And so it has traveled, mostly unhindered, in the approximately 13.8 billion years since that time of “last scattering.” These photons carry a snapshot of the 380,000-year-old universe.

Since the 1960s, telescopes on Earth have captured that glow in every direction of the sky. While the light 380,000 years into the universe’s history would have been visible to human eyes if we were around, cosmic expansion has since stretched the light into the longer wavelengths of microwaves — at least, that’s the wavelength astronomers had predicted. But would observations match theory?

The three probes

The Cosmic Background Explorer (COBE) launched in 1989. One of its instruments measured the intensity of the microwave glow at wavelengths ranging from 0.1 to 10 millimeters across the entire sky. The COBE science team’s first announcement, in 1990, was the result of that measurement. The radiation’s intensity plotted by wavelength makes it obvious that the CMB has a very specific intensity curve, where the strongest signal is at 2 mm. That wavelength corresponds to a temperature of 2.725 K. (The wavelength of light, and thus how much energy that light carries, is directly related to its temperature; redder light has less energy and a lower temperature than bluer light.)

COBE’s other instrument broke apart the seemingly uniform 2.725 K glow into more detail, looking for spots where the temperature is warmer or colder than average. It turned out there is a difference of only a tiny fraction of a degree, about 0.00001 K, between hotter and colder spots.

Each successive cosmological probe has improved astronomers’ view of the CMB with better resolution, revealing ever-finer details (anisotropies in temperature and density) that hold the key to assembling an accurate picture of our young universe.

This nearly identical cosmic glow with exactly the right temperature was concrete evidence that the entire sky — the entire observable universe — began in a Big Bang. With such tiny temperature differences across vast regions of sky, those spots must have been in contact at early times. COBE leaders John Mather and George Smoot won the 2006 Nobel Prize in Physics for their work.

But there is so much more that scientists can do with the CMB than confirm the Big Bang. “From the anisotropies, the hot and cold spots, we get the initial conditions — how bumpy was the early universe and also what is its composition,” says Mather.

The next CMB satellite was designed to improve upon these anisotropy measurements, mapping them at finer angular resolutions. COBE could map hot and cold spots of about 7° on the sky, while the Wilkinson Microwave Anisotropy Probe (WMAP), launched in 2001 and operated until 2010, could zoom in to a resolution of better than 0.5°. Planck, the CMB satellite that operated from 2009 to 2013, zoomed in even further, to 0.16°.

All of these missions mapped temperatures to the order of 0.00001 to 0.000001 K. To minimize measurement errors related to such small signals, the spacecrafts’ detectors pointed toward two spots on the sky at the same time and measured the temperature difference between them. The satellites swept the entire sky in this fashion, and software generated a map of all those tiny differences. That map holds a treasure-trove of cosmic secrets.

The CMB represents the moment at which the universe became “transparent.” Immediately after its birth, the universe was hot and dense. As it expanded, it cooled, and its density dropped. Within the young universe, photons couldn’t travel very far — a few inches — before colliding with a nearby particle. As the matter in the universe transitioned from plasma (left) to atomic hydrogen (right) 380,000 years after the Big Bang, photons could travel much farther — the width of the universe — without necessarily experiencing a collision. This moment, also called the surface of last scattering, is encoded in the CMB we see today.

Unlocking the early universe
To reveal those secrets, cosmologists study the pattern of hot and cold spots frozen into the CMB and decompose those spots into their constituent sizes. While most of the hot and cold spots are about 1° on the sky, they are overlaid on fluctuations with larger sizes.

“Imagine looking at a smooth pond of water that we might drop pebbles into,” says Dunkley. “If you drop a whole bunch of pebbles in, the ripples will sort of combine together, and you see a whole pattern of ripples across the water. We think of this pattern of slightly different temperatures of this light on the sky a little bit like the pond after it’s covered in ripples.”

The size breakdown of the CMB’s temperature spots, or fluctuations, is like a cosmic Rosetta Stone. The strength of the fluctuations’ signals at different scales is associated with the universe’s age, its ingredients, its expansion rate, and when the first stars lit up the cosmos. By comparing computer models to the signal strengths (which astronomers obtained from analyzing WMAP and Planck data), researchers can piece together what the early universe looked like and how it has evolved.

Thanks to these three cosmic probes, we know the universe began in a Big Bang, and around 380,000 years later, electrons and protons combined, letting light roam free. We know our cosmos is 13.8 billion years old and how fast it is expanding. We know that 31 percent of the universe is matter, but only 5 percent is made of ordinary matter like you and me, while 26 percent is invisible dark matter. Much more of the cosmos is composed of a mysterious, repulsive dark energy — 69 percent.

And perhaps most importantly, astronomers now have a way to find out pieces of information not literally encoded in the CMB itself. That’s because the CMB maps and their statistics have led to the so-called standard model of cosmology.

“We now have a really simple model that describes basically all of our observations,” says Dunkley. “We can track from the very first moments of time all the way through today and make predictions about how large-scale structure evolved. And it has remarkable success. That’s the big thing these satellite missions have given the community.”

Mirror Universe


Our Universe May Be a Giant Hologram

Physicist Brian Greene explains how properties at the black hole’s surface—its event horizon—suggest the unsettling theory that our world is a mere representation of another universe, a shadow of the realm where real events take place.

Brian Greene


Two monster black holes may lie within the double bright area at the center of galaxy NGC 6240. NASA

If, when I was growing up, my room had been adorned with only a single mirror, my childhood daydreams might have been very different. But it had two. And each morning when I opened the closet to get my clothes, the one built into its door aligned with the one on the wall, creating a seemingly endless series of reflections of anything situated between them. It was mesmerizing. All the reflections seemed to move in unison—but that, I knew, was a mere limitation of human perception; at a young age I had learned of light’s finite speed. So in my mind’s eye, I would watch the light’s round-trip journeys. The bob of my head, the sweep of my arm silently echoed between the mirrors, each reflected image nudging the next. Sometimes I would imagine an irreverent me way down the line who refused to fall into place, disrupting the steady progression and creating a new reality that informed the ones that followed. During lulls at school, I would sometimes think about the light I had shed that morning, still endlessly bouncing between the mirrors, and I would join one of my reflected selves, entering an imaginary parallel world constructed of light and driven by fantasy.

To be sure, reflected images don’t have minds of their own. But these youthful flights of fancy, with their imagined parallel realities, resonate with an increasingly prominent theme in modern science—the possibility of worlds lying beyond the one we know.

There was a time when the word universe meant “all there is.” Everything. The whole shebang. The notion of more than one universe, more than one everything, would seemingly be a contradiction in terms. Yet a range of theoretical developments has gradually qualified the interpretation of universe. The word’s meaning now depends on context. Sometimes universe still connotes absolutely everything. Sometimes it refers only to those parts of everything that someone such as you or I could, in principle, have access to. Sometimes it’s applied to separate realms, ones that are partly or fully, temporarily or permanently, inaccessible to us; in this sense, the word relegates our universe 
to membership in a large, perhaps infinitely large, collection.

With its hegemony diminished, universe has given way to other terms that capture the wider canvas on which the totality of reality may be painted. Parallel worlds or parallel universes or multiple universes or alternate universes or the metaverse, megaverse, or multiverse—they’re all synonymous, and they’re all among the words used to embrace not just our universe but a spectrum of others that may be out there.

The strangest version of all parallel universe proposals is one that emerged gradually over 30 years of theoretical studies on the quantum properties of black holes. The work culminated in the last decade, and it suggests, remarkably, that all we experience is nothing but a holographic projection of processes taking place on some distant surface that surrounds us. You can pinch yourself, and what you feel will be real, but it mirrors a parallel process taking place in a different, distant reality.

Plato likened our view of the world to that of an ancient forebear watching shadows meander across a dimly lit cave wall. He imagined our perceptions to be but a faint inkling of a far richer reality that flickers beyond reach. Two millennia later, Plato’s cave may be more than a metaphor. To turn his suggestion on its head, reality—not its mere shadow—may take place on a distant boundary surface, while everything we witness in the three common spatial dimensions is a projection of that faraway unfolding. Reality, that is, may be akin to a hologram. Or, really, a holographic movie.

The journey to this peculiar possibility combines developments deep and far-flung—insights from general relativity; from research on black holes; from thermodynamics, quantum mechanics, and, most recently, string theory. The thread linking these diverse areas is the nature of information in a quantum universe.

Physicists Jacob Bekenstein and Stephen Hawking established that, for a black hole, the information storage capacity is determined not by the volume of its interior but by the area of its surface. But when the math says that a black hole’s store of information is measured by its surface area, does that merely reflect a numerical accounting, or does it mean that the black hole’s surface is where the information is actually stored? It’s a deep issue and has been pursued for decades by some of the most renowned physicists. The answer depends on whether you view the black hole from the outside or from the inside—and from the outside, there’s good reason to believe that information is indeed stored at the event horizon. This doesn’t merely highlight a peculiar feature of black holes. Black holes don’t just tell us about how black holes store information. 
Black holes inform us about information storage 
in any context.

Think of any region of space, such as the room in which you’re reading. Imagine that whatever happens in the region amounts to information processing—information regarding how things are right now is transformed by the laws of physics into information regarding how they will be in a second or a minute or an hour. Since the physical processes we witness, as well as those by which we’re governed, seemingly take place within the region, it’s natural to expect that the information those processes carry is also found within the region. But for black holes, we’ve found that the link between information and surface area goes beyond mere numerical accounting; there’s a concrete sense in which information is stored on their surfaces. Physicists Leonard Susskind and Gerard ’t Hooft stressed that the lesson should be general: Since the information required to describe physical phenomena within any given region of space can be fully encoded by data on a surface that surrounds the region, then there’s reason to think that the surface is where the fundamental physical processes actually happen. Our familiar three-dimensional reality, these bold thinkers suggest, would then be likened to a holographic projection of those distant two-dimensional physical processes.

If this line of reasoning is correct, then there are physical processes taking place on some distant surface that, much as a puppeteer pulls strings, are fully linked to the processes taking place in my fingers, arms, and brain as I type these words at my desk. Our experiences here and that distant reality there would form the most interlocked of parallel worlds. Phenomena in the two—I’ll call them Holographic Parallel Universes—would be so fully joined that their respective evolutions would be as connected as me and my shadow.

 Excerpted from The Hidden Reality by Brian Greene. Copyright © 2011 by Brian Greene. Reprinted with permission by Alfred A. Knopf, a division of Random House, Inc. All rights reserved.

 See the related DISCOVER feature, “The Strange Physicsand SightsInside Black Holes.”

My Related Posts

Shape of the Universe

Mind, Consciousness and Quantum Entanglement

Myth of Invariance: Sound, Music, and Recurrent Events and Structures

On Anticipation: Going Beyond Forecasts and Scenarios

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Geometry of Consciousness

The Great Chain of Being

Maha Vakyas: Great Aphorisms in Vedanta

Indra’s Net: On Interconnectedness

On Synchronicity

Color Science of Gem Stones

Key Sources of Research

CPT-Symmetric Universe

Latham Boyle,1 Kieran Finn,1,2 and Neil Turok1
1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5
2School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom

PHYSICAL REVIEW LETTERS 121, 251301 (2018)



Why some physicists really think there’s a ‘mirror universe’ hiding in space-time

By Rafi Letzter – Staff Writer June 22, 2020


Cosmology of the Mirror Universe

Paolo Ciarcelluti

April 2003 PhD Thesis

Mirror dark matter:
Cosmology, galaxy structure and direct detection

R. Foot

ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, University of Melbourne,
Victoria 3010 Australia


Mirror dark matter cosmology and structure formation

Roux, Jean-Samuel

PhD Thesis McGill Univ


White holes: Do black holes have mirror images? 

These black hole opposites would spew energy, be impossible to enter, and might even answer some of the universe’s fundamental questions.By Bill Andrews  |  Published: Friday, June 28, 2019


A cosmic hall of mirrors

Jean-Pierre Luminet
Laboratoire Univers et Théories (LUTH) – CNRS UMR Observatoire de Paris, 92195 Meudon (France) Jean-pierre.luminet@obspm.fr

The fractal universe

SEPTEMBER 12, 2018

Love the Reflections in the Cosmic Mirror


The Shape of the Universe: Ten Possibilities

Is the universe a dodecahedron?




Planck and the cosmic microwave background


The Atacama Cosmology Telescope ACT


Wilkinson Microwave Anisotropy Probe WMAP


Cosmic Topology : Twenty Years After

Jean-Pierre Luminet,

Laboratoire Univers et Th ́eories Observatoire de Paris-CNRS-Universit ́e Paris Diderot (France) email : jean-pierre.luminet@obspm.fr

October 15, 2013

Cosmic microwave background anisotropies in multi-connected flat spaces

Alain Riazuelo∗
Service de Physique Th ́eorique, CEA/DSM/SPhT, Unit ́e de recherche associ ́ee au CNRS, CEA/Saclay F–91191 Gif-sur-Yvette c ́edex, France

Jeffrey Weeks†
15 Farmer St., Canton NY 13617-1120, USA

Jean-Philippe Uzan‡
Institut d’Astrophysique de Paris, GRεCO, FRE 2435-CNRS, 98bis boulevard Arago, 75014 Paris, France Laboratoire de Physique Th ́eorique, CNRS-UMR 8627,
Universit ́e Paris Sud, Bˆatiment 210, F–91405 Orsay c ́edex, France

Roland Lehoucq§
CE-Saclay, DSM/DAPNIA/Service d’Astrophysique, F–91191 Gif-sur-Yvette c ́edex, France, Laboratoire Univers et Th ́eories, CNRS-UMR 8102,
Observatoire de Paris, F–92195 Meudon c ́edex, France

Jean-Pierre Luminet¶
Laboratoire Univers et Th ́eories, CNRS-UMR 8102, Observatoire de Paris, F–92195 Meudon c ́edex, France (Dated: 13 November 2003)



The Shape of Space after WMAP data

Jean-Pierre Luminet
Laboratoire Univers et Th ́eories, CNRS-UMR 8102, Observatoire de Paris, F–92195 Meudon c ́edex, France.


Click to access a02v361b.pdf

Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background



Geometry and Topology in Relativistic Cosmology

Jean-Pierre Luminet

Laboratoire Univers et Théories, CNRS-UMR 8102, Observatoire de Paris, F-92195 Meudon cedex, France


The Spectral Action and Cosmic Topology

Matilde Marcolli

MAT1314HS Winter 2019, University of Toronto T 12-2 and W 12 BA6180

Click to access IntroNCGToronto10.pdf

Cosmic Topology

M. Lachieze-Rey (1), J.P.Luminet


Cosmic crystallography

R. Lehoucq1, M. Lachi`eze–Rey1,2 and J.P. Luminet3

  1. 1  CE-Saclay, DSM/DAPNIA/Service d’Astrophysique, F-91191 Gif sur Yvette cedex, France
  2. 2  CE-Saclay, DSM/DAPNIA/Service d’Astrophysique, CNRS–URA 2052, F-91191 Gif sur Yvette cedex, France
  3. 3  D ́epartement d’Astrophysique Relativiste et de Cosmologie, CNRS–UPR 176, Observatoire de Paris–Meudon, France

september 1995

The Status of Cosmic Topology after Planck Data 

Jean-Pierre Luminet 1,2

Received: 19 November 2015; Accepted: 7 January 2016; Published: 15 January 2016 Academic Editors: Stephon Alexander, Jean-Michel Alimi, Elias C. Vagenas and Lorenzo Iorio



Cosmic Topology: A Brief Overview

M. J. Rebouc ̧as

Centro Brasileiro de Pesquisas F ́ısicas, Departamento de Relatividade e Part ́ıculas Rua Dr. Xavier Sigaud, 150 , 22290-180 Rio de Janeiro – RJ, Brazil

and G. I. Gomero

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XVIII. Background geometry and topology of the Universe

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Janna Levin1, Evan Scannapieco1, Giancarlo de Gasperis1, Joseph Silk1 and John D. Barrow2 1Center for Particle Astrophysics, UC Berkeley
Berkeley, CA 94720-7304
2Astronomy Centre, University of Sussex
Brighton BN1 9QJ, U.K.


The Conformal Singularity as a Cosmological Mirror: Classical Theory

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Physics Letters B
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24. Cosmological Parameters

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2.4. The Cosmic Microwave Background


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Is the universe a dodecahedron?

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Geometry of the Universe :


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Mirror World, E(6) Unification and Cosmology

C.R. Das 1 ∗, L.V. Laperashvili 2 †,
1 Institute of Mathematical Sciences, Chennai, India

2 The Institute of Theoretical and Experimental Physics, Moscow, Russia

Click to access mirror_world_e6_unification_and_cosmology.pdf

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17 Dec 2019


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  • October 2017

Leah Broussard
Oak Ridge National Laboratory

Joshua Lawrence Barrow
Fermi National Accelerator Laboratory (Fermilab)

B. Chance

Christopher Crawford
University of Kentucky


Radiation as Self-Action via a Cosmological Mirror

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09 Nov 2015

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Niloy Chattaraj

September 19, 2020

The Holographic Universe Explained

A Thin Sheet of Reality: The Universe as a Hologram

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The Pillar of Celestial Fire

The Pillar of Celestical Fire


On top, sits Brahma

At bottom, Hari (Vishnu) in Boar (Varaha) Avatar


Key Terms

  • Mount Meru
  • Alif
  • Lingam
  • Danda
  • Meru Danda
  • Staff
  • Skambha
  • Stambha
  • Yupa-Stambha
  • Pillar
  • Fire
  • Jwala Linga
  • Agni Linga
  • One
  • 1
  • Ek Seedhi Lakir
  • Straight Vertical Line
  • Danda and the Serpent
  • Verticalism and Horizontalism
  • +
  • Ek Onkar
  • Om
  • Pranav
  • Fire Ascends
  • Yagya
  • Yajna
  • Ascending God
  • Water Descends
  • Descending God
  • Avatar
  • It rains
  • Falling Rains
  • Fall
  • Water flows Horizontally
  • Fire leaps Vertically
  • A Torus
  • An Apple
  • Atharv Veda, The Fourth Veda
  • Ida Lunar Vishnu
  • Pingala Solar Brahma
  • Sushumana  Agni Shiva Central Column
  • Sukhmani
  • 4 – 3 – 2 – 1
  • 432
  • 4321
  • 4/3 Fourth
  • 3/2 Fifth
  • 2/1 Octave
  • 1 great Monad
  • Pingala and Panini
  • Fire and Water
  • Mahat Tattva
  • Sa Re Ga Ma Pa Dha Ni
  • Saptak
  • Pancham




Skambha, the Pillar or Fulcrum of all existenc

Atharva Veda ( X – 7,8) — Skambha Suktam

The origins of the worship of the Shiva-Linga are unknown. Shiva-Linga has one complete purana which is dedicated to its form and origin. It may be a symbolic representation of self (Atma Linga) or of everything. Some associate it with the physical form of Pranava (Om). Oval form represents even the shape of the Universe including the existing space. The beginning of the oval form is A in OM and prolonged part is U in OM and M is the ending part of the linga. It is single shape of Trimurti. Praying Shiva Linga is considered as praying the Thrimurti in absolute form. Linga represents absolute and Single power of this universe. Some associate them with the famous hymn in the Atharva-Veda Samhitâ sung in praise of the Yupa-Stambha, the sacrificial post. In that hymn a description is found of the beginningless and endless Stambha or Skambha and it is shown that the said Skambha is put in place of the eternal Brahman. As afterwards the Yajna (sacrificial) fire, its smoke, ashes and flames, the Soma plant and the ox that used to carry on its back the wood for the Vedic sacrifice gave place to the conceptions of the brightness of Shiva’s body, his tawny matted-hair, his blue throat and the riding on the bull of the Shiva. The Yupa-Skambha gave place in time to the Shiva-Linga. In the Linga Purâna the same hymn is expanded in the shape of stories, meant to establish the glory of the great Stambha and the superiority of Mahâdeva.

In the context of Hindu mythology, stambha, also spelt as Skambha, is believed to a cosmic column. It is believed that the stambha functions as a bond, which joins the heaven (Svarga) and the earth (prithvi). A number of Hindu scriptures, including the Atharva Veda, have references to stambha. In the Atharva Veda, a celestial stambha has been mentioned, and that has been described as a scaffold, which supports the cosmos and material creation

Skambha Sukta ( Atharva Veda X-7 )

kásminn áṅge tápo asyā́dhi tiṣṭhati kásminn áṅga r̥tám asyā́dhy ā́hitam
kvà vratáṃ kvà śraddhā́sya tiṣṭhati kásminn áṅge satyám asya prátiṣṭhitam 1

kásmād áṅgād dīpyate agnír asya kásmād áṅgāt pavate mātaríśva
kásmād áṅgād ví mimīté ‘dhi candrámā mahá skambhásya mímāno áṅgam 2

kásminn áṅge tiṣṭhati bhū́mir asya kásminn áṅge tiṣṭhaty antárikṣam
kásminn áṅge tiṣṭhaty ā́hitā dyáuḥ kásminn áṅge tiṣṭhaty úttaraṃ diváḥ 3

kvà prépsan dīpyata ūrdhvó agníḥ kvà prépsan pavate mātaríśvā
yátra prépsantīr abhiyánty āvŕ̥taḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 4

kvā̀rdhamāsā́ḥ kvà yanti mā́sāḥ saṃvatsaréṇa sahá saṃvidānā́ḥ
yátra yánty r̥távo yátrārtavā́ḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 5

kvà prépsantī yuvatī́ vírūpe ahorātré dravataḥ saṃvidāné
yátra prépsantīr abhiyánty ā́paḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 6

yásmint stabdhvā́ prajā́patir lokā́nt sárvām̐ ádhārayat
skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 7

yát paramám avamám yác ca madhyamáṃ prajā́patiḥ sasr̥jé viśvárūpam
kíyatā skambháḥ prá viveśa tátra yán ná prā́viśat kíyat tád babhūva 8

kíyatā skambháḥ prá viveśa bhūtám kíyad bhaviṣyád anvā́śaye ‘sya
ékaṃ yád áṅgam ákr̥ṇot sahasradhā́ kíyatā skambháḥ prá viveśa tátra 9

yátra lokā́mś ca kóśāṃś cā́po bráhma jánā vidúḥ
ásac ca yátra sác cāntá skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 10

yátra tápaḥ parākrámya vratáṃ dhāráyaty úttaram
r̥táṃ ca yátra śraddhā́ cā́po bráhma samā́hitāḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 11

yásmin bhū́mir antárikṣaṃ dyáur yásminn ádhy ā́hitā
yátrāgníś candrámāḥ sū́ryo vā́tas tiṣṭhanty ā́rpitāḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 12

yásya tráyastriṃśad devā́ áṅge sárve samā́hitāḥ
skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 13

yátra ŕ̥ṣayaḥ prathamajā́ ŕ̥caḥ sā́ma yájur mahī́
ekarṣír yásminn ā́rpitaḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 14

yátrāmŕ̥taṃ ca mr̥tyúś ca púruṣé ‘dhi samā́hite
samudró yásya nāḍyàḥ púruṣé ‘dhi samā́hitāḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 15

yásya cátasraḥ pradíśo nāḍyàs tíṣṭhanti prathamā́ḥ
yajñó yátra párākrāntaḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 16

yé púruṣe bráhma vidús té viduḥ parameṣṭhínam
yó véda parameṣṭhínaṃ yáś ca véda prajā́patim
jyeṣṭháṃ yé brā́hmaṇaṃ vidús te skambhám anusáṃviduḥ 17

yásya śíro vaiśvānaráś cákṣur áṅgirasó ‘bhavan
áṅgāni yásya yātávaḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 18

yásya bráhma múkham āhúr jihvā́ṃ madhukaśā́m utá
virā́jam ū́dho yásyāhúḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 19

yásmād ŕ̥co apā́takṣan yájur yásmād apā́kaṣan
sā́māni yásya lómāny atharvāṅgiráso múkhaṃ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 20

asaccākhā́ṃ pratíṣṭhantīṃ paramám iva jánā viduḥ
utó sán manyanté ‘vare yé te śā́khām upā́sate 21

yátrādityā́ś ca rudrā́ś ca vásavaś ca samā́hítāḥ
bhūtáṃ ca yátra bhávyaṃ ca sárve lokā́ḥ prátiṣṭhitāḥ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 22

yásya tráyastriṃśad devā́ nidhíṃ rákṣanti sarvadā́
nidhíṃ tám adyá kó veda yáṃ devā abhirákṣatha 23

yátra devā́ brahmavído bráhma jyeṣṭhám upā́sate
yó vái tā́n vidyā́t pratyákṣaṃ sá brahmā́ véditā syāt 24

br̥hánto nā́ma té devā́ yé ‘sataḥ pári jajñiré
ékaṃ tád áṅgaṃ skambhásyā́sad āhuḥ paró jánāḥ 25

yátra skambháḥ prajanáyan purāṇáṃ vyávartayat
ékaṃ tád áṅgaṃ skambhásya purāṇám anusáṃviduḥ 26

yásya tráyastriṃśad devā́ áṅge gā́trā vibhejiré
tā́n vái tráyastriṃśad devā́n éke brahamvído viduḥ 27

hiraṇyagarbhám paramám anatyudyáṃ jánā viduḥ
skambhás tád ágre prā́siñcad dhíraṇyaṃ loké antarā́ 28

skambhé lokā́ḥ skambhé tápaḥ skambhé ‘dhy r̥tám ā́hitam
skámbha tvā́ veda pratyákṣam índre sárvaṃ samā́hitam 29

índre lokā́ índre tápa índre ‘dhy r̥tám ā́hitam
índraṃ tvā́ veda pratyákṣaṃ skambhé sárvaṃ prátiṣṭhitam 30

nā́ma nā́mnā johavīti purā́ sū́ryāt puróṣásaḥ
yád ajáḥ prathamáṃ saṃbabhū́va sá ha tát svarā́jyam iyāya yásmān nā́nyát páram ásti bhūtám 31

yásya bhū́miḥ pramā́ntárikṣam utódáram
dívaṃ yáś cakré mūrdhā́naṃ tásmai jyeṣṭhā́ya bráhmaṇe námaḥ 32

yásya sū́ryaś cákṣuś candrámāś ca púnarṇavaḥ
agníṃ yáś cakrá āsyàṃ tásmai jyeṣṭhā́ya bráhmaṇe námaḥ 33

yásya vā́taḥ prāṇāpānáu cákṣur áṅgirasó ‘bhavan
díśo yáś cakré prajñā́nīs tásmai jyeṣṭhā́ya bráhmaṇe námaḥ 34

skambhó dādhāra dyā́vāpr̥thivī́ ubhé imé skambhó dādhārorv àntárikṣam
skambhó dādhāra pradíśaḥ ṣáḍ urvī́ḥ skambhá idáṃ víśvaṃ bhúvanam ā́ viveśa 35

yáḥ śrámāt tápaso jātó lokā́nt sárvānt samānaśé
sómaṃ yáś cakré kévalaṃ tásmai jyeṣṭhā́ya bráhmaṇe námaḥ 36

katháṃ vā́to nélayati katháṃ ná ramate mánaḥ
kím ā́paḥ satyáṃ prépsantīr nélayanti kadā́ caná 37

mahád yakṣáṃ bhúvanasya mádhye tápasi krāntáṃ salilásya pr̥ṣṭhé
tásmin chrayante yá u ké ca devā́ vr̥kṣásya skándhaḥ paríta iva śā́khāḥ 38

yásmai hástābhyāṃ pā́dābhyāṃ vācā́ śrótreṇa cákṣuṣā
yásmai devā́ḥ sádā balíṃ prayáchanti vímité ‘mitaṃ skambháṃ táṃ brūhi katamáḥ svid evá sáḥ 39

ápa tásya hatáṃ támo vyā́vr̥ttaḥ sá pāpmánā
sárvāṇi tásmin jyótīṃṣi yā́ni trī́ṇi prajā́patau 40

yó vetasáṃ hiraṇyáyaṃ tiṣṭhantaṃ salilé véda
sá vái gúhyaḥ prajā́patiḥ 41

tantrám éke yuvatī́ vírūpe abhyākrā́maṃ vayataḥ ṣáṇmayūkham prā́nyā́ tántūṃs tiráte dhatté anyā́ nā́pa vr̥ñjāte ná gamāto ántam 42

táyor aháṃ parinŕ̥tyantyor iva ná ví jānāmi yatarā́ parástāt
púmān enad vayaty úd gr̥ṇanti púmān enad ví jabhārā́dhi nā́ke 43

imé mayū́khā úpa tastabhur dívaṃ sā́māni cakrus tásarāṇi vā́tave 44


1)Which of his members is the seat of Fervour: Which is the base of Ceremonial Order? Where in him standeth Faith? Where Holy Duty? Where, in what part of him is truth implanted?

2)Out of which member glows the light of Agni? Form which proceeds the breath of Mātarisvan? From which doth Chandra measure out his journey, travelling over Skambha’s mighty body?

3)Which of his members is the earth’s upholder? Which gives the middle air a base to rest on? Where, in which member is the sky established? Where hath the space above the sky its dwelling?

4)Whitherward yearning blazeth Agni upward? Whitherward yearning bloweth Mātarisvan? Who out of many, tell me, is that Skambha to whom with longing go the turning pathways?

5)Whitheward go the half-months, and, accordant with the full year, the months in their procession? Who out of many, tell me, is that Skambha to whom go seasons and the groups of seasons?

6)Whitherward yearning speed the two young Damsels, accordant, Day and Night, of different colour? Who out of many, tell me, is that Skambha to whom the Waters take their way with longing?

7)Who out of many, tell me, is that Skambha, On whom Prajāpati set up and firmly stablished all the worlds?

8)That universe which Prajāpati created, wearing all forms,, the highest, midmost, lowest, How far did Skambha penetrate within it? What portion did he leave unpenetrated?

9)How far within the past hath Skambha entered? How much of him hath reached into the future? That one part which he set in thousand places,—how far did Skambha penetrate within it?

10)Who out of many, tell me, is that Skambha in whom men recognize the Waters, Brahma, In whom they know the worlds and their enclosures, in whom are non-existence and existence?

11)Declare that. Skambha, who is he of many, In whom, exerting every power, Fervour maintains her loftiest vow; In whom are comprehended Law, Waters, Devotion and Belief

12)Who out of many, tell me, is that Skambha On whom as their foundation earth and firmament and sky are set; In whom as their appointed place rest Fire and Moon and Sun and Wind?

13)Who out of many, tell me, is that Skambha He in whose body are contained all three-and-thirty Deities?

14)Who out of many, tell me, is that Skambha. In whom the Sages earliest born, the Richas, Sāman, Yajus, Earth, and the one highest Sage abide?

15)Who out of many, tell me, is the Skambha. Who comprehendeth, for mankind, both immortality and death, He who containeth for mankind the gathered waters as his veins?

16)Who out of many, tell me, is that Skambha, He whose chief arteries stand there, the sky’s four regions, he irk whom Sacrifice putteth forth its might?

17)They who in Purusha understand Brahma know Him who is. Supreme. He who knows Him who is Supreme, and he who knows the Lord of Life, These know the loftiest Power Divine, and thence know Skam- bha thoroughly.

18)Who out of many, tell me, is that Skambha Of whom Vaisvānara became the head, the Angirases his eye, and Yātus his corporeal parts?

19)Who out of many, tell me, is that Skambha Whose mouth they say is Holy Lore, his tongue the Honey- sweetened Whip, his udder is Virāj, they say?

20)Who out of many, tell me, is that Skambha From whom they hewed the lichas off, from whom they chipped the Yajus, he Whose hairs are Sāma-verses and his mouth the Atharvāngi- rases?

21)Men count as ’twere a thing supreme nonentity’s conspicuous branch; And lower man who serve thy branch regard it as an entity.

22)Who out of many, tell me, is that Skambha In whom Ādityas dwell, in whom Rudras and Vasus are contained, In whom the future and the past and all the worlds are firmly set;

23)Whose secret treasure evermore the three-and thirty Gods protect? Who knoweth now the treasure which, O Deities ye watch and guard?

24)Where the Gods, versed in Sacred Lore, worship the loftiest Power Divine The priest who knows them face to face may be a sage who knows the truth.

25)Great, verily, are those Gods who sprang from non-existence into life. Further, men say that that one part of Skambha is nonentity.

26)Where Skambha generating gave the Ancient World its shape and form, They recognized that single part of Skambha as the Ancient World,

27)The three-and-thirty Gods within his body were disposed as limbs: Some, deeply versed in Holy Lore, some know those three-and- thirty Gods.

28)Men know Hiranyagarbha as supreme and inexpressible: In the beginning, in the midst of the world, Skambha poured that gold.

29)On Skambha Fervour rests, the worlds and Holy Law repose on him. Skambha, I clearly know that all of thee on Indra is imposed.

30)On Indra Fervour rests, on him the worlds and Holy Law recline. Indra, I clearly know that all of thee on Skambha findeth rest.

31)Ere sun and dawn man calls and calls one Deity by the other’s name. When the Unborn first sprang into existence he reached that independent sovran lordship; than which aught higher never hath arisen.

32)Be reverence paid to him, that highest Brahma, whose base is Earth, his belly Air, who made the sky to be his head.

33)Homage to highest Brahma, him whose eye is Sūrya and the Moon who groweth young and new again, him who made Agni for his mouth.

34)Homage to highest Brahma, him whose two life-breathings were the Wind, The Angirases his sight: who made the regions be his means of sense.

35)Skambha set fast these two, the earth and heaven, Skambha maintained the ample air between them. Skambha established the six spacious regions: this whole world Skambha entered and pervaded.

36)Homage to highest Brahma, him who, sprung from Fervour and from toil, Filled all the worlds completely, who made Soma for himself alone.

37)Why doth the Wind move ceaselessly? Why doth the spirit take no rest? Why do the Waters, seeking truth, never at any time repose?

38)Absorbed in Fervour, is the mighty Being, in the world’s centre, on the waters’ surface. To him the Deities, one and all betake them. So stand the tree- trunk with the branches round it.

39)Who out of many, tell me, is that Skambha. To whom the Deities with hands, with feet, and voice, and ear, and eye. Present unmeasured tribute in the measured hall of sacrifice?

40)Darkness is chased away from him: he is exempt from all dist- ress. In him are all the lights, the three abiding in Prajāpati.

41)He verily who knows the Reed of Gold that stands amid the flood, is the mysterious Lord of Life.

42)Singly the two young Maids of different colours approach the six-pegged warp in turns and weave it. The one draws out the threads, the other lays them: they break them not, they reach no end of labour.

43)Of these two, dancing round as ’twere, I cannot distinguish whether ranks before the other. A Male in weaves this web, a Male divides it: a Male hath stretched it to the cope of heaven

44)These pegs have buttressed up the sky. The Sāmans have turned them into shuttles for the weaving.



yó bhūtáṃ ca bhávyaṃ ca sárvaṃ yáś cādhitíṣṭhati
sv àryásya ca kévalaṃ tásmai jyeṣṭhā́ya bráhmaṇe námaḥ 1
Worship to loftiest Brahma, Lord of what hath been and what shall be, To him who rules the universe, and heavenly light is all his own!
skambhénemé víṣṭabhite dyáuś ca bhū́miś ca tiṣṭhataḥ
skambhá idáṃ sárvam ātmanvád yát prāṇán nimiṣác ca yát 2
Upheld by Skambha’s power these two, the heaven and the earth,stand fast. Skambha is all this world of life, whatever breathes or shuts eye.
tisró ha prajā́ atyāyám āyan ny ànyā́ arkám abhíto ‘viśanta

br̥hán ha tasthau rájaso vimā́no hárito háriṇīr ā́ viveśa 3

Three generations have gone by and vanished and others near have entered into sunlight. There stood on high he who metes out the region into green, plants hath passed the Golden-coloured

dvā́daśa pradháyaś cakrám ékaṃ trī́ṇi nábhyāni ká u tác ciketa tátrā́hatās trī́ṇi śatā́ni śaṅkávaḥ ṣaṣṭíś ca khī́lā ávicācalā yé 4

One is the wheel, the tires are twelve in number, the naves are three What man hath understood it?Three hundred spokes have thereupon been hammered, and sixty pins set firmly in their places.

idáṃ savitar ví jānīhi ṣáḍ yamā́ éka ekajáḥ tásmin hāpitvám ichante yá eṣām éka ekajáḥ 5

Discern thou this, O Savitar. Six are the twins, one singly born.They claim relationship in that among them which is born alone.

āvíḥ sán níhitaṃ gúhā járan nā́ma mahát padám

tátredáṃ sárvam ā́rpitam éjat prāṇát prátiṣṭhitam 6

Though manifest, it lies concealed in the vast place they call the old:Therein is firmly stationed all the moving, breathing universe.

ékacakraṃ vartata ékanemi sahásrākṣaraṃ prá puró ní paścā
ardhéna víśvaṃ bhúvanaṃ jajā́na yád asyārdháṃ kvà tád babhūva 7

Up, eastward downward in the west, ‘it rolleth, with countless elements, one-wheeled, single-fellied.With half it hath begotten all creation. Where hath the other half become unnoticed?

pañcavāhī́ vahatyágram eṣāṃ práṣṭayo yuktā́ anusáṃvahanti
áyātam asya dadr̥śé ná yātáṃ páraṃ nédīyó ‘varaṃ dávīyaḥ 8

In front of these the five-horsed car moves onward: side-horses, harnessed with the others draw it. No one hath seen its hither course untravelled; the height sees it more near, the depth more distant.

tiryágbilaś camasá ūrdhvábudhnas tásmin yáśo níhitaṃ viśvárūpam
tád āsata ŕ̥ṣayaḥ saptá sākáṃ yé asyá gopā́ maható babhūvúḥ 9

The bowl with mouth inclined and bottom upward holds stored within it every form of glory.Thereon together sit the Seven Rishis who have become thismighty One’s protectors

yā́ purástād yujyáte yā́ ca paścā́d yā́ viśváto yujyáte yā́ ca sarvátaḥ
yáyā yajñáḥ prā́ṅ tāyáte tā́ṃ tvā pr̥chāmi katamā́ sā́ r̥cā́m 10

The Verse employed at opening and conclusion, the Verseemployed in each and every portion;That by which sacrifice proceedeth onward. I ask thee which is that of all the Verses

yád éjati pátati yác ca tíṣṭhati prāṇád áprāṇan nimiṣác ca yád bhúvat
tád dādhāra pr̥thivī́ṃ viśvárūpaṃ tát saṃbhū́ya bhavaty ékam evá 11

That which hath power of motion, that which flies, or stands,which breathes or breathes not, which, existing, shuts the eyeWearing all forms that entity upholds the earth, and in its closeconsistence still is only one.

anantáṃ vítataṃ purutrā́nantám ántavac cā sámante
té nākapāláś carati vicinván vidvā́n bhūtám utá bhávyam asya 12

The infinite to every side extended, the finite and the infinite around us,These twain Heaven’s Lord divides as he advances, knowing the past hereof and all the future

prajā́patiś carati gárbhe antár ádr̥śyamāno bahudhā́ ví jāyate
ardhéna víśvaṃ bhúvanaṃ jajā́na yád asyārdháṃ katamáḥ sá ketúḥ 13

Within the womb Prajapati is moving: he, though unseen, is born in sundry places. He with one half engendered all creation. What sign is there to tell us of the other?

ūrdhváṃ bhárantam udakáṃ kumbhénevodahāryàm
páśyanti sárve cákṣuṣā ná sárve mánasā viduḥ 14

All men behold him with the eye, but with the mind they know not him.Holding aloft the water as a water-bearer in her jar.

dūré pūrṇéna vasati dūrá ūnéna hīyate
mahád yakṣáṃ bhúvanasya mádhye tásmai balíṃ rāṣṭrabhŕ̥to bharanti 15

With the full vase he dwells afar, is left far off what time it fails, A mighty Being in creation’s centre: to him the rulers of the realms bring tribute.

yátaḥ sū́ryaḥ udéty ástaṃ yátra ca gáchati
tád evá manye ‘háṃ jyeṣṭháṃ tád u nā́ty eti kíṃ caná 16

That, whence the Sun arises, that whither he goes to take his rest,That verily I hold supreme: naught in the world surpasses it.

yé arvā́ṅ mádhya utá vā purāṇáṃ védaṃ vidvā́ṃsam abhíto vádanti
ādityám evá té pári vadanti sárve agníṃ dvitī́yaṃ trivŕ̥taṃ ca haṃsám 17

Those who in recent times, midmost, or ancient, on all sides.greet the sage who knows the Veda,One and all, verily discuss Aditya, the second Agni, and the threefold Hansa.

sahasrāhṇyáṃ víyatāv asya pakṣáu hárer haṃsásya pátataḥ svargám
sá devā́nt sárvān úrasy upadádya saṃpáśyan yāti bhúvanāni víśvā 18

This gold-hued Haiisa’s wings, flying to heaven, spread o’er athousand days’ continued journey.Supporting all the Gods upon his bosom, he goes his way beholding every creature.

satyénordhvás tapati bráhmaṇārvā́ṅ ví paśyati
prāṇéna tiryáṅ prā́ṇati yásmin jyeṣṭhám ádhi śritám 19

By truth he blazes up aloft by Brahma, he looks down below: He breathes obliquely with his breath, he on whom what is highest rests.

yó vái té vidyā́d aráṇī yā́bhyāṃ nirmathyáte vásu
sá vidvā́n jyeṣṭháṃ manyeta sá vidyād brā́hmaṇaṃ mahát 20

The sage who knows the kindling-sticks whence by attrition wealth is drawn,Will comprehend what is most high, will know the mighty Brahmana.

apā́d ágre sám abhavat só ágre svàr ā́bharat
cátuṣpād bhūtvā́ bhógyaḥ sárvam ā́datta bhójanam 21

Footless at first was he produced, footless he brought celestiallight. Four-footed grown, and meet for use, he seized each thing enjoyable.

bhógyo bhavad átho ánnam adad bahú
yó devám uttarā́vantam upā́sātai sanātánam 22

Useful will he become, and then will he consume great store of food The man who humbly worshippeth the eternal and victorious God.

sanātánam enam āhur utā́dyá syāt púnarṇavaḥ
ahorātré prá jāyete anyó anyásya rūpáyoḥ 23

Him too they call eternal; he may become new again to-day.Day and Night reproduce themselves, each from the form the other wears.

śatáṃ sahásram ayútaṃ nyàrbudam asaṃkhyeyáṃ svám asmin níviṣṭam
tád asya ghnanty abhipáśyata evá tásmād devó rocat eṣá etát 24

A hundred, thousand, myriad, yea a hundred million stores of wealth that passes count are laid in him.This wealth they kill as he looks on, and now this God shines bright therefrom.

bā́lād ékam aṇīyaskám utáikaṃ néva dr̥śyate
tátaḥ páriṣvajīyasī devátā sā́ máma priyā́ 25

One is yet finer than a hair, one is not even visible. And hence the Deity who grasps with firmer hold is dear to me.

iyáṃ kalyāṇy àjárā mártyasyāmŕ̥tā gr̥hé
yásmai kr̥tā́ śáye sá yáś cakā́ra jajā́ra sáḥ 26

This fair one is untouched by age, immortal in a mortal’s house. He for whom she was made lies low, and he who formed her hath grown old.

tváṃ strī́ tváṃ púmān asi tváṃ kumārá utá vā kumārī́
tváṃ jīrṇó daṇḍéna vañcasi tváṃ jātó bhavasi viśvátomukhaḥ 27

Thou art a woman, and a man; thou art a damsel and a boy. Grown old thou totterest with a staff, new-born thou lookest every way.

utáiṣāṃ pitótá vā putrá eṣām utáiṣāṃ jyeṣṭhá utá vā kaniṣṭháḥ
éko ha devó mánasi práviṣṭaḥ prathamó jātáḥ sá u gárbhe antáḥ 28

Either the sire or son of these, the eldest or the youngest child.As sole God dwelling in the mind, first born, he still is in the womb.

pūrṇā́t pūrṇám úd acati pūrṇáṃ pūrṇéna sicyate
utó tád adyá vidyāma yátas tát pariṣicyáte 29

Forth from the full he lifts the full, the full he sprinkles withthe full.Now also may we know the source from which the stream is sprinkled round.

eṣā́ sanátnī sánam evá jātáiṣā́ purāṇī́ pári sárvaṃ babhūva
mahī́ devy ùṣáso vibhātī́ sáikenaikena miṣatā́ ví caṣṭe 30

Brought forth in olden time, the everlasting, high over all that is was she, the Ancient. The mighty Goddess of the Morn, refulgent with one eye, looketh round with one that winketh,

ávir vái nā́ma devátarténāste párīvr̥tā
tásyā rūpéṇemé vr̥kṣā́ háritā háritasrajaḥ 31

Known by the name of Guardian Grace the Deity sits girt by Right.The trees have taken from her hue, green-garlanded, their robe of green.

ánti sántaṃ ná jahāty ánti sántaṃ ná paśyati
devásya paśya kā́vyaṃ ná mamāra ná jīryati 32

When he is near she leaves him not, she sees him not though he is near. Behold the wisdom of the God; he hath not died, he grows not old.

apūrvéṇeṣitā́ vā́cas tā́ vadanti yathāyathám
vádantīr yátra gáchanti tád āhur brā́hmaṇaṃ mahát 33

Voices that never were before emitted speak as fitteth them. Whither they go and speak, they say there is the mighty Brahmana.

yátra devā́ś ca manuṣyā̀ś cārā́ nā́bhāv iva śritā́ḥ
apā́ṃ tvā púṣpaṃ pr̥chāmi yátra tán māyáyā hitám 34

I ask thee where the waters’ flower by wondrous magic art was placed,Thereon the Gods and men are set as spokes are fastened in the nave.

yébhir vā́ta iṣitáḥ pravā́ti yé dádante páñca díśaḥ sadhrī́cīḥ
yá ā́hutim atyámanyanta devā́ apā́ṃ netā́raḥ katamé tá āsan 35

Who gave command unto the wind that blowet! Who ranged the five united heavenly regions? Who were the Gods who cared not for oblations! Which of them brought the sacrificial waters?

imā́m eṣāṃ pr̥thivī́ṃ vásta éko ‘ntárikṣaṃ páry éko babhūva
dívam eṣāṃ dadate yó vidhartā́ víśvā ā́śāḥ práti rakṣanty éke 36

One God inhabiteth the earth we live on; another hath encompassed air’s mid-region. One, the Supporter, takes the heaven and bears it: some keeping watch guard all the quarters safely.

yó vidyā́t sū́traṃ vítataṃ yásminn ótāḥ prajā́ imā́ḥ
sū́traṃ sū́trasya yó vidyā́d sá vidyād brā́hmaṇaṃ mahát 37

The man who knows the drawn-out string on which these creatures all are strung,The man who knows the thread’s thread, he may know the mighty Brahmana.

védāháṃ sū́traṃ vítataṃ yásminn ótāḥ prajā́ imā́ḥ
sū́traṃ sū́trasyāháṃ vedā́tho yád brā́hmaṇaṃ mahád 38

I know the drawn-out string, the thread whereon these creatures all are strung. I know the thread’s thread also, thus I know the mighty Brahmana.

yád antarā́ dyā́vāpr̥thivī́ agnír áit pradáhan viśvadāvyàḥ
yátrā́tiṣṭhann ékapatnīḥ parástāt kvèvāsīn mātaríśvā tadā́nīm 39

When Agni passed between the earth and heaven devouring with his flame the all-consumer,Where dwelt afar the spouses of one husband, where at that moment, where was Matarisvan?

apsv ā̀sīn mātaríśvā práviṣṭaḥ práviṣṭā devā́ḥ salilā́ny āsan
br̥hán ha tasthau rájaso vimā́naḥ pávamāno haríta ā́ viveśa 40

Into the floods had Matarisvan entered, the deities had past into the waters. There stood the mighty measurer of the region: into the verdant plants went Pavamana.

úttareṇeva gayatrī́m amŕ̥té ‘dhi ví cakrame
sā́mnā yé sā́ma saṃvidúr ajás tád dadr̥śe kvà 41

Over the Gayatri, above the immortal world he strode away.Those who by Song discovered Song–where did the Unborn see that thing?

nivéśanaḥ saṃgámano vásūnāṃ devá iva savitā́ satyádharmā
índro ná tasthau samaré dhánānām 42

Luller to rest, and gatherer-up of treasures, Savitar like a God whose laws are constant, hath stood like Indra in the war for riches.

puṇḍárīkaṃ návadvāraṃ tribhír guṇébhir ā́vr̥tam
tásmin yád yakṣám ātmanvát tád vái brahmavído viduḥ 43

Men versed in sacred knowledge know that living Being that abides. In the nine-portalled Lotus Flower, enclosed with triple bands and bonds.

akāmó dhī́ro amŕ̥taḥ svayaṃbhū́ rásena tr̥ptó ná kútaś canónaḥ
tám evá vidvā́n ná bibhāya mr̥tyór ātmā́naṃ dhī́ram ajáraṃ yúvānam 44

Desireless, firm, immortal, self-existent, contented with the essence, lacking nothing, Free from the fear of Death is he who knoweth that Soul courageous, youthful, undecaying.

Please see my related posts

The Great Chain of Being


Key Sources of Research

Atharva Veda ( X – 7,8) — Skambha Suktam


The unassailable glory of lord Bhuvaneshwara – the primordial Skambha supporting the worlds



The Lamps And The Cosmic Pillar




Vedas and Torah


Click to access 52791.pdf

Knot Theory and Recursion: Louis H. Kauffman

Knot Theory and Recursion: Louis H. Kauffman


Some knots are tied forever.


Key Terms

  • Louis H Kauffman
  • Heinz Von Foerster
  • George Spencer Brown
  • Francisco Varela
  • Charles Sanders Peirce
  • Recursion
  • Reflexivity
  • Knots
  • Laws of Form
  • Shape of Process
  • Trefoil Knots
  • Triplicity
  • Nonduality
  • Self Reference
  • Eigen Form
  • Form Dynamics
  • Recursive Forms
  • Knot Logic
  • Bio Logic
  • Distinctions
  • Topology
  • Topological Recursion
  • Ganth
  • Granthi – Brahma, Vishnu, Rudra
  • Chakra
  • Braids
  • Bandhu
  • Mitra
  • Vishvamitra
  • Friend
  • Relation
  • Sambandh
  • Love
  • True Love
  • Its a Knotty problem.



In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a “braid” rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.

A knot can be generalized to a link, which is simply a knotted collection of one or more closed strands.

The study of knots and their properties is known as knot theory. Knot theorywas given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations (Thompson 1867). P. G. Tait then cataloged possible knots by trial and error. Much progress has been made in the intervening years.

Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed (Livingston 1993, p. 5; Adams 1994, pp. 8-9). Knots that can be so decomposed are then known as composite knots. The total number (prime plus composite) of distinct knots (treating mirror images as equivalent) having k=0, 1, … crossings are 1, 0, 0, 1, 1, 2, 5, 8, 25, … (OEIS A086825).

Klein proved that knots cannot exist in an even-dimensional space >=4. It has since been shown that a knot cannot exist in any dimension >=4. Two distinct knots cannot have the same knot complement (Gordon and Luecke 1989), but two links can! (Adams 1994, p. 261).

Knots are most commonly cataloged based on the minimum number of crossings present (the so-called link crossing number). Thistlethwaite has used Dowker notation to enumerate the number of prime knots of up to 13 crossings, and alternating knots up to 14 crossings. In this compilation, mirror images are counted as a single knot type. Hoste et al. (1998) subsequently tabulated all prime knots up to 16 crossings. Hoste and Weeks subsequently began compiling a list of 17-crossing prime knots (Hoste et al. 1998).

Another possible representation for knots uses the braid group. A knot with n+1 crossings is a member of the braid group n.

There is no general algorithm to determine if a tangled curve is a knot or if two given knots are interlocked. Haken (1961) and Hemion (1979) have given algorithms for rigorously determining if two knots are equivalent, but they are too complex to apply even in simple cases (Hoste et al. 1998).


LH Kauffman with Trefoil Knot in the back.

LH Kauffman


From Reflexivity

A Knot

Screen Shot 2020-01-06 at 12.49.45 PM


Trefoil Knot



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From Reflexivity

This slide show has been only an introduction to certain mathematical and conceptual points of view about reflexivity.

In the worlds of scientific, political and economic action these principles come into play in the way structures rise and fall in the play of realities that are created from (almost) nothing by the participants in their desire to profit, have power or even just to have clarity and understanding. Beneath the remarkable and unpredictable structures that arise from such interplay is a lambent simplicity to which we may return, as to the source of the world.


From Laws of Form and the Logic of Non-Duality

This talk will trace how a mathematics of distinction arises directly from the process of discrimination and how that language, understood rightly as an opportunity to join as well as to divide, can aid in the movement between duality and non-duality that is our heritage as human beings on this planet.The purpose of this talk is to express this language and invite your participation in it and to present the possiblity that all our resources physical, scientific, logical, intellectual, empathic are our allies in the journey to transcend separation.

From Laws of Form and the Logic of Non-Duality

True Love.  It is a knotty problem.

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Wikipedia on Knot Theory




Please see my related posts:

Reflexivity, Recursion, and Self Reference

Jay W. Forrester and System Dynamics

Steps to an Ecology of Mind: Recursive Vision of Gregory Bateson

Second Order Cybernetics of Heinz Von Foerster

Cybernetics Group: A Brief History of American Cybernetics

Cybernetics, Autopoiesis, and Social Systems Theory

Cyber-Semiotics: Why Information is not enough

Ratio Club: A Brief History of British Cyberneticians

Autocatalysis, Autopoiesis and Relational Biology

Feedback Thought in Economics and Finance

Increasing Returns and Path Dependence in Economics

Boundaries and Distinctions

Boundaries and Relational Sociology

Boundaries and Networks

Socio-Cybernetics and Constructivist Approaches

Society as Communication: Social Systems Theory of Niklas Luhmann

Semiotics, Bio-Semiotics and Cyber Semiotics

Meta Integral Theories: Integral Theory, Critical Realism, and Complex Thought

Networks and Hierarchies


Key Sources of Research:


Home Page of Louis H. Kauffman


Recursive Distinctioning

By Joel Isaacson and Louis H. Kauffman


Click to access JSP-Spr-2016-8_Kauffman-Isaacson-Final-v2.pdf



Knot Logic – Logical Connection and Topological Connection

by Louis H. Kauffman

Click to access 1508.06028.pdf




by Louis H. Kauffman


Click to access KNOTS.pdf





Louis H. Kaufman, UIC

Click to access BioL.pdf

New Invariants in the Theory of Knots

Louis H. Kaufman, UIC





Eigenform – An Introduction

by Louis H. Kauffman

Click to access 2007_813_Kauffman.pdf



Knot Logic and Topological Quantum Computing with Majorana Fermions

Louis H. Kauffman


Click to access arXiv%3A1301.6214.pdf




by Louis H. Kauffman

Click to access videoLKss-slides.pdf




Eigenforms, Discrete Processes and Quantum Processes

Louis H Kauffman 2012 J. Phys.: Conf. Ser. 361 012034





Eigenforms — Objects as Tokens for Eigenbehaviors

by Louis H. Kauffman

Click to access 1817.pdf




Reflexivity and Eigenform The Shape of Process

Louis H. Kauffman A University of


Click to access ReflexPublished.pdf






Louis H. Kauffman


Click to access Eigen.pdf





Louis H. Kauffman UIC, Chicago


Click to access Eigenform.pdf



Form Dynamics

Click to access FormDynamics.pdf



Arithmetics in the Form

Click to access ArithForm.pdf




Self Reference and Recursive Forms

Click to access SelfRefRecurForm.pdf

Click to access Relativity.pdf




Laws of Form and the Logic of Non-Duality

Louis H. Kauffman, UIC


Click to access KauffSAND.pdf




Laws of Form – An Exploration in Mathematics and Foundations

by Louis H. Kauffman UIC


Click to access Laws.pdf




The Mathematics of Charles Sanders Peirce

Louis H. Kauffman1


Click to access Peirce.pdf




A Recursive Approach to the Kauffman Bracket

Abdul Rauf Nizami, Mobeen Munir, Umer Saleem, Ansa Ramzan

Division of Science and Technology, University of Education, Lahore, Pakistan



On Synchronicity

On Synchronicity

There are invisible ties between us.


Click to access Synchronicity2010.pdf

Causality has to do with events that happen in sequence, a cause producing an effect, whereas synchronicity has to do with events that happen together.

Synchronicism is the prejudice of the East, causality is the modern prejudice of the West.  – Carl Jung, 1929



Key Terms

  • Synchronicity
  • Serendipity
  • Interconnected Universe
  • Quantum Entanglements
  • Fractal Universe
  • Recursive Universe
  • Platonic Solids
  • Carl G Jung
  • Events in Sequence
  • Events in Parallel
  • Space Structure
  • Ether
  • Geometry of space
  • Complex Numbers
  • Shri Yantra Geometry
  • Mind and Matter
  • Brain and Mind
  • Parapsychology
  • Occult
  • Esoteric




The history of coincidence studies can be told through the stories of thefour people who coined words for the types of coincidences they noticed in their lives. The most famous is Carl Jung and synchronicity, but he was not the first.

Serendipity: Horace Walpole (1717-1797)


Horace Walpole, a member of the British House of Commons in the 18th century, recognized in himself a talent for finding what he needed just when he needed it.A gift in the form of a portrait of a Grand Duchess whom Walpole had long admired arrived from his distant cousin in Florence, Italy. Walpole needed a coat of arms to decorate the new picture frame and just happened to find it an old book. On January 28, 1754, Walpole, thrilled with this coincidence, wrote to his cousin Horace, giving a name to his ability to find things unexpectedly—serendipity.

He got the name from a fairy tale called “The Travels and Adventures of Three Princes of Sarendip.” Sarendip (or Serendib) is an ancient name for the island nation Sri Lanka off India’s southern coast. The king of the fable recognized that education requires more than learning from books, so he sent his sons out of the country to broaden their experience. Throughout the story, the clever princes carefully observed their surroundings, and then used their observations in ways that saved them from danger and death.

For Walpole, serendipity meant finding something by informed observation (sagacity as he called it) and by accident. The main ingredients of serendipity include luck, chance, active searching, and informed observation.

Seriality: Paul Kammerer (1880-1926)


Biologist Paul Kammerer spent hours sitting on benches in various public parks in Vienna noting repetitions among the people who passed by. He classified them by sex, age, dress, whether they carried umbrellas or parcels, and by many other details. He did the same during the long train rides from his home to his office in Vienna. Kammerer was not particularly interested in meaning—only repeated sequences of numbers, names, words, and letters. Two examples can illustrate his thinking: His wife was in a waiting room reading about a painter named Schwalbach when a patient named Mrs. Schwalbach was called into the consultation room.A second example involved his friend Prince Rohan. On the train his wife was reading a novel with a character “Mrs. Rohan.” She then saw a man get on the train who looked like Prince Rohan. Later that night the Prince himself unexpectedly dropped by their house for a visit.

He defined “seriality” as “a recurrence of the same or similar things or events in time and space” which, “are not connected by the same acting cause.”  To him these repetitions were simply natural phenomena.

Kammerer thought these similarities were part of the structure of natural law, and in his 1919 book Das Gesetz Der Serie outlined what he thought these laws to be along with a broad set of classifications of their types and qualities.

Synchronicity: Carl Jung (1875-1961)

Psychologist Carl G. Jung

Carl Jung grew up in Swiss family that, on his mother’s side, embraced the paranormal. His personal experiences included apparitions (the disembodied figure of another person) and poltergeists (troublesome ghosts), spiritualistic communications (communication with people after their deaths), and materializations (creation of matter from unknown sources). His experiences also included telepathic, clairvoyant, and precognitive dreams, prophetic visions, psychokinetic events, and out-of-body and near-death experiences.

He invented the word synchronicity from the Greek syn—with, together—andchronos—time. Synchronicity means moving-together-in-time. Its fundamental characteristic is the surprise that occurs when a thought in the mind is mirrored by an external event to which it has no apparent causal connection. He also used the word synchronicity to refer to “an acausal connecting principle” that he placed on equal status with causation.

He included many strange events under the synchronicity umbrella including telepathy, precognition, and clairvoyance, along with poltergeists, apparitions, divination (e.g. the I Ching), and astrology. The definition of synchronicity has been stretched in many different directions.

Simulpathity: Bernard Beitman (1942–) Founder of Coincidence Studies


The term “simulpathity” defines a specific subclass of meaningful coincidences: the simultaneous experience at a distance by one person of another person’s distress. The experience occurs without the two people being together in the same place and sometimes without conscious awareness of its source. One person is in pain and another person feels distress for no apparent reason. Sometimes the distress is very similar to the other person’s pain. Often, the two people share a strong emotional bond. The largest number of simulpathity reports comes from twins, although reports involving mothers and their children are also prominent.

Simulpathity suggests that the individuals are more closely bonded than current scientific thought holds possible.

Simulpathity — from the Latin simul (simultaneous) and the Greek pathos (suffering) — differs from “sympathy.” The sympathetic person is aware of the suffering of the other but does not usually feel it. In the experience of simulpathity, one person suffers along with the other person and can experiences some form of that suffering. Only later is the simultaneity of the distress recognized, although some twins know just why they are feeling pain—the other twin is now feeling it.

Please see my related posts:

Interconnected Pythagorean Triples using Central Squares Theory

Indra’s Net: On Interconnectedness

The Great Chain of Being

On Holons and Holarchy

Mind, Consciousness and Quantum Entanglement

Geometry of Consciousness

Systems View of Life: A Synthesis by Fritjof Capra

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Shape of the Universe

Reflexivity, Recursion, and Self Reference

Key Sources of Research


Synchronicity: Nature and Psyche in an Interconnected Universe

Joseph Cambray




Synchronicity and Healing



Click to access 18-Beitman-Chap18.pdf




An Acausal Connecting Principle

CG Jung






Christopher Jargodzki


Click to access Synchronicity2010.pdf

Synchronicity, Mind, and Matter

Wlodzislaw Duch






Synchronicity: did Jung have it right?

Kurt Forrer


Click to access bb26804cdc465b355e2f2e09c574d50dc4bb.pdf




C.G. Jung’s Synchronicity and Quantum Entanglement:  Schrodinger’s Cat ‘Wanders’ Between Chromosomes


Click to access Limar.Synchronicity.pdf

Synchronicity: The Bridge Between Matter and Mind

F. David Peat
Bantam Books (1987)

Emotions Over Time: Synchronicity and Development of Subjective, Physiological, and Facial Affective Reactions to Music

C. G. Jung’s Psychology of Religion and Synchronicity

By Robert Aziz

Jung on Synchronicity and the Paranormal

By Carl Gustav Jung



Synchronicity and Emergence


Click to access Cambray.pdf

Jung, synchronicity, & human destiny: Noncausal dimensions of human experience.

Synchronicity: Science, Myth, and the Trickster

Allan Combs
Marlowe & Co. (1996)

Synchronicity, Science and Soul-Making: Understanding Jungian Synchronicity …

By Victor Mansfield

Synchronicity: Nature and Psyche in an Interconnected Universe

By Joseph Cambray

On Beauty

On Beauty

Perspectives on Beauty.


Classical conception of beauty is mathematical.  It is based on:
  • Proportions
  • Symmetry
  • Harmony
  • Golden Ratio


in Architecture, Sculpture, Music. and Literature.

In the West, classical conception of Beauty is dominant.



Plotinus was critical of Plato’s view on Beauty,

Read Idealist conception of beauty to get views of Plotinus.

From Stanford encyclopedia of Philosophy

First published Tue Sep 4, 2012; substantive revision Wed Oct 5, 2016


The nature of beauty is one of the most enduring and controversial themes in Western philosophy, and is—with the nature of art—one of the two fundamental issues in philosophical aesthetics. Beauty has traditionally been counted among the ultimate values, with goodness, truth, and justice. It is a primary theme among ancient Greek, Hellenistic, and medieval philosophers, and was central to eighteenth and nineteenth-century thought, as represented in treatments by such thinkers as Shaftesbury, Hutcheson, Hume, Burke, Kant, Schiller, Hegel, Schopenhauer, Hanslick, and Santayana. By the beginning of the twentieth century, beauty was in decline as a subject of philosophical inquiry, and also as a primary goal of the arts. However, there were signs of revived interest by the early 2000s.

This article will begin with a sketch of the debate over whether beauty is objective or subjective, which is perhaps the single most-prosecuted disagreement in the literature. It will proceed to set out some of the major approaches to or theories of beauty developed within Western philosophical and artistic traditions.


1. Objectivity and Subjectivity

Perhaps the most familiar basic issue in the theory of beauty is whether beauty is subjective—located ‘in the eye of the beholder’—or whether it is an objective feature of beautiful things. A pure version of either of these positions seems implausible, for reasons we will examine, and many attempts have been made to split the difference or incorporate insights of both subjectivist and objectivist accounts. Ancient and medieval accounts for the most part located beauty outside of anyone’s particular experiences. Nevertheless, that beauty is subjective was also a commonplace from the time of the sophists. By the eighteenth century, Hume could write as follows, expressing one ‘species of philosophy’:

Beauty is no quality in things themselves: It exists merely in the mind which contemplates them; and each mind perceives a different beauty. One person may even perceive deformity, where another is sensible of beauty; and every individual ought to acquiesce in his own sentiment, without pretending to regulate those of others. (Hume 1757, 136)

And Kant launches his discussion of the matter in The Critique of Judgment(the Third Critique) at least as emphatically:

The judgment of taste is therefore not a judgment of cognition, and is consequently not logical but aesthetical, by which we understand that whose determining ground can be no other than subjective. Every reference of representations, even that of sensations, may be objective (and then it signifies the real [element] of an empirical representation), save only the reference to the feeling of pleasure and pain, by which nothing in the object is signified, but through which there is a feeling in the subject as it is affected by the representation. (Kant 1790, section 1)

However, if beauty is entirely subjective—that is, if anything that anyone holds to be or experiences as beautiful is beautiful (as James Kirwan, for example, asserts)—then it seems that the word has no meaning, or that we are not communicating anything when we call something beautiful except perhaps an approving personal attitude. In addition, though different persons can of course differ in particular judgments, it is also obvious that our judgments coincide to a remarkable extent: it would be odd or perverse for any person to deny that a perfect rose or a dramatic sunset was beautiful. And it is possible actually to disagree and argue about whether something is beautiful, or to try to show someone that something is beautiful, or learn from someone else why it is.

On the other hand, it seems senseless to say that beauty has no connection to subjective response or that it is entirely objective. That would seem to entail, for example, that a world with no perceivers could be beautiful or ugly, or perhaps that beauty could be detected by scientific instruments. Even if it could be, beauty would seem to be connected to subjective response, and though we may argue about whether something is beautiful, the idea that one’s experiences of beauty might be disqualified as simply inaccurate or false might arouse puzzlement as well as hostility. We often regard other people’s taste, even when it differs from our own, as provisionally entitled to some respect, as we may not, for example, in cases of moral, political, or factual opinions. All plausible accounts of beauty connect it to a pleasurable or profound or loving response, even if they do not locate beauty purely in the eye of the beholder.

Until the eighteenth century, most philosophical accounts of beauty treated it as an objective quality: they located it in the beautiful object itself or in the qualities of that object. In De Veritate Religione, Augustine asks explicitly whether things are beautiful because they give delight, or whether they give delight because they are beautiful; he emphatically opts for the second (Augustine, 247). Plato’s account in the Symposium and Plotinus’s in theEnneads connect beauty to a response of love and desire, but locate beauty itself in the realm of the Forms, and the beauty of particular objects in their participation in the Form. Indeed, Plotinus’s account in one of its moments makes beauty a matter of what we might term ‘formedness’: having the definite shape characteristic of the kind of thing the object is.

We hold that all the loveliness of this world comes by communion in Ideal-Form. All shapelessness whose kind admits of pattern and form, as long as it remains outside of Reason and Idea, is ugly from that very isolation from the Divine-Thought. And this is the Absolute Ugly: an ugly thing is something that has not been entirely mastered by pattern, that is by Reason, the Matter not yielding at all points and in all respects to Ideal-Form. But where the Ideal-Form has entered, it has grouped and coordinated what from a diversity of parts was to become a unity: it has rallied confusion into co-operation: it has made the sum one harmonious coherence: for the Idea is a unity and what it moulds must come into unity as far as multiplicity may. (Plotinus, 22 [Ennead I, 6])

In this account, beauty is at least as objective as any other concept, or indeed takes on a certain ontological priority as more real than particular Forms: it is a sort of Form of Forms.

Though Plato and Aristotle disagree on what beauty is, they both regard it as objective in the sense that it is not localized in the response of the beholder. The classical conception (see below) treats beauty as a matter of instantiating definite proportions or relations among parts, sometimes expressed in mathematical ratios, for example the ‘golden section.’ The sculpture known as ‘The Canon,’ by Polykleitos (fifth/fourth century BCE), was held up as a model of harmonious proportion to be emulated by students and masters alike: beauty could be reliably achieved by reproducing its objective proportions. Nevertheless, it is conventional in ancient treatments of the topic also to pay tribute to the pleasures of beauty, often described in quite ecstatic terms, as in Plotinus: “This is the spirit that Beauty must ever induce: wonderment and a delicious trouble, longing and love and a trembling that is all delight” (Plotinus 23, [Ennead 1, 3]).

At latest by the eighteenth century, however, and particularly in the British Isles, beauty was associated with pleasure in a somewhat different way: pleasure was held to be not the effect but the origin of beauty. This was influenced, for example, by Locke’s distinction between primary and secondary qualities. Locke and the other empiricists treated color (which is certainly one source or locus of beauty), for example, as a ‘phantasm’ of the mind, as a set of qualities dependent on subjective response, located in the perceiving mind rather than of the world outside the mind. Without perceivers of a certain sort, there would be no colors. One argument for this was the variation in color experiences between people. For example, some people are color-blind, and to a person with jaundice much of the world takes on a yellow cast. In addition, the same object is perceived as having different colors by the same the person under different conditions: at noon and midnight, for example. Such variations are conspicuous in experiences of beauty as well.

Nevertheless, eighteenth-century philosophers such as Hume and Kant perceived that something important was lost when beauty was treated merely as a subjective state. They saw, for example, that controversies often arise about the beauty of particular things, such as works of art and literature, and that in such controversies, reasons can sometimes be given and will sometimes be found convincing. They saw, as well, that if beauty is completely relative to individual experiencers, it ceases to be a paramount value, or even recognizable as a value at all across persons or societies.

Hume’s “Of the Standard of Taste” and Kant’s Critique Of Judgment attempt to find ways through what has been termed ‘the antinomy of taste.’ Taste is proverbially subjective: de gustibus non est disputandum (about taste there is no disputing). On the other hand, we do frequently dispute about matters of taste, and some persons are held up as exemplars of good taste or of tastelessness. Some people’s tastes appear vulgar or ostentatious, for example. Some people’s taste is too exquisitely refined, while that of others is crude, naive, or non-existent. Taste, that is, appears to be both subjective and objective: that is the antinomy.

Both Hume and Kant, as we have seen, begin by acknowledging that taste or the ability to detect or experience beauty is fundamentally subjective, that there is no standard of taste in the sense that the Canon was held to be, that if people did not experience certain kinds of pleasure, there would be no beauty. Both acknowledge that reasons can count, however, and that some tastes are better than others. In different ways, they both treat judgments of beauty neither precisely as purely subjective nor precisely as objective but, as we might put it, as inter-subjective or as having a social and cultural aspect, or as conceptually entailing an inter-subjective claim to validity.

Hume’s account focuses on the history and condition of the observer as he or she makes the judgment of taste. Our practices with regard to assessing people’s taste entail that judgments of taste that reflect idiosyncratic bias, ignorance, or superficiality are not as good as judgments that reflect wide-ranging acquaintance with various objects of judgment and are unaffected by arbitrary prejudices. “Strong sense, united to delicate sentiment, improved by practice, perfected by comparison, and cleared of all prejudice, can alone entitle critics to this valuable character; and the joint verdict of such, wherever they are to found, is the true standard of taste and beauty” (“Of the Standard of Taste” 1757, 144).

Hume argues further that the verdicts of critics who possess those qualities tend to coincide, and approach unanimity in the long run, which accounts, for example, for the enduring veneration of the works of Homer or Milton. So the test of time, as assessed by the verdicts of the best critics, functions as something analogous to an objective standard. Though judgments of taste remain fundamentally subjective, and though certain contemporary works or objects may appear irremediably controversial, the long-run consensus of people who are in a good position to judge functions analogously to an objective standard and renders such standards unnecessary even if they could be identified. Though we cannot directly find a standard of beauty that sets out the qualities that a thing must possess in order to be beautiful, we can describe the qualities of a good critic or a tasteful person. Then the long-run consensus of such persons is the practical standard of taste and the means of justifying judgments about beauty.

Kant similarly concedes that taste is fundamentally subjective, that every judgment of beauty is based on a personal experience, and that such judgments vary from person to person.

By a principle of taste I mean a principle under the condition of which we could subsume the concept of the object, and thus infer, by means of a syllogism, that the object is beautiful. But that is absolutely impossible. For I must immediately feel the pleasure in the representation of the object, and of that I can be persuaded by no grounds of proof whatever. Although, as Hume says, all critics can reason more plausibly than cooks, yet the same fate awaits them. They cannot expect the determining ground of their judgment [to be derived] from the force of the proofs, but only from the reflection of the subject upon its own proper state of pleasure or pain. (Kant 1790, section 34)

But the claim that something is beautiful has more content merely than that it gives me pleasure. Something might please me for reasons entirely eccentric to myself: I might enjoy a bittersweet experience before a portrait of my grandmother, for example, or the architecture of a house might remind me of where I grew up. “No one cares about that,” says Kant (1790, section 7): no one begrudges me such experiences, but they make no claim to guide or correspond to the experiences of others.

By contrast, the judgment that something is beautiful, Kant argues, is a disinterested judgment. It does not respond to my idiosyncrasies, or at any rate if I am aware that it does, I will no longer take myself to be experiencing the beauty per se of the thing in question. Somewhat as in Hume—whose treatment Kant evidently had in mind—one must be unprejudiced to come to a genuine judgment of taste, and Kant gives that idea a very elaborate interpretation: the judgment must be made independently of the normal range of human desires—economic and sexual desires, for instance, which are examples of our ‘interests’ in this sense. If one is walking through a museum and admiring the paintings because they would be extremely expensive were they to come up for auction, for example, or wondering whether one could steal and fence them, one is not having an experience of the beauty of the paintings at all. One must focus on the form of the mental representation of the object for its own sake, as it is in itself. Kant summarizes this as the thought that insofar as one is having an experience of the beauty of something, one is indifferent to its existence. One takes pleasure, rather, in its sheer representation in one’s experience:

Now, when the question is whether something is beautiful, we do not want to know whether anything depends or can depend on the existence of the thing, either for myself or anyone else, but how we judge it by mere observation (intuition or reflection). … We easily see that, in saying it is beautiful, and in showing that I have taste, I am concerned, not with that in which I depend on the existence of the object, but with that which I make out of this representation in myself. Everyone must admit that a judgement about beauty, in which the least interest mingles, is very partial and is not a pure judgement of taste. (Kant 1790, section 2)

One important source of the concept of aesthetic disinterestedness is the Third Earl of Shaftesbury’s dialogue The Moralists, where the argument is framed in terms of a natural landscape: if you are looking at a beautiful valley primarily as a valuable real estate opportunity, you are not seeing it for its own sake, and cannot fully experience its beauty. If you are looking at a lovely woman and considering her as a possible sexual conquest, you are not able to experience her beauty in the fullest or purest sense; you are distracted from the form as represented in your experience. And Shaftesbury, too, localizes beauty to the representational capacity of the mind. (Shaftesbury 1738, 222)

For Kant, some beauties are dependent—relative to the sort of thing the object is—and others are free or absolute. A beautiful ox would be an ugly horse, but abstract textile designs, for example, may be beautiful in themselves without a reference group or “concept,” and flowers please whether or not we connect them to their practical purposes or functions in plant reproduction (Kant 1790, section 16). The idea in particular that free beauty is completely separated from practical use and that the experiencer of it is not concerned with the actual existence of the object leads Kant to conclude that absolute or free beauty is found in the form or design of the object, or as Clive Bell put it, in the arrangement of lines and colors (in the case of painting) (Bell 1914). By the time Bell writes in the early twentieth century, however, beauty is out of fashion in the arts, and Bell frames his view not in terms of beauty but in terms of a general formalist conception of aesthetic value.

Since in reaching a genuine judgment of taste one is aware that one is not responding to anything idiosyncratic in oneself, Kant asserts (1790, section 8), one will reach the conclusion that anyone similarly situated should have the same experience: that is, one will presume that there ought to be nothing to distinguish one person’s judgment from another’s (though in fact there may be). Built conceptually into the judgment of taste is the assertion that anyone similarly situated ought to have the same experience and reach the same judgment. Thus, built into judgments of taste is a ‘universalization’ somewhat analogous to the universalization that Kant associates with ethical judgments. In ethical judgments, however, the universalization is objective: if the judgment is true, then it is objectively the case that everyone ought to act on the maxim according to which one acts. In the case of aesthetic judgments, however, the judgment remains subjective, but necessarily contains the ‘demand’ that everyone should reach the same judgment. The judgment conceptually entails a claim to inter-subjective validity. This accounts for the fact that we do very often argue about judgments of taste, and that we find tastes that are different than our own defective.

The influence of this series of thoughts on philosophical aesthetics has been immense. One might mention related approaches taken by such figures as Schopenhauer, Hanslick, Bullough, and Croce, for example. A somewhat similar though more adamantly subjectivist line is taken by Santayana, who defines beauty as ‘objectified pleasure.’ The judgment of something that it is beautiful responds to the fact that it induces a certain sort of pleasure; but this pleasure is attributed to the object, as though the object itself were having subjective states.

We have now reached our definition of beauty, which, in the terms of our successive analysis and narrowing of the conception, is value positive, intrinsic, and objectified. Or, in less technical language, Beauty is pleasure regarded as the quality of a thing. … Beauty is a value, that is, it is not a perception of a matter of fact or of a relation: it is an emotion, an affection of our volitional and appreciative nature. An object cannot be beautiful if it can give pleasure to nobody: a beauty to which all men were forever indifferent is a contradiction in terms. … Beauty is therefore a positive value that is intrinsic; it is a pleasure. (Santayana 1896, 50–51)

It is much as though one were attributing malice to a balky object or device. The object causes certain frustrations and is then ascribed an agency or a kind of subjective agenda that would account for its causing those effects. Now though Santayana thought the experience of beauty could be profound or could even be the meaning of life, this account appears to make beauty a sort of mistake: one attributes subjective states (indeed, one’s own) to a thing which in many instances is not capable of having subjective states.

It is worth saying that Santayana’s treatment of the topic in The Sense of Beauty (1896) was the last major account offered in English for some time, possibly because, once beauty has been admitted to be entirely subjective, much less when it is held to rest on a sort of mistake, there seems little more to be said. What stuck from Hume’s and Kant’s treatments was the subjectivity, not the heroic attempts to temper it. If beauty is a subjective pleasure, it would seem to have no higher status than anything that entertains, amuses, or distracts; it seems odd or ridiculous to regard it as being comparable in importance to truth or justice, for example. And the twentieth century also abandoned beauty as the dominant goal of the arts, again possibly in part because its trivialization in theory led artists to believe that they ought to pursue more real and more serious projects. This decline is explored eloquently in Arthur Danto’s book The Abuse of Beauty (2003).

However, there has been a revival of interest in beauty in both art and philosophy in recent years, and several theorists have made new attempts to address the antinomy of taste. To some extent, such approaches echo G.E. Moore’s: “To say that a thing is beautiful is to say, not indeed that it is itself good, but that it is a necessary element in something which is: to prove that a thing is truly beautiful is to prove that a whole, to which it bears a particular relation as a part, is truly good” (Moore 1903, 201). One interpretation of this would be that what is fundamentally valuable is the situation in which the object and the person experiencing are both embedded; the value of beauty might include both features of the beautiful object and the pleasures of the experiencer.

Similarly, Crispin Sartwell in his book Six Names of Beauty (2004), attributes beauty neither exclusively to the subject nor to the object, but to the relation between them, and even more widely also to the situation or environment in which they are both embedded. He points out that when we attribute beauty to the night sky, for instance, we do not take ourselves simply to be reporting a state of pleasure in ourselves; we are turned outward toward it; we are celebrating the real world. On the other hand, if there were no perceivers capable of experiencing such things, there would be no beauty. Beauty, rather, emerges in situations in which subject and object are juxtaposed and connected.

Alexander Nehamas, in Only a Promise of Happiness (2007), characterizes beauty as an invitation to further experiences, a way that things invite us in, while also possibly fending us off. The beautiful object invites us to explore and interpret, but it also requires us to explore and interpret: beauty is not to be regarded as an instantaneously apprehensible feature of surface. And Nehamas, like Hume and Kant, though in another register, considers beauty to have an irreducibly social dimension. Beauty is something we share, or something we want to share, and shared experiences of beauty are particularly intense forms of communication. Thus, the experience of beauty is not primarily within the skull of the experiencer, but connects observers and objects such as works of art and literature in communities of appreciation.

Aesthetic judgment, I believe, never commands universal agreement, and neither a beautiful object nor a work of art ever engages a catholic community. Beauty creates smaller societies, no less important or serious because they are partial, and, from the point of view of its members, each one is orthodox—orthodox, however, without thinking of all others as heresies. … What is involved is less a matter of understanding and more a matter of hope, of establishing a community that centers around it—a community, to be sure, whose boundaries are constantly shifting and whose edges are never stable. (Nehamas 2007, 80–81)

2. Philosophical Conceptions of Beauty

Each of the views sketched below has many expressions, some of which may be incompatible with one another. In many or perhaps most of the actual formulations, elements of more than one such account are present. For example, Kant’s treatment of beauty in terms of disinterested pleasure has obvious elements of hedonism, while the ecstatic neo-Platonism of Plotinus includes not only the unity of the object, but also the fact that beauty calls out love or adoration. However, it is also worth remarking how divergent or even incompatible with one another many of these views are: for example, some philosophers associate beauty exclusively with use, others precisely with uselessness.

2.1 The Classical Conception

The art historian Heinrich Wölfflin gives a fundamental description of the classical conception of beauty, as embodied in Italian Renaissance painting and architecture:

The central idea of the Italian Renaissance is that of perfect proportion. In the human figure as in the edifice, this epoch strove to achieve the image of perfection at rest within itself. Every form developed to self-existent being, the whole freely co-ordinated: nothing but independently living parts…. In the system of a classic composition, the single parts, however firmly they may be rooted in the whole, maintain a certain independence. It is not the anarchy of primitive art: the part is conditioned by the whole, and yet does not cease to have its own life. For the spectator, that presupposes an articulation, a progress from part to part, which is a very different operation from perception as a whole. (Wölfflin 1932, 9–10, 15)

The classical conception is that beauty consists of an arrangement of integral parts into a coherent whole, according to proportion, harmony, symmetry, and similar notions. This is a primordial Western conception of beauty, and is embodied in classical and neo-classical architecture, sculpture, literature, and music wherever they appear. Aristotle says in the Poetics that “to be beautiful, a living creature, and every whole made up of parts, must … present a certain order in its arrangement of parts” (Aristotle, volume 2, 2322 [1450b34]). And in the Metaphysics: “The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree” (Aristotle, volume 2 1705 [1078a36]). This view, as Aristotle implies, is sometimes boiled down to a mathematical formula, such as the golden section, but it need not be thought of in such strict terms. The conception is exemplified above all in such texts as Euclid’s Elements and such works of architecture as the Parthenon, and, again, by the Canon of the sculptor Polykleitos (late fifth/early fourth century BCE).

The Canon was not only a statue deigned to display perfect proportion, but a now-lost treatise on beauty. The physician Galen characterizes the text as specifying, for example, the proportions of “the finger to the finger, and of all the fingers to the metacarpus, and the wrist, and of all these to the forearm, and of the forearm to the arm, in fact of everything to everything…. For having taught us in that treatise all the symmetriae of the body, Polyclitus supported his treatise with a work, having made the statue of a man according to his treatise, and having called the statue itself, like the treatise, the Canon” (quoted in Pollitt 1974, 15). It is important to note that the concept of ‘symmetry’ in classical texts is distinct from and richer than its current use to indicate bilateral mirroring. It also refers precisely to the sorts of harmonious and measurable proportions among the parts characteristic of objects that are beautiful in the classical sense, which carried also a moral weight. For example, in the Sophist (228c-e), Plato describes virtuous souls as symmetrical.

The ancient Roman architect Vitruvius gives as good a characterization of the classical conception as any, both in its complexities and, appropriately enough, in its underlying unity:

Architecture consists of Order, which in Greek is called taxis, and arrangement, which the Greeks name diathesis, and of Proportion and Symmetry and Decor and Distribution which in the Greeks is called oeconomia.

Order is the balanced adjustment of the details of the work separately, and as to the whole, the arrangement of the proportion with a view to a symmetrical result.

Proportion implies a graceful semblance: the suitable display of details in their context. This is attained when the details of the work are of a height suitable to their breadth, of a breadth suitable to their length; in a word, when everything has a symmetrical correspondence.

Symmetry also is the appropriate harmony arising out of the details of the work itself: the correspondence of each given detail to the form of the design as a whole. As in the human body, from cubit, foot, palm, inch and other small parts come the symmetric quality of eurhythmy. (Vitruvius, 26–27)

Aquinas, in a typically Aristotelian pluralist formulation, says that “There are three requirements for beauty. Firstly, integrity or perfection—for if something is impaired it is ugly. Then there is due proportion or consonance. And also clarity: whence things that are brightly coloured are called beautiful” (Summa Theologica I, 39, 8).

Francis Hutcheson in the eighteenth century gives what may well be the clearest expression of the view: “What we call Beautiful in Objects, to speak in the Mathematical Style, seems to be in a compound Ratio of Uniformity and Variety; so that where the Uniformity of Bodys is equal, the Beauty is as the Variety; and where the Variety is equal, the Beauty is as the Uniformity” (Hutcheson 1725, 29). Indeed, proponents of the view often speak “in the Mathematical Style.” Hutcheson goes on to adduce mathematical formulae, and specifically the propositions of Euclid, as the most beautiful objects (in another echo of Aristotle), though he also rapturously praises nature, with its massive complexity underlain by universal physical laws as revealed, for example, by Newton. There is beauty, he says, “In the Knowledge of some great Principles, or universal Forces, from which innumerable Effects do flow. Such is Gravitation, in Sir Isaac Newton’s Scheme” (Hutcheson 1725, 38).

A very compelling series of refutations of and counter-examples to the idea that beauty can be a matter of any specific proportions between parts, and hence to the classical conception, is given by Edmund Burke in A Philosophical Enquiry into the Origin of our Ideas of the Beautiful and the Sublime:

Turning our eyes to the vegetable kingdom, we find nothing there so beautiful as flowers; but flowers are of every sort of shape, and every sort of disposition; they are turned and fashioned into an infinite variety of forms. … The rose is a large flower, yet it grows upon a small shrub; the flower of the apple is very small, and it grows upon a large tree; yet the rose and the apple blossom are both beautiful. … The swan, confessedly a beautiful bird, has a neck longer than the rest of its body, and but a very short tail; is this a beautiful proportion? we must allow that it is. But what shall we say of the peacock, who has comparatively but a short neck, with a tail longer than the neck and the rest of the body taken together? … There are some parts of the human body, that are observed to hold certain proportions to each other; but before it can be proved, that the efficient cause of beauty lies in these, it must be shewn, that wherever these are found exact, the person to whom they belong is beautiful. … For my part, I have at several times very carefully examined many of these proportions, and found them to hold very nearly, or altogether alike in many subjects, which were not only very different from one another, but where one has been very beautiful, and the other very remote from beauty. … You may assign any proportions you please to every part of the of the human body; and I undertake, that a painter shall observe them all, and notwithstanding produce, if he pleases, a very ugly figure. (Burke 1757, 84–89)

2.2 The Idealist Conception

There are many ways to interpret Plato’s relation to classical aesthetics. The political system sketched in The Republic characterizes justice in terms of the relation of part and whole. But Plato was also no doubt a dissident in classical culture, and the account of beauty that is expressed specifically in The Symposium—perhaps the key Socratic text for neo-Platonism and for the idealist conception of beauty—expresses an aspiration toward beauty as perfect unity.

In the midst of a drinking party, Socrates recounts the teachings of his instructress, one Diotima, on matters of love. She connects the experience of beauty to the erotic or the desire to reproduce (Plato, 558–59 [Symposium 206c–207e]). But the desire to reproduce is associated in turn with a desire for the immortal or eternal: ‘And why all this longing for propagation? Because this is the one deathless and eternal element in our mortality. And since we have agreed that the lover longs for the good to be his own forever, it follows that we are bound to long for immortality as well as for the good—which is to say that Love is a longing for immortality” (Plato, 559, [Symposium 206e–207a]). What follows is, if not classical, at any rate classic:

The candidate for this initiation cannot, if his efforts are to be rewarded, begin too early to devote himself to the beauties of the body. First of all, if his preceptor instructs him as he should, he will fall in love with the beauty of one individual body, so that his passion may give life to noble discourse. Next he must consider how nearly related the beauty of any one body is to the beauty of any other, and he will see that if he is to devote himself to loveliness of form it will be absurd to deny that the beauty of each and every body is the same. Having reached this point, he must set himself to be the lover of every lovely body, and bring his passion for the one into due proportion by deeming it of little or no importance.

Next he must grasp that the beauties of the body are as nothing to the beauties of the soul, so that wherever he meets with spiritual loveliness, even in the husk of an unlovely body, he will find it beautiful enough to fall in love with and cherish—and beautiful enough to quicken in his heart a longing for such discourse as tends toward the building of a noble nature. And from this he will be led to contemplate the beauty of laws and institutions. And when he discovers how every kind of beauty is akin to every other he will conclude that the beauty of the body is not, after all, of so great moment. …

And so, when his prescribed devotion to boyish beauties has carried our candidate so far that the universal beauty dawns upon his inward sight, he is almost within reach of the final revelation. … Starting from individual beauties, the quest for universal beauty must find him mounting the heavenly ladder, stepping from rung to rung—that is, from one to two, and from two to every lovely body, and from bodily beauty to the beauty of institutions, from institutions to learning, and from learning in general to the special lore that pertains to nothing but the beautiful itself—until at last he comes to know what beauty is.

And if, my dear Socrates, Diotima went on, man’s life is ever worth living, it is when he has attained this vision of the very soul of beauty. (Plato, 561–63 [Symposium 210a–211d])

Beauty here is conceived—perhaps explicitly in contrast to the classical aesthetics of integral parts and coherent whole—as perfect unity, or indeed as the principle of unity itself.

Plotinus, as we have already seen, comes close to equating beauty with formedness per se: it is the source of unity among disparate things, and it is itself perfect unity. Plotinus specifically attacks what we have called the classical conception of beauty:

Almost everyone declares that the symmetry of parts towards each other and towards a whole, with, besides, a certain charm of colour, constitutes the beauty recognized by the eye, that in visible things, as indeed in all else, universally, the beautiful thing is essentially symmetrical, patterned.

But think what this means.

Only a compound can be beautiful, never anything devoid of parts; and only a whole; the several parts will have beauty, not in themselves, but only as working together to give a comely total. Yet beauty in an aggregate demands beauty in details; it cannot be constructed out of ugliness; its law must run throughout.

All the loveliness of colour and even the light of the sun, being devoid of parts and so not beautiful by symmetry, must be ruled out of the realm of beauty. And how comes gold to be a beautiful thing? And lightning by night, and the stars, why are these so fair?

In sounds also the simple must be proscribed, though often in a whole noble composition each several tone is delicious in itself. (Plotinus, 21 [Ennead 1.6])

And Plotinus declares that fire is the most beautiful physical thing, “making ever upwards, the subtlest and sprightliest of all bodies, as very near to the unembodied. … Hence the splendour of its light, the splendour that belongs to the Idea” (Plotinus, 22 [Ennead 1.3]). For Plotinus as for Plato, all multiplicity must be immolated finally into unity, and all roads of inquiry and experience lead toward the Good/Beautiful/True/Divine.

This gave rise to a basically mystical vision of the beauty of God that, as Umberto Eco has argued, persisted alongside an anti-aesthetic asceticism throughout the Middle Ages: a delight in profusion that finally merges into a single spiritual unity. In the sixth century, Pseudo-Dionysius the Areopagite characterized the whole of creation as yearning toward God; the universe is called into being by love of God as beauty (Pseudo-Dionysius, 4.7; see Kirwan 1999, 29). Sensual/aesthetic pleasures could be considered the expressions of the immense, beautiful profusion of God and our ravishment thereby. Eco quotes Suger, Abbot of St Denis in the twelfth century, describing a richly-appointed church:

Thus, when—out of my delight in the beauty of the house of God—the loveliness of the many-colored gems has called me away from external cares, and worthy meditation has induced me to reflect, transferring that which is material to that which is immaterial, on the diversity of the sacred virtues: then it seems to me that I see myself dwelling, as it were, in some strange region of the universe which neither exists entirely in the slime of the earth nor entirely in the purity of Heaven; and that, by the grace of God, I can be transported from this inferior to that higher world in an anagogical manner. (Eco 1959, 14)

This conception has had many expressions in the modern era, including in such figures as Shaftesbury, Schiller, and Hegel, according to whom the aesthetic or the experience of art and beauty is a primary bridge (or to use the Platonic image, stairway or ladder) between the material and the spiritual. For Shaftesbury, there are three levels of beauty: what God makes (nature); what human beings make from nature or what is transformed by human intelligence (art, for example); and finally what makes even the maker of such things as us (that is, God). Shaftesbury’s character Theocles describes “the third order of beauty,”

which forms not only such as we call mere forms but even the forms which form. For we ourselves are notable architects in matter, and can show lifeless bodies brought into form, and fashioned by our own hands, but that which fashions even minds themselves, contains in itself all the beauties fashioned by those minds, and is consequently the principle, source, and fountain of all beauty. … Whatever appears in our second order of forms, or whatever is derived or produced from thence, all this is eminently, principally, and originally in this last order of supreme and sovereign beauty. … Thus architecture, music, and all which is of human invention, resolves itself into this last order. (Shaftesbury 1738, 228–29)

Schiller’s expression of a similar series of thoughts was fundamentally influential on the conceptions of beauty developed within German Idealism:

The pre-rational concept of Beauty, if such a thing be adduced, can be drawn from no actual case—rather does itself correct and guide our judgement concerning every actual case; it must therefore be sought along the path of abstraction, and it can be inferred simply from the possibility of a nature that is both sensuous and rational; in a word, Beauty must be exhibited as a necessary condition of humanity. Beauty … makes of man a whole, complete in himself. (1795, 59–60, 86)

For Schiller, beauty or play or art (he uses the words, rather cavalierly, almost interchangeably) performs the process of integrating or rendering compatible the natural and the spiritual, or the sensuous and the rational: only in such a state of integration are we—who exist simultaneously on both these levels—free. This is quite similar to Plato’s ‘ladder’: beauty as a way to ascend to the abstract or spiritual. But Schiller—though this is at times unclear—is more concerned with integrating the realms of nature and spirit than with transcending the level of physical reality entirely, a la Plato. It is beauty and art that performs this integration.

In this and in other ways—including the tripartite dialectical structure of the view—Schiller strikingly anticipates Hegel, who writes as follows.

The philosophical Concept of the beautiful, to indicate its true nature at least in a preliminary way, must contain, reconciled within itself, both the extremes which have been mentioned [the ideal and the empirical] because it unites metaphysical universality with real particularity. (Hegel 1835, 22)

Beauty, we might say, or artistic beauty at any rate, is a route from the sensuous and particular to the Absolute and to freedom, from finitude to the infinite, formulations that—while they are influenced by Schiller—strikingly recall Shaftesbury, Plotinus, and Plato.

Both Hegel and Shaftesbury, who associate beauty and art with mind and spirit, hold that the beauty of art is higher than the beauty of nature, on the grounds that, as Hegel puts it, “the beauty of art is born of the spirit and born again” (Hegel 1835, 2). That is, the natural world is born of God, but the beauty of art transforms that material again by the spirit of the artist. This idea reaches is apogee in Benedetto Croce, who very nearly denies that nature can ever be beautiful, or at any rate asserts that the beauty of nature is a reflection of the beauty of art. “The real meaning of ‘natural beauty’ is that certain persons, things, places are, by the effect which they exert upon one, comparable with poetry, painting, sculpture, and the other arts” (Croce 1928, 230).

2.3 Love and Longing

Edmund Burke, expressing an ancient tradition, writes that, “by beauty I mean, that quality or those qualities in bodies, by which they cause love, or some passion similar to it” (Burke 1757, 83). As we have seen, in almost all treatments of beauty, even the most apparently object or objectively-oriented, there is a moment in which the subjective qualities of the experience of beauty are emphasized: rhapsodically, perhaps, or in terms of pleasure or ataraxia, as in Schopenhauer. For example, we have already seen Plotinus, for whom beauty is certainly not subjective, describe the experience of beauty ecstatically. In the idealist tradition, the human soul, as it were, recognizes in beauty its true origin and destiny. Among the Greeks, the connection of beauty with love is proverbial from early myth, and Aphrodite the goddess of love won the Judgment of Paris by promising Paris the most beautiful woman in the world.

There is an historical connection between idealist accounts of beauty and those that connect it to love and longing, though there would seem to be no entailment either way. We have Sappho’s famous fragment 16: “Some say thronging cavalry, some say foot soldiers, others call a fleet the most beautiful sights the dark world offers, but I say it’s whatever you love best” (Sappho, 16). (Indeed, at Phaedrus 236c, Socrates appears to defer to “the fair Sappho” as having had greater insight than himself on love [Plato, 483].)

Plato’s discussions of beauty in the Symposium and the Phaedrus occur in the context of the theme of erotic love. In the former, love is portrayed as the ‘child’ of poverty and plenty. “Nor is he delicate and lovely as most of us believe, but harsh and arid, barefoot and homeless” (Plato, 556 [Symposium 203b–d]). Love is portrayed as a lack or absence that seeks its own fulfillment in beauty: a picture of mortality as an infinite longing. Love is always in a state of lack and hence of desire: the desire to possess the beautiful. Then if this state of infinite longing could be trained on the truth, we would have a path to wisdom. The basic idea has been recovered many times, for example by the Romantics. It fueled the cult of idealized or courtly love through the Middle Ages, in which the beloved became a symbol of the infinite.

Recent work on the theory of beauty has revived this idea, and turning away from pleasure has turned toward love or longing (which are not necessarily entirely pleasurable experiences) as the experiential correlate of beauty. Both Sartwell and Nehamas use Sappho’s fragment 16 as an epigraph. Sartwell defines beauty as “the object of longing” and characterizes longing as intense and unfulfilled desire. He calls it a fundamental condition of a finite being in time, where we are always in the process of losing whatever we have, and are thus irremediably in a state of longing. And Nehamas writes

I think of beauty as the emblem of what we lack, the mark of an art that speaks to our desire. … Beautiful things don’t stand aloof, but direct our attention and our desire to everything else we must learn or acquire in order to understand and possess, and they quicken the sense of life, giving it new shape and direction. (Nehamas 2007, 77)

2.4 Hedonist Conceptions

Thinkers of the 18th century—many of them oriented toward empiricism—accounted for beauty in terms of pleasure. The Italian historian Ludovico Antonio Muratori, for example, in quite a typical formulation, says that “By beautiful we generally understand whatever, when seen, heard, or understood, delights, pleases, and ravishes us by causing within us agreeable sensations” (see Carritt 1931, 60). In Hutcheson it is not clear whether we ought to conceive beauty primarily in terms of classical formal elements or in terms of the viewer’s pleasurable response. He begins the Inquiry Into the Original of Our Ideas of Beauty and Virtue with a discussion of pleasure. And he appears to assert that objects which instantiate his “compound ratio of uniformity and variety’ are peculiarly or necessarily capable of producing pleasure:

The only Pleasure of sense, which our Philosophers seem to consider, is that which accompanys the simple Ideas of Sensation; But there are vastly greater Pleasures in those complex Ideas of objects, which obtain the Names of Beautiful, Regular, Harmonious. Thus every one acknowledges he is more delighted with a fine Face, a just Picture, than with the View of any one Colour, were it as strong and lively as possible; and more pleased with a Prospect of the Sun arising among settled Clouds, and colouring their Edges, with a starry Hemisphere, a fine Landskip, a regular Building, than with a clear blue Sky, a smooth Sea, or a large open Plain, not diversify’d by Woods, Hills, Waters, Buildings: And yet even these latter Appearances are not quite simple. So in Musick, the Pleasure of fine Composition is incomparably greater than that of any one Note, how sweet, full, or swelling soever. (Hutcheson 1725, 22)

When Hutcheson then goes on to describe ‘original or absolute beauty,’ he does it, as we have seen, in terms of the qualities of the beautiful thing, and yet throughout, he insists that beauty is centered in the human experience of pleasure. But of course the idea of pleasure could come apart from Hutcheson’s particular aesthetic preferences, which are poised precisely opposite Plotinus’s, for example. That we find pleasure in a symmetrical rather than an asymmetrical building (if we do) is contingent. But that beauty is connected to pleasure appears, according to Hutcheson, to be necessary, and the pleasure which is the locus of beauty itself has ideas rather than things as its object.

Hume writes in a similar vein in the Treatise of Human Nature:

Beauty is such an order and construction of parts as, either by the primary constitution of our nature, by custom, or by caprice, is fitted to give a pleasure and satisfaction to the soul. … Pleasure and pain, therefore, are not only necessary attendants of beauty and deformity, but constitute their very essence. (Hume 1740, 299)

Though this appears ambiguous as between locating the beauty in the pleasure or in the impression or idea that causes it, Hume is soon talking about the ‘sentiment of beauty,’ where sentiment is, roughly, a pleasurable or painful response to impressions or ideas, though beauty is a matter of cultivated or delicate pleasures. Indeed, by the time of Kant’s Third Critique and after that for perhaps two centuries, the direct connection of beauty to pleasure is taken as a commonplace, to the point where thinkers are frequently identifying beauty as a certain sort of pleasure. Santayana, for example, as we have seen, while still gesturing in the direction of the object or experience that causes pleasure, emphatically identifies beauty as a certain sort of pleasure.

One result of this approach to beauty—or perhaps an extreme expression of this orientation—is the assertion of the positivists that words such as ‘beauty’ are meaningless or without cognitive content, or are mere expressions of subjective approval. Hume and Kant were no sooner declaring beauty to be a matter of sentiment or pleasure and therefore to be subjective than they were trying to ameliorate the sting, largely by emphasizing critical consensus. But once this fundamental admission is made, any consensus is contingent. Another way to formulate this is that it appears to certain thinkers after Hume and Kant that there can be no reasons to prefer the consensus to a counter-consensus assessment. A.J. Ayer writes:

Such aesthetic words as ‘beautiful’ and ‘hideous’ are employed … not to make statements of fact, but simply to express certain feelings and evoke a certain response. It follows…that there is no sense attributing objective validity to aesthetic judgments, and no possibility of arguing about questions of value in aesthetics. (Ayer 1952, 113)

All meaningful claims either concern the meaning of terms or are empirical, in which case they are meaningful because observations could confirm or disconfirm them. ‘That song is beautiful’ has neither status, and hence has no empirical or conceptual content. It merely expresses a positive attitude of a particular viewer; it is an expression of pleasure, like a satisfied sigh. The question of beauty is not a genuine question, and we can safely leave it behind or alone. Most twentieth-century philosophers did just that.

2.5 Use and Uselessness

Philosophers in the Kantian tradition identify the experience of beauty with disinterested pleasure, psychical distance, and the like, and contrast the aesthetic with the practical. “Taste is the faculty of judging an object or mode of representing it by an entirely disinterested satisfaction or dissatisfaction. The object of such satisfaction is called beautiful” (Kant 1790, 45). Edward Bullough distinguishes the beautiful from the merely agreeable on the grounds that the former requires a distance from practical concerns: “Distance is produced in the first instance by putting the phenomenon, so to speak, out of gear with our practical, actual self; by allowing it to stand outside the context of our personal needs and ends.“ (Bullough 1912, 244)

On the other hand, many philosophers have gone in the opposite direction and have identified beauty with suitedness to use. ‘Beauty’ is perhaps one of the few terms that could plausibly sustain such entirely opposed interpretations.

According to Diogenes Laertius, the ancient hedonist Aristippus of Cyrene took a rather direct approach.

Is not then, also, a beautiful woman useful in proportion as she is beautiful; and a boy and a youth useful in proportion to their beauty? Well then, a handsome boy and a handsome youth must be useful exactly in proportion as they are handsome. Now the use of beauty is, to be embraced. If then a man embraces a woman just as it is useful that he should, he does not do wrong; nor, again, will he be doing wrong in employing beauty for the purposes for which it is useful. (Diogenes Laertius, 94)

In some ways, Aristippus is portrayed parodically: as the very worst of the sophists, though supposedly a follower of Socrates. And yet the idea of beauty as suitedness to use finds expression in a number of thinkers. Xenophon’s Memorabilia puts the view in the mouth of Socrates, with Aristippus as interlocutor:

Socrates: In short everything which we use is considered both good and beautiful from the same point of view, namely its use.

Aristippus: Why then, is a dung-basket a beautiful thing?

Socrates: Of course it is, and a golden shield is ugly, if the one be beautifully fitted to its purpose and the other ill. (Xenophon, Book III, viii)

Berkeley expresses a similar view in his dialogue Alciphron, though he begins with the hedonist conception: “Every one knows that beauty is what pleases” (Berkeley 1732, 174, see Carritt 1931, 75). But it pleases for reasons of usefulness. Thus, as Xenophon suggests, on this view, things are beautiful only in relation to the uses for which they are intended or to which they are properly applied. The proper proportions of an object depend on what kind of object it is, and again a beautiful ox would make an ugly horse. “The parts, therefore, in true proportions, must be so related, and adjusted to one another, as they may best conspire to the use and operation of the whole” (Berkeley 1732, 174–75, see Carritt 1931, 76). One result of this is that, though beauty remains tied to pleasure, it is not an immediate sensible experience. It essentially requires intellection and practical activity: one has to know the use of a thing, and assess its suitedness to that use.

This treatment of beauty is often used, for example, to criticize the distinction between fine art and craft, and it avoids sheer philistinism by enriching the concept of ‘use,’ so that it might encompass not only performing a practical task, but performing it especially well or with an especial satisfaction. Ananda Coomaraswamy, the Ceylonese-British scholar of Indian and European medieval arts, adds that a beautiful work of art or craft expresses as well as serves its purpose.

A cathedral is not as such more beautiful than an airplane, … a hymn than a mathematical equation. … A well-made sword is not more beautiful than a well-made scalpel, though one is used to slay, the other to heal. Works of art are only good or bad, beautiful or ugly in themselves, to the extent that they are or are not well and truly made, that is, do or do not express, or do or do not serve their purpose. (Coomaraswamy 1977, 75)

Roger Scruton, in his book Beauty (2009) returns to a modified Kantianism with regard to both beauty and sublimity, enriched by many and varied examples. “We call something beautiful,” writes Scruton, “when we gain pleasure from contemplating it as an individual object, for its own sake, and in its presented form.” (Scruton 2009, 26)

Despite the Kantian framework, Scruton, like Sartwell and Nehamas, throws the subjective/objective distinction into question. He compares experiencing a beautiful thing to a kiss. To kiss someone that one loves is not merely to place one body part on another, “but to touch the other person in his very self. Hence the kiss is compromising – it is a move from one self toward another, and a summoning of the other into the surface of his being.” (Scruton 2009, 48)


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  • Bullough, Edward, 1912. “‘Psychical Distance’ as a Factor in Art and as an Aesthetic Principle,” British Journal of Psychology, 5. Widely anthologized, e.g., in Cahn, Steven and Meskin, Aaron, 2008. Aesthetics: A Comprehensive Anthology, Malden, MA: Blackwell.
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  • Carritt, E.F., 1931, Philosophies of Beauty, London: Oxford University Press.
  • Coomaraswamy, Ananda, 1977, Traditional Art and Symbolism (Selected Papers, volume 1), Princeton: Bollingen.
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Please see my related post:

Shapes and Patterns in Nature





Interconnected Pythagorean Triples using Central Squares Theory

Interconnected Pythagorean Triples using Central Squares Theory



Triples are connected through Squares.



Key Terms

  • Pythagorean Theorem
  • Pythagorean Triples
  • Pythagorean Family of Triples
  • Plato family of Triples
  • 3D Pythagorean Theorem
  • Octahedron in 3D
  • Square in 2D
  • Square and Triangles
  • Pythagorean Tree
  • Geometrical Gear
  • 3D Geometrical Gear
  • Interconnected Triples
  • Tripartite Universe
  • Tri-loka
  • Great Pyramids
  • Set, Osiris, Isis
  • IS RA EL
  • Herma-polis and Helio-polis
  • Set/Thoth and RA/Atum
  • Shadows and Pillers
  • Saturn and Jupiter
  • Moon and Sun
  • Silver and Gold
  • Tamas and Rajas
  • Serpent and Eagle
  • Set and Horus
  • Fourth and Fifth
  • Descent and Ascent
  • Ganesh and Hanuman
  • Mars Planet
  • Sun, Earth, Moon
  • Three Parts of the Soul
  • Fundamental Triplicity
  • Inverted Pipal Tree in Hindu Cosmology
  • Vastu Purush Mandala
  • Brahma in Vedic Physics
  • Brahma, Vishnu, Mahesh/Shiv
  • Ram, Lakshman, Sita
  • Krishna, Baldev/Balram, Subhadra
  • Surya Vanshi  and Chandra Vanshi – Solar and Lunar Dynasties
  • Tripitaka in Buddhism
  • Eightfold path in Buddhism
  • Lotus and Diamond Sutra in Buddhism
  • Vajra and Lotus
  • Indra, Brahma, Buddha
  • Three Gunas – Satva, Rajas, Tamas
  • Three Sheaths – Gross, Subtle, Causal
  • Three Nadis: Pingala(Sun), Ida(Moon), Sushumna(Fire)
  • Three Doshas: Vata, Kapha, Pitta
  • Three Gods: Brahma, Rudra, Vishnu
  • Path: Ascending, Descending, Bidirectional
  • Three inner layers of Sushumna Nadi: Chitrini, Vajra, Brahma
  • Plimpton 322


Pythagorean Family of Triples

  • 3 4 5
  • 5 12 13
  • 7 24 25
  • 9 40 41


Plato Family of Triples

  • 3 4 5
  • 8 15 17
  • 12 35 37
  • 16 63 65


Pythagoras triples explained via central squares

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Anatomy of the Pythagoras’ tree

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Key sources of Research

Pythagoras triples explained via central squares

Luis Teia Gomes


Click to access EJ1093370.pdf



Anatomy of the Pythagoras’ tree

Luis Teia

The University of Lund, Sweden



Click to access EJ1121416.pdf

Click to access Anatomy-of-the-Pythagoras-Tree.pdf




Fermat’s Theorem – a Geometrical View


Article · March 2017


Click to access Fermats-Theorem-a-Geometrical-View.pdf


The Pythagorean geometric gear


Luis Teia

University of Lund, Sweden



Click to access The-Pythagorean-Geometric-Gear.pdf



Geometry of the 3 D Pythagoras ’ Theorem

  • Luis Teia
  • Published 2016






Special Case of the Three-Dimensional Pythagorean Gear

(Australian Senior Mathematics Journal)

Article in Journal of the Australian Mathematical Society ·

December 2018


Click to access Special-Case-of-the-Three-Dimensional-Pythagorean-Gear-Australian-Senior-Mathematics-Journal.pdf


Geometry of 3D Pythagorean Theorem






The Pythagorean Tree: A New Species

H. Lee Price

September, 2008


Click to access 0809.4324.pdf




Pythagoras’ garden, revisited

  • Frank R. Bernhart, H. Lee Price
  • Published 2012