Dasa (Ten) Maha Vidyas

Dasa (Ten) Maha Vidyas

Source: The Dasa Mahavidyas/Chamunda Swami ji

काली तारा महाविद्या षोडशी भुवनेश्वरी।
भैरवी छिन्नमस्ता च विद्या धूमावती तथा।
बगला सिद्ध विद्या च मातंगी कमलात्मिका
एता दशमहाविद्याः सिद्धविद्या प्रकीर्तिताः॥

Kali Tara Mahavidya Shorashi Bhuvaneshwari
Bhairavi Chinnamasta cha Vidya Dhumavati Tatha
Vagala Sidhdhavidya cha Matangi Kamalatmika
Iti Das Mahavidya Sidhdhavidya Prakirtita

Source: Dus Mahavidyas

Key Terms

  • Kali Maa
  • Durga
  • Parvati
  • Parvati – Durga – Kali
  • 8 – 16 – 64
  • Shiva – Rudra – Bhairav
  • 10 – 18 – 64
  • Expansion of Series
  • Sun, Moon, and Earth
  • Series to Chamunda
  • Series to Kali
  • Kali Kul
  • Sri Kul
  • Dasa Avatars of Vishnu
  • Dasa Maha Vidyas
  • Aspects of Maa Kali
  • Bhadra Kali
  • Great Knowledge
  • Wisdom Goddesses
  • Terrible and Beautiful
  • Evolution of Soul
  • Evolution of Nature
  • Angry and Peaceful
  • Shakti
  • Shaktism
  • Tantra Science
  • Mothers
  • Dik Pals
  • Ten Directions
  • 9 Planets
  • Sri Vidya

Ten Maha Vidyas

The 10 Mahavidyas are usually named in the following sequence:

  • Kali
  • Tara
  • Tripura Sundari
  • Bhuvaneshvari
  • Bhairavi
  • Chhinnamasta
  • Dhumavati
  • Bagalamukhi
  • Matangi
  • Kamala

Source: The Ten Great Cosmic Powers (Dasa Mahavidyas)

Chapter 1 Disciplines of Knowledge

He had studied the Vedas. Like all vaidiks of his time he took to priesthood and was eking out a living. Driven by poverty he approached someone to teach him a Mantra for getting rich quickly. Having got the Mantra, he took his seat in the front courtyard of his house and began practicing the repetition of the Mantra. Two hours would have passed and the Vaidik saw a beggar woman at attempt to drive her away as he did not want his japa to be interrupted by some words spoken in between. The beggar woman who was in tatters stooped down, patiently untied the knots of a bundle of rags, took out the small coins kept there and before the Vaidik knew what she was doing, threw the coins at his feet. The Vaidik naturally was taken aback and began to remonstrate. “Why, you have been asking for this for the past two hours” said she and went her way. The Vaidik stopped repeating his Mantra in sheer disgust and vowed that never again would he go after such pursuits. 

This incident which happened years ago is quoted here for the flood of light it throws on certain basic principles of Tantra Shastra. Evidently, the man contacted with the help of the formula given to him an entity or a spirit, a low class of deity which caught hold of the human vehicle of the beggar woman and within two hours made her part with the pittance she had in favour of the man. The deity responded quickly, as being in the lower rungs of the cosmic ladder nearest to the earth-plane it was within easy reach of human beings; and its power was limited to grant only so much of money to the one who called for its assistance. 

The Tantra recognizes the one Supreme Deity presiding over everything as the Highest, at the same time admitting the existence as the Highest, at the same time admitting the existence of various Gods and Goddesses. In the words of Sri Kapali Sastriar: “The sages of the Tantra do not see any inconsistency in the position, for they recognise that this creation is not a unitary system but a gradation of worlds spread over a rising tier of consciousness and planes and the various Gods and Goddesses are higher beings, powers and entities, deriving their authority from the Supreme to take their part and act or preside over their spheres of domain. There is a regular hierarchy of Gods some of whom are far above the highest heavens of human reach. But there are also Gods and Goddesses closer to the human level. They are more readily accessible to those who aspire to them and in some cases the seeker on the Tantric path looks to the aid and lead of these deities in his effort. They are endowed with capacities and powers beyond normal human possibility, but they are not all for that reason divine in nature. There are higher and lower classes of them, UCCA and ksudra devatas. Those that are nearest to the earthplane, swarming in the vital world overtopping the physical, are usually of the latter type. They respond very readily to the approaches of those who seek their help but they do so mainly for their own purpose, namely, to get hold of the particular human vehicle and convert it into a centre for their activity on the earth. They may and do answer the call of the seeker in the beginning but in the end they let him down, rather roughly, once their purpose is fulfilled. The seeker is misled; his inner progress comes to a standstill if it does not end in disaster. The Kshudra Devatas mislead the seeker with petty glamorous gifts, induce a false sense of progress and siddhi, prevent the dawn of real jnana which would expose their whole game and succeed in enslaving the man for their purpose at the cost of his soul which is betrayed into misadventure”.

Bu there are also entities of a higher order, benevolent deities, Uccha Devatas and as they occupy higher levels, the seeker has to make an effort to ascent to them. But they take the seeker on the path steadily and safely and ultimately do him the utmost good. There are still higher deities with cosmic functionings nearer to the Supreme Cosmic Godhead who presides over the myriad worlds that are created. Then at the top of this pyramidal structure of the cosmos spreading over an ascending tier of consciousness, at the summit of the innumerable levels and planes of existence there are certain cardinal Godheads, so many facets of the One truth, the Supreme Deity that correspond to the Brahman of the Upanishads. The spiritual disciplines leading to such cardinal Deities are known as Brahma Vidyas. These are also popular as Maha Vidyas, the great paths of discipline or Siddha Vidyas, the lines of quest where fulfillment is assured.

The Sanskrit word vidya is formed from the root vid to know or to understand. Vidya means learning or knowledge and also denotes the way to understanding, the path of knowledge, the Teaching. The mystics, all over the world, in their quest for the highest knowledge followed certain paths, undertook certain disciplines which were kept secret and were revealed only in the esoteric hierarchy of master and disciple. In India, the Upanishads, the repository of ancient wisdom and secret knowledge, mention in the body of their texts certain Vidyas, disciplines of knowledge. It has to be remembered that the Upanishads are not merely texts outlining the philosophical speculations of our ancient seers, as popularly held, but also manuals of Sadhana, practical guide-books on spiritual quest. These are the records of jottings of the reaching the ancient seekers had from their masters, an aid-memoire to remember the direct realizations they had in pursuing various disciplines of knowledge in their Sadhana. Likewise in the Tantra which is acclaimed as the great Sadhana Shastra, a practical manual, we find the great disciplines of knowledge, Maha Vidyas, occupying an important place. Especially where the Supreme is adored as the Great Primordial Goddess, the Tantra classifies the disciplines leading to the cardinal Deities as dasa maha vidyas, the ten great paths of knowledge. These cardinal deities are the ten outstanding personalities of the Divine Mother. Their great names are: Kali, Tara, Tripurasundari, Bhuvaneshwari, Tripura Bhairavi, Chinnamasta, Dhumavati, Bahalamukhi, Matangi and Kamalatmika. 

What may be the precise significance of classifying the Vidyas into ten will be a difficult question to answer. But we can indicate the following: The Supreme Mother is the Transcendent Absolute ineffable immutable. In the act of creation, she subjects herself to time and space. Though the space is actually one vast stretch, for our grasp and understanding, we demarcate the indivisible and infinite Space into ten directions, east, west, south, north, south-east, south-west, north-east, north-west, above and below. Similarly the one infinite Mother is delineated as ten outstanding personalities. Again, knowledge is one and the consciousness is one and the same everywhere. But it is grasped and understood in ten different ways by the ten senses, skin, eye, ear, tongue, nose, mouth, foot, hand, anus and genital. Likewise, the one Truth is sensed in its ten different facets; the Divine Mother is adored and approached as the ten cosmic Personalities, Dasa Maha Vidyas. Each of these ten great Vidyas is a Brahma Vidya. The Sadhaka of any one of these Vidyas attains ultimately, if his aspiration is such, the supreme purpose of life, parama purusartha viz, self-realisation and God-realisation, realizing the Goddess as not different from one’s self. All these Vidyas are benevolent deities of the highest order and so do the utmost good to the seeker of the Vidya. For anyone who takes to any of these ten Vidyas, the Sadhana proceeds on sound lines and is safe and sure. It is not necessary at the beginning for the aspirant to have as his goal the highest aim of life. His aim most probably is the fulfillment of his immediate wants and for that he approaches the Deity. Once an aspirant takes to the Deity, the Deity takes upon itself the Sadhana. This is the characteristic of these Maha Vidyas. Whatever the seeker desires the Divine Mother fulfils it. In the process his devotion to the Deity becomes stronger and stronger and he learns to look upon the Deity for even the most trivial things in life, seeks its guidance at every step and knows to wait on its grace. There starts a living concourse, a concrete intimacy between the devotee and the Deity which is so absorbing and so enthralling that all desires, all aims of the devotee with which he started the Sadhana pale into significance. The whole perspective becomes different and there comes about a change in the attitude of the Sadhaka to life and things. There come about visible signs of communion, concrete evidences of contact, irresistible proofs of the Presence and the unmistakable touches of the Divine’s gracious hand in every happening and in every circumstance. What were at one time miracles become now common-place things and the whole life becomes a happy hastening towards the Supreme Goal. 

Thus nothing short of Self-Realisation, atma saksatkara is the goal of the Vidyas. Of course much depends on the seeker and on his active participation, But even if the seeker stops in the middle of the path, he does not come to any harm. Only his progress is delayed. In the Sadhana there are many pitfalls. The baser emotions may hold sway or the ego may interfere at each stage. Once an aspirant has taken to any of these ten Vidyas, he has to succeed ultimately. He may fail in the present birth. The Vidya will make him take the thread of the Sadhana in the next birth and will continue to give the necessary push to the soul of the aspirant on its onward march. Even one step forward in the path of these Vidyas goes a long way towards the Goal. Nothing goes in vain. In fact, the Tantra categorically declares that only those who have been sufficiently prepared in the previous births can approach the precincts of these Vidyas. For them alone are these well-proven carefully laid-out paths. They are the chosen ones, the men with a mission, the indefatigable toilers on the uphill path of these disciplines of knowledge. 

Because all these Vidyas lead to the ultimate Reality, it does not mean that they are all one and the same. Each Vidya is distinct and distinguishable from the Other. Each is a particular Cosmic function and each leads to a special realization of the One Reality. The might of Kali, the sound-force of Tara, the beauty and bliss of Sundari, the vast vision of Bhuvaneshwari, the effulgent charm of Bhairavi, the striking force of Chinnamasta, the silent inertness of Dhumavati, the paralysing power of Bagalamukhi, the expressive play of Matangi and the concord and harmony of Kamalatmika are the various characteristics, the distinct manifestations of the Supreme Consciousness that has made this creation possible. The Tantra says that the Supreme can be realised at these various points. According to one’s ability and aptitude, one realises the great Goddess, becomes identified with her in her might, in her striking force, in her paralysing power or in her beauty and bliss, in her concord and harmony. If one wants to bathe in a river, one cannot bathe in the whole river; one has to bathe at a spot. Similarly if an aspirant wants to reach the Divine, circumscribed as he is by his receptivity and capacity, he chooses one particular path, takes up for adoration one aspect of the Divine. As his pursuit is exclusive, his progress is quick and his approach becomes direct. Ultimately he attains a perfect identification with the Divine. 

But it is not the intention of the Tantra to limit the capacity of the individual to a particular realisation however perfect it may be. It is true that there are ten Maha Vidyas directly leading to the Supreme Truth, ten chalked out paths, safe and sure to have an immediate and direct approach. For instance if one takes to the Sadhana of the Maha Vidya, Chinnamasta, ultimately one attains a perfect identification with the Supreme. All the same it is a partial identification for Chinnamasta is one facet of the many-faceted Supreme. Not a particular realisation, but a global realisation is required to attain the Total Divine. Pursuing our analogy of bathing in a river, if one wants to have conception of the ebb and flow, the eddies and currents of a river one has to bathe at various spots in the river. This is exactly what one does when one takes a holy bath in a sacred place. For instance, in Kashi, baths in so many bathing ghats are prescribed in order that the bather may have a full realisation of the grandeur of the Divine Ganges. Similarly in these ten great disciplines of knowledge, the seeker can aspire for an integral knowledge and he may, depending on his capacity, come nearer the Total Divine, by having as many realisations as he can.

In fact, the Tantra which is justly famous as a science of synthesis holds that one Maha Vidya leads its Sadhaka to another depending on the need and aspiration of the Sadhaka. The worshipper of Kali has a unique realisation of the Divine, the Terrible. At the same time, the realisation of the Divine, the Auspiscious is available to him if he understands the concept of bhadra kali. Then he goes on to appreciate the correspondence between the other two Vidyas, Tripurasundari and Tripurabhairavi, Tripura the beautiful and Tripura the terrific. In the Sundari Vidya itself worship is prescribed for Mantrini and Dandanatha, the attendants of Lalita Tripurasundari. Mantrini is Syamala or Matangi and Dandanatha corresponds to Bagalamukhi. 

Thus the Vidya of Tripurasundari brings to the aspirant the realisations pertaining to the Vidyas of Matangi and Bagalamukhi.

Here a word of caution is necessary. When it is said that the Vidyas of Matangi and Bagalamukhi are implied in the Vidya of Sundari, the human mind immediately jumps to the conclusion that the Vidya of Sundari is superior to the other two Vidyas. The mind of man subject to the limitations of time and space can understand anything only in relation to time and space. Any new knowledge is immediately related to the old, classified and docketed as anterior or posterior, higher or lower. We reiterate that each of the ten Vidyas leads the seeker to the Supreme Reality. Each is great in its own right and each is equal in all respects to each of the other nine Vidyas. The practices of certain disciplines are widely prevalent, others are less known. For that matter they are not less important. 

Again, an integral realisation is possible in these Vidyas because though they are distince and unique, they have among themselves many characteristics in common. Kali, Chinnamasta, Dhumavati and Bagalamukhi have the common characteristics of Power and Force, active or dormant. Sundari Bhuvaneshwari, Bhairavi, Matangi and Kamalatmika share the qualities of Light, Delight and Beauty. Tara has certain characteristics of Kali and certain others of Sundari and is correlated to Bhairavi, Bagalamukhi and Matangi in the aspect of Sound-Force expressed or impeded. Thus the ten Maha Vidyas fall into three broad divisions of discipline. The Veda lauds three Goddesses, producers of delight, tisro devir mayobhuvah. The Upanishads mention the One unborn, red, white and dark ajam ekam lohita sukla krsnam. The Tantras speak of Kali, the dark, Tara the white and Sundari the red. 

In this ancient land, for ages the worship of these great mighty Personalities of the Mother has been prevalent. The other Vidyas have been practised but they have not come into the lime light. In the southern part of India the Vidya of Sundari, Sri Vidya, has been much in vogue. In the far north and north-west, in Tibet and in Kashmir adoration of Tara is popular. In the north-east parts of the country, especially in Bengal, the cult of Kali is famous. Thus the whole of India is full of adoration for the Divine Mother and the spirit of India has been eternally sustained by the Force Supreme, para sakti. The might of Kali, the wisdom of Tara and the beauty of Sundari have forged and fashioned this ancient race where the first man, the offspring of manu, the thinker, dared to peer with his earthly eyes into the portals of the Beyond.

We shall now take up for the study the respective Vidyas.

Mahavidyas and Avatars of Vishnu

Source: http://shanmatha.blogspot.com/2009/09/dasavatharam-dasamahavidya.html

You wont find any puranic lore for the relation between these Vidyas and Avatharas but the logic of putting the Dasamahavidya with the Dasavataram is related thru perception and thru the Tanric interpretation.
If watched carefully one can find the theory of evolution in the Dasaavathara.
The same way the Dasamahavidyas represent the evolution of the Soul.

The First Avatara in an 100% aquatic fish and the Last is the Man of Purnathvam(complete) – Kalki.
– from a fish to a complete man..

Similarly as Nirguna Swaroopi, she is Kali of Darness;
As Saguna roopi she is Sundari of ultimate Gnana(wisdom)

A sloka from Lalita Sahasranama says “karanguli nakhotpanna Narayana dasa kritih” that from the nails of the ten fingers of the Devi emerged the ten manifestations of Narayana as Dasavataram.

The ten manifestations of devi as Kali, Tara, Tripurasundari, Bhairavi, Bhagalamuki, CHinnamasta, Doomavati, Kamala, Bhuvaneswari and Matangi embody a whole range of attitudes gracious and awesome, benign and destructive. The same force called “Shakti” is the motivational force for the ten incarnations of Vishnu.

She is Vishnu and He is Devi; There is no difference between them.

These comparisons only mean that Amba’s being non-different from Narayana and need not be taken literally.

https://sarwamangala.com/Dasamaha.html

Todala Tantra equates Vishnu‟s ten incarnations with the ten Mahavidyas as follows: “Shri Devi said: Lord of Gods, Guru of the universe, tell me of the ten avatars. Now I want to hear of this, tell me of their true nature. Paramesvara, reveal to me which avatar goes with which Devi. “Shri Shiva said: Tara Devi is the blue form, Bagala is the tortoise incarnation, Dhumavati is the boar, Chinnamasta is Nrisimha, Bhuvaneshvari is Vamana, Matangi is the Rama form, Tripura is Jamadagni, Bhairavi is Balabhadra, Mahalakshmi is Buddha, and Durga is the Kalki form. BhagavatÌ Kali is the Krishna murti.” (Todalatantra 10)

The worship of these is also prescribed as an astrological remedy – for the 9 planets and the Lagna as follows: Kali for Saturn, Tara for Jupiter, Maha Tripura Sundari (or Shodasi-Sri Vidya) for Mercury, Bhuvaneshvari for Moon, Chinnamasta for Rahu, Bhairavi for Lagna, Dhumavati for Ketu, Bagalamukhi for Mars, Matangi for Sun, and Kamala for Venus.

Source: my own compilation

Mahavidya Yantras

Source: Dasa Mahavidya Yantras

Source: Dasa Mahavidya Yantras

Source: Dasa Mahavidya Yantras

Source: Dasa Mahavidya Yantras

Source: Dasa Mahavidya Yantras

Source: Author’s own work

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Key Sources of Research

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2010, Exotic India Art

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Dasa Mahavidya & Tantra Sastra


Sarbeswar Satpathy
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Śakti Iconography in Tantric Mahāvidyās


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Book

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Yogiraj Yashpal Bharti

Book, Hindi

Dasamahavidya-Mimamsa

Author(s): Vishnu Datta Rakesh

ISBN: 9788174531445
Year of Publication: 2004
Bibliographic Information: 218p
Language: English

Book

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samatabooks@gmail.com

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  • Language – Hindi and Sanskrit

Dasa Mahavidyas: the Manifestations of Cosmic Female Energy

Atmanism

Dasa Mahavidya Yantras

Varahamihira

http://varahamihira.blogspot.com/2004/09/dasa-mahavidya-yantras.html

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Learn Kriya Yoga

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Hindu Online

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Dasha Maha Vidya: Wisdom from the Ten Directions

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Indira’s Pearls: Apollonian Gasket, Circle and Sphere Packing

Indira’s Pearls: Apollonian Gasket, Circle and Sphere Packing

Source: Matrikas

Key Terms

  • Circle Packing
  • Disk Packing
  • Sphere Packing
  • Indira’s Net
  • Indira’s Pearls
  • Apollonian Gasket
  • Fractals
  • Square and Circle
  • Circular Temple
  • Earth and Heaven
  • 64 Yogini Hindu Shakti Temples
  • 64 Bhairav
  • 64 Yogini
  • 7 Matrikas
  • Mandala
  • Curvilinear Sierpinski Gasket
  • Poincare Disc
  • Jagannath Puri, Odisha
  • Bhuvneswar, Odisha
  • Konark, Odisha
  • Non Euclidian Geometry
  • Euclidian Geometry
  • Fractal Geometry
  • Sacred Geometry
  • Hyperbolic Geometry
  • 1, 2, 3 where is the 4th
  • 1 2 3 4
  • Fifth and Fourth in Music Theory

Fractals, Circle, and Sphere Packing

Source: Circle Packing Explorations

Source: Circle Packing Explorations

Source: On a Diophantine Equation That Generates AllIntegral Apollonian Gaskets

Source:

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: On the cover: Apollonian packing

What can you do with this space? So asks Andrew Stacey. ‘Fill it’ is the prompt reply, but fill it with what? Maybe like Andrew you want to use a single curve, but I want to use circles. If you do this in the way shown above in blue, the result is called an Apollonian packing, a variant of which can be seen on the cover of this issue.

Here we shall explore the history of this entrancing object, which spans over 2000 years, and percolates into a surprising variety of mathematical disciplines. Starting in the familiar world of Euclidean geometry, Apollonian packings extend into fractal geometry and measure theory; Möbius transformations and the hyperbolic plane; and then on into the distant reaches of geometric group theory, number theory, orbital mechanics, and even ship navigation.

At times we may wonder off into thickets of more obscure mathematics, so those readers who get lost should feel free to skip ahead to the next section.

Apollonius of Perga

Apollonius (c 230 BC) was a Hellenistic mathematician, considered one of the greatest after Euclid and Archimedes. Perhaps his most important work was his eight book treaties Κωνικα on conic sections—once lost to European civilisation, but fortuitously preserved by the more enlightened Middle Eastern scholars and later reintroduced by Edmund Halley in 1710. The same unfortunately cannot be said of Έπαφαι (De Tractionibus or Tangencies). Although now lost, we have accounts of the work from other ancient authors, particularly in the writings of Pappus of Alexandria. In it, Apollonius posed and solved the following problem.

Problem: Given three geometric objects in the plane (points, lines, and/or circles), find all circles which meet all three simultaneously (ie which pass through any points, and are tangent to any lines or circles).

So for example, given three points which don’t lie on the same line, there is exactly one circle which passes through all three. The case which interests us at present is when we are given three circles, each of which is tangent to the other two. In the very special case that all three are tangent at the same point there are infinitely many circles tangent to all three. Usually, however, the circles will be pairwise tangent at three distinct points, in which case there are exactly two other circles tangent to all three simultaneously.

Given three mutually tangent circles (black) there are usually exactly two others (blue) tangent to all three.

This is as far as Apollonius went; the next step would not be taken until 1643, when René Descartes discovered a formula for the size of the two tangent circles, which he wrote in a letter to Princess Elizabeth of the Palatinate. The same formula was later rediscovered by Frederick Soddy and published as a poem in Nature in 1936.

The size of a circle is determined by its radius rr. If rr is small, the circle will be small, but it will also be very curved. We can define the curvature of the circle to be k=1/rk=1/r. Descartes showed that if three given circles are mutually tangent at three distinct points, and have curvatures k1k1, k2k2, and k3k3, then a fourth circle which is tangent to all three has curvature k4k4 satisfying

(k1+k2+k3+k4)2=2(k21+k22+k23+k24)(1)(1)(k1+k2+k3+k4)2=2(k12+k22+k32+k42)

For technical algebraic reasons, sometimes this equation gives negative values for the curvature k4k4, which we can interpret as corresponding to a circle with curvature |k||k|which contains the other circles in its interior. Notice that this equation is quadratic in the variable k4k4, so there are two solutions; these will correspond to the two possibilities for the fourth circle found by Apollonius.

Apollonian packings

So far we have constructed at most 5 mutually tangent circles. The step to infinity may seem obvious, but took another 63 years and some 1900 years after Apollonius. The earliest description seems to appear in a letter from Leibniz to des Bosses (11 March 1706):

Imagine a circle; in it draw three other circles that are the same size and as large as possible, and in any new circle and in the space between circles again draw the three largest circles of the same size that are possible. Imagine proceeding to infinity in this way…

A finite iteration of a nested Apollonian packing similar to the one described by Leibniz. Image: adapted from Todd Stedl, CC BY-SA 4.0

What Leibniz is describing is in fact a nested Apollonian packing, since at each step he fills in every circle as well as the gaps between circles. This early description makes the nested Apollonian packing one of the first fractals, although it wasn’t studied properly until mathematicians like Cantor, Weierstrass, von Koch, and Sierpinski started discovering other fractals in the late nineteenth and early twentieth centuries. This may be because Leibniz was not interested in the mathematical construction, but rather was trying to draw an analogy to argue against the existence in infinitesimals in nature.

Henceforth we shall only consider the un-nested Apollonian packing. As a fractal, it has a number of interesting properties: it is a set of measure 0, which means that if you tried to make it by starting with a disc of metal, and then drilled out infinitely many ever smaller holes (and if you ignore that metal is made out of atoms), then you would finish up with a single piece of metal (you haven’t removed everything), but nevertheless with exactly 0 mass. It has fractal dimension approximately 1.30568, which means that mathematically it lives somewhere between a 1D curve and a 2D area. Finally, if you look at just the portion of an Apollonian packing which lies in the triangular region between three tangent circles, this is homeomorphic to the Sierpinski triangle, which means that one can be bent and stretched to look like the other.

A portion of an Apollonian packing is homeomorphic to the Sierpinski triangle—just squash all of the circles to make them triangular. Images: adapted from Todd Stedl, CC BY-SA 4.0 and Beojan Stanislaus, CC BY-SA 3.0.

There is a curious combinatorial consequence of Descartes’ formula for Apollonian packings. If we start with three mutually tangent circles with curvatures k1k1, k2k2, and k3k3, we can solve (1)(1) to find that the curvatures k+4k4+ and k−4k4− of the other two circles are

k±4=k1+k2+k3±2√k1k2+k2k3+k3k1(2)(2)k4±=k1+k2+k3±2k1k2+k2k3+k3k1

The integral Apollonian packing starting with curvatures -10, 18, 23, and 27. Image: adapted from Todd Stedl, CC BY-SA 4.0.

Now suppose we start constructing an Apollonian packing by drawing four mutually tangent circles whose curvatures k1k1, k2k2, k3k3, and k+4k4+ are all integers. From equation (2)(2) it follows that 2√k1k2+k2k3+k3k12k1k2+k2k3+k3k1 must be an integer since k+4k4+ is an integer, and so k−4k4− is also an integer. Now we can build the packing by filling in a fifth circle wherever we see four mutually tangent circles. By the observation above, if the four circles have integer curvatures, the fifth circle will also have integer curvature. Inductively therefore we will end up with an Apollonian packing consisting of infinitely many tangent circles, all of which have integer curvatures.

Hyperbolic geometry

All these scorpions have the same hyperbolic size.

If you have some familiarity with non-Euclidean geometry, Apollonian packings may remind you of the Poincaré model of the hyperbolic plane. The hyperbolic plane H2H2 is a 2D surface on which we can do geometry just like we can on the flat Euclidean plane. Whereas a sphere has constant positive curvature (it curves the same way in all directions), and the Euclidean plane has constant zero curvature (it’s flat), H2H2 is an infinite surface which has constant negative curvature, which means that at every point it curves in the same way as a Pringle. This negative curvature makes the surface crinkle up on itself more and more as you move out towards infinity, which is inconvenient when we try to work with it. Usually then we represent it on a flat surface so we can draw pictures of it in magazines and the like. One way to do this is with the Poincaré model. This views the hyperbolic plane as a disc. In order to fit the whole infinity of H2H2 into a finite disc, we have to shrink distances as we move out towards the edge of the disc. Using this skewed way of measuring distances, the circular edge of the disc is infinitely far away from its centre.

We can think of an Apollonian packing as living in the Poincaré disc, with the outermost circle of the packing as the boundary circle of H2H2. Then the circles in the packing which are not tangent to this boundary are also circles in the strange hyperbolic way of measuring distance, that is, all points are equidistant from some other point in the plane—the circle’s hyperbolic centre. Circles in the packing which are tangent to the boundary are called horocycles (in Greek this literally means border circle), which are circles with infinite radius in the hyperbolic metric. Horocycles have no analogue in the Euclidean plane.

Something interesting happens when we see what an Apollonian packing looks like in the upper half-plane (UHP) model for H2H2. This model is similar to the Poincaré model, but instead of using a disc, we use the half-plane above the xx-axis {(x,y)∈R2:y>0}{(x,y)∈R2:y>0}, where the xx-axis behaves like the boundary circle and should be thought of as at infinity. There is a problem, in that in the Poincaré disc, the boundary of H2H2 was a circle, and so it closed up on itself. In the UHP, the boundary is a line which doesn’t close up on itself, but these are supposed to be models for the same thing. To fix this, we imagine there is a point at infinity ∞∞ which joins up the two ends of the boundary to form an infinite diameter circle.

If we start with any Apollonian packing living in the Poincaré disc, there is a map from the disc to the UHP preserving hyperbolic distances, under which the outer circle of the packing becomes the xx-axis (together with the point at infinity), and exactly one of the horocycles (one of the circles tangent to the outer circle in the packing) becomes the horizontal line y=1y=1. All other circles and horocycles in the packing are sent to circles which are tangent to each other as before, but are now sandwiched between the lines y=0y=0 and y=1y=1.

If we focus on just those circles which meet the xx-axis we get what are called Ford circles. Remarkably each of these circles is tangent to the xx-axis at a rational number p/qp/q, and has radius 1/2q21/2q2. Moreover every rational number is the point of tangency of one of the circles (see below). Now some magic happens: suppose the Ford circles at a/ba/b and c/dc/d are tangent to each other, then there is a unique circle sandwiched between these two circles and the xx-axis. The rational point at which this circle meets the xx-axis is given by the Farey sum of a/ba/b and c/dc/d

ab⊕cd=a+bc+dab⊕cd=a+bc+d

Note that for this to be well-defined, a/ba/b and c/dc/d must be written in their simplest form. This Farey sum, and the associated Farey sequences FnFn you get by looking at all rational numbers between 0 and 1 which can be written as a fraction with denominator at most nn, turn up in several places across number theory. These include rational approximation of irrational numbers and the Riemann Hypothesis.

Möbius transformations

If you haven’t seen hyperbolic geometry before, you may wonder how we can map the Poincaré disc model to the UHP model, and in such a way that the strange distance measure in the two models is preserved—for a start one is a finite region while the other is an infinite half-plane. The answer is to view both models as living inside the complex plane CC (or more accurately the extended complex plane ˆC=C∪{∞}C^=C∪{∞}): the Poincaré disc is the unit disc {z∈C:|z|<1}{z∈C:|z|<1}, and the UHP is the region above the real axis {z∈C:{z∈C: Im(z)>0}(z)>0}. Then a function like

z↦−iz+1z−1=−iz−iz−1(3)(3)z↦−iz+1z−1=−iz−iz−1

will do the trick. This function is an example of a Möbius transformation, which in general is a complex function of the form

z↦az+bbz+dz↦az+bbz+d

were we require ad−bc≠0ad−bc≠0 so that this function is invertible. The function (3)(3) sends the unit disc to the UHP, but it is not the only Möbius transformation which does this. In fact there are infinitely many such functions, all of which preserve the hyperbolic metric. In the previous section I claimed that starting with any Apollonian packing, we could choose one of these Möbius transformations such that the image had a very specific form, sandwiched between the lines Im(z)=0(z)=0 and Im(z)=1(z)=1.

An exercise: If you have seen Möbius transformations before, you may wish to try and prove that the purported mapping exists yourself. (Hint: remember that Möbius transformations send circles and lines to circles and lines, and are completely determined by their image on 3 distinct points.)

The upshot of this is that all Apollonian packings are the same in the hyperbolic plane, because they can all be mapped to the same packing by (invertible) functions which preserve hyperbolic distance. Once we have started thinking about the Apollonian packing living in the complex plane, the whole world of complex functions is open to us, and we can start to do crazy things. If we don’t restrict ourselves to just Möbius transformations, but see what happens when we apply holomorphic or anti-holomorphic functions to the packing (these are complex functions with a good notion of derivative in the sense of calculus, which in particular have the property that they preserve angles between intersecting curves), we can get some very pretty designs. We need not even require (anti-)holomorphicity. The patterns featured on the front and back covers were drawn in this way.

Beyond the packing

Let us return to Apollonius of Perga. Remember that his treaties Έπαφαι, where he stated and solved the problem of finding tangent circles, is lost to history—how then do we know what he proved and how? The answer is that we don’t. The only record we have appears in the writings of Pappus of Alexandria, who lived some 400 years after Apollonius, but who references many of Apollonius’ works, including six which are no longer extant. All he says of Tangencies is the general problem which Apollonius was interested in, and that he solved it by solving many simple special cases and working up from there.

The first person to reprove Apollonius’ results in ‘modern’ times was Adriaan van Roomen in 1596. His solution, however, does not use ruler and compass constructions, so cannot have been the one Apollonius used. The result was later proved using methods available to Apollonius, and in the way described by Pappus, by van Roomen’s friend François Viéte.

A ship’s location determined by its distance from three points.

The method of Viéte was later reworked and simplified by several mathematicians, including Isaac Newton in his Principia. Newton related the position of the centre of the fourth circle to its distance from the centres of the three circles to which it is supposed to be tangent. This is called hyperbolic positioning or trilateration. Newton used this viewpoint to describe the orbits of planets in the solar system, but it can also be used to help navigate ships, and to locate the source of a signal based on the different times the signal is received at three different locations. In the first world war this was used to locate artillery based on when shots were heard. This is also how modern GPS works (not by triangulation as is commonly believed).

So this 2000-year-old problem in abstract geometry turned out to have extremely useful applications in the real world. The Apollonian packing also shows up in lots of different areas of mathematics. For example, Ford circles inspired the Hardy–Littlewood circle method, an important tool in analytic number theory which was used to solve Waring’s Problem: for an integer kk, can every integer be written as a sum of at most nn kkth powers for some value of nn? This is true: for example, every integer is the sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 2013, Harald Helfgott used the circle method to prove the weak Goldbach conjecture: every odd number greater than 5 is the sum of 3 primes.

To infinity

As a final application, I am a geometric group theorist, and I cannot help but talk about one place the Apollonian packing shows up in my field. Be warned: there is definitely some advanced maths coming up, but if you don’t mind skipping over some of the details, there are some very pretty pictures to make it worthwhile.

It turns out that the extended complex plane ˆCC^ can be thought of as the boundary of 3 dimensional hyperbolic space H3H3. If we model H3H3 as the upper half-space {(x,y,z)∈R3∣z≥0}∪{∞}{(x,y,z)∈R3∣z≥0}∪{∞} then ˆCC^ is identified with the plane {(x,y,z)∈R3∣z=0}∪{∞}{(x,y,z)∈R3∣z=0}∪{∞}. When Möbius transformations act on ˆCC^, they also act on the whole of H3H3, and preserve hyperbolic distance. If we start by choosing just a few Möbius transformations, these generate a group which acts on H3H3. In doing so, the group creates a pattern on the complex plane called its limit set. This is a picture of how the group acts ‘at infinity’. Choosing the Möbius transformations carefully gives a group whose limit set is precisely the Apollonian packing.

A valid arrangement of circles, with their interiors shaded. Note that the straight line is an infinite radius circle through ∞∞, so it still counts.

Let’s be a bit more precise; pick a point p∈ˆCp∈C^ and choose gg pairs of circles (C+i,C−i)gi=1(Ci+,Ci−)i=1g, each of which doesn’t intersect pp. Each circle cuts ˆCC^ into two regions, call the region containing pp the exterior of that circle, and the complementary region the circle’s interior. We also want to arrange things so that no two circles have overlapping interiors (although two circles are allowed to be tangent). Next, for each pair of circles (C+i,C−i)(Ci+,Ci−) choose a Möbius transformation mimiwhich maps C+iCi+ to C−iCi− and which sends the interior of C+iCi+ to the exterior of C−iCi−. The group G=⟨m1,…,mg⟩G=⟨m1,…,mg⟩ generated by these transformations is called a (classical) Schottky group and it acts as a subgroup of the group of isometries of H3H3. Since we chose the circles to have non-overlapping interiors, we can use the delightfully named ‘Ping-Pong Lemma’ to prove that GG is abstractly isomorphic to the free group on gggenerators.

A packing with the four starting circles emphasised in black, and the Schottky group generators shown in pink and blue.

So how do we get a Schottky group whose limit set is the Apollonian packing? We can cheat slightly by working backwards; starting off with the picture we want to create, then we will choose the pairs of circles in the right way. Remember that one way we thought about constructing the Apollonian packing was to start off with four mutually tangent circles and then inductively draw the fifth circle wherever we can. Our strategy will be to choose Möbius transformations which do the same thing. We are helped by the following curious fact which you may want to try and prove yourself (again using Möbius transformations): given any three mutually tangent circles, there is a unique circle (possibly through ∞∞) which passes through all three circles at right angles. Given the four initial circles, there are (43)=4(43)=4 triples of mutually tangent circles, so we let C±1C1± and C±2C2± be the four circles orthogonal to each of these triples, as shown on the left. The corresponding Möbius transformations are :

m1:z↦z−2iz+1m2:z↦(1−i)z+1z+(1+i)m1:z↦z−2iz+1m2:z↦(1−i)z+1z+(1+i)

The limit set of G=⟨m1,m2⟩G=⟨m1,m2⟩ is indeed the Apollonian packing we started with. If we perturb the starting Möbius transformations just slightly by varying the matrix entries (while being careful to ensure that the resulting group acts nicely on H3H3), we get a group whose limit set is a twisted Apollonian packing.

The limit sets of a one parameter family of groups which contains the Schottky group GG. Click the image to view a higher quality version.

Even though some of these perturbed limit sets look like they are still made up more or less of circles, they are in fact made up of a single continuous closed curve which is fractal, and does not intersect itself anywhere. They are examples of Jordan curves and illustrate why the Jordan Curve Theorem is so difficult to prove despite being ‘obvious’. Playing around more with different choices of Möbius generators we can produce even more beautiful examples of fractal limit sets; below are just a few to finish off. If you want to learn more about Schottky groups, their limit sets, and how to draw these pictures, I highly recommend the book Indra’s pearls: the vision of Felix Klein. It is the basis of this final section of this article, and gives details on exactly how you can draw these and many other pictures yourself.

David Sheard

Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry

Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry

Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry

Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry

Source: Circle patterns in Gothic Architecture

My Related Posts

Indra’s Net: On Interconnectedness

Charles Sanders Peirce’s Visual Logic: Diagrams and Existential Graphs

A Calculus for Self Reference, Autopoiesis, and Indications

64 Yogini Hindu Temples Architecture

Fractal Geometry and Hindu Temple Architecture

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Cosmic Mirror Theory

Shape of the Universe

Geometry of Consciousness

Mind, Consciousness and Quantum Entanglement

Cantor Sets, Sierpinski Carpets, Menger Sponges

Interconnected Pythagorean Triples using Central Squares Theory

Key Sources of Research

ON A DIOPHANTINE EQUATION THAT GENERATES ALL APOLLONIAN GASKETS

JERZY KOCIK

Volume 2012 | Article ID 348618 | https://doi.org/10.5402/2012/348618

Click to access Diophantine.pdf

https://www.academia.edu/24595555/On_a_Diophantine_Equation_That_Generates_All_Integral_Apollonian_Gaskets

Circle Packing Explorations.

Francesco De Comite ́
Laboratoire d’Informatique Fondamentale de Lille

University of Sciences and Technology of Lille , France

Francesco.De-Comite@univ-lille1.fr

Bridges 2013: Mathematics, Music, Art, Architecture, Culture, 2013,

Enschede, Netherlands. pp.399–402. hal-00861402

https://www.researchgate.net/publication/278769035_Circle_Packing_Explorations

Circle patterns in Gothic Architecture

Tiffany C. Inglis and Craig S. Kaplan
Published 2012

David R. Cheriton School of Computer Science

University of Waterloo

piffany@gmail.com

Bridges 2012: Mathematics, Music, Art, Architecture, Culture

https://www.semanticscholar.org/paper/Circle-patterns-in-Gothic-Architecture-Kaplan/da91338728b153b9785913d43199a4227853365a

Concept cathedral and “squaring the circle”: Interpreting the Gothic cathedral of Notre Dame de Paris as a standing hymn

Nelly Shafik Ramzy
Department of Architectural Engineering, Faculty of Engineering, Benha University, El Kalyobia, Egypt

Frontiers of Architectural Research

Volume 10, Issue 2, June 2021, Pages 369-393

https://doi.org/10.1016/j.foar.2021.02.001

https://www.sciencedirect.com/science/article/pii/S209526352100008X

https://www.academia.edu/45236439/Concept_Cathedral_and_Squaring_the_Circle_Interpreting_the_Gothic_cathedral_of_Notre_Dame_de_Paris_as_a_standing_hymn

The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry

Nelly Shafik Ramzy

Sinai University

Peregrinations: Journal of Medieval Art and Architecture
Volume 5 Issue 2 135-172
2015

QUADRALECTIC ARCHITECTURE – A Panoramic Review

by Marten Kuilman
Posted on 26 Aug 2013 by quadralectics
Falcon Press (2011) – ISBN 978-90-814420-0-8

https://quadralectics.wordpress.com

An Introduction to the Apollonian Fractal

Paul Bourke
Email: pdb@swin.edu.au
Swinburne University of Technology P. O. Box 218, Hawthorn Melbourne, Vic 3122, Australia.

Click to access apollony.pdf

http://paulbourke.net/papers/apollony/

http://paulbourke.net/fractals/apollony/

Apollonian gaskets and circle inversion fractals

Estimate for the fractal dimension of the Apollonian gasket in d dimensions.

Farr RS, Griffiths E.

Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Jun;81(6 Pt 1):061403. doi: 10.1103/PhysRevE.81.061403. Epub 2010 Jun 24. PMID: 20866417.

https://pubmed.ncbi.nlm.nih.gov/20866417/

On the cover: Apollonian packing

David Sheard explores the rich mathematics and history behind the Apollonian packing, and the cover of issue 11

David Sheard
17 April 2020

On the cover: Apollonian packing

Recursive Apollonian Gasket

Photostream on Flickr

by FDeComite

2011

Recursive Apollonian Gasket

Quadrilaterals & Triangles

Malin Christersson
2019 

http://www.malinc.se/math/geometry/pythagorasen.php

Apollonian circle packings: Dynamics and Number theory

Hee Oh

Yale University
ICWM, 2014

SINGLE LINE APOLLONIAN GASKETS:
IS THE LIMIT A SPACE FILLING FRACTAL CURVE?

L.M.G. FEIJS

University of Technology Eindhoven and LAURENTIUS LAB. Sittard (The Netherlands) l.m.g.feijs@tue.nl

Spatial Statistics of Apollonian Gaskets, 

Weiru Chen, Mo Jiao, Calvin Kessler, Amita Malik & Xin Zhang (2019) 

Experimental Mathematics, 28:3, 263-270,

 DOI: 10.1080/10586458.2017.1385037

Revisiting Apollonian gaskets

Published: Sat 06 August 2022
By Alden Bradford
In Blog.

https://aldenbradford.com/revisiting-apollonian-gaskets.html

Matrikas

Wikipedia

https://en.wikipedia.org/wiki/Matrikas

Indra’s Pearls: The Vision of Felix Klein.

David Mumford, Caroline Series, and David Wright. 

Cambridge University Press,Cambridge, 2002.

Geometry in Art and Architecture

Paul Calter

https://math.dartmouth.edu/~matc/math5.geometry/syllabus.html

The Circular Church Plan

Quadralectic Architecture