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

My Related Posts

Chausath (64) Yogini Hindu Temples Architecture

Sapta Matrikas (Seven Mothers) and Cosmology

On Holons and Holarchy

Hierarchy Theory in Biology, Ecology and Evolution

Networks and Hierarchies

The Great Chain of Being

Key Sources of Research

Das Mahavidyas – The 10 Tantric Goddesses of Wisdom

Written By Swami Ayyappa Giri

Kriya Tantra Institute

https://kriyatantrainstitute.com/articles/mahavidyasadhana

Goddess Worship in Hinduism: The Ten Wisdom Goddesses of Shaktism

Aishwarya Javalgekar

2012

https://www.academia.edu/32388291/Goddess_Worship_in_Hinduism_The_Ten_Wisdom_Goddesses_of_Shaktism

Dus Mahavidyas – the Ten Forms of the Devi

Ganesh Kumar

https://www.academia.edu/5923232/Dus_Mahavidyas_the_Ten_Forms_of_the_Devi

Ten Mahavidyas-Manifestations Of Cosmic Female Energy

Exotic India
2010, Exotic India Art

https://www.academia.edu/50890236/Ten_Mahavidyas_Manifestations_Of_Cosmic_Female_Energy

Wisdom Goddesses – Mahavidyas and the Assertion of Femininity in Indian Thought

May 2002

Nitin Kumar

Editor http://www.exoticindia.com

https://www.exoticindiaart.com/article/mahavidyas/

About Mahavidya The 10 Forms of Goddess Shakti | Adi Parashakti

https://templesinindiainfo.com/about-mahavidya-the-10-forms-of-goddess-shakti/

Tantric Visions of the Divine Feminine: The Ten Mahāvidyās

By David R. Kinsley

Book

Dasa Mahavidya & Tantra Sastra


Sarbeswar Satpathy
Punthi Pustak, 1992

Book

https://www.exoticindiaart.com/book/details/dasamahavidya-ten-mahavidyas-and-tantrasastra-nag414/

Śakti Iconography in Tantric Mahāvidyās


Sarbeswar Satpathy
Punthi Pustak, 1991

Book

Dasa Mahavidya (Tantrasara)  

Yogiraj Yashpal Bharti

Book, Hindi

Dasamahavidya-Mimamsa

Author(s): Vishnu Datta Rakesh

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

Book

https://www.vedicbooks.net/dasamahavidyamimamsa-p-16126.html

Tantric Yoga and the Wisdom Goddesses: Spiritual Secrets of Ayurveda


David Frawley
Lotus Press, 1994

Book

Change, Continuity and Complexity: The Mahāvidyās in East Indian Śākta Traditions

By Jae-Eun Shin

Book

Transformations and Transfer of Tantra in Asia and Beyond

edited by István Keul

Book

The Mahavidyas: The Powers of Consciousness Conceptualized – Part 1

By Devadatta Kali

Vedanta Society

The Mahavidyas: The Powers of Consciousness Conceptualized – Part 2

By Devadatta Kali

Vedanta Society


Dasa Mahavidya – The Ten Great Wisdoms

Posted on June 19, 2019 by Rati Mehrotra

Das Mahavidyas Presentation

Story of the Origin of the Das Mahavidyas

Dasa Mahavidya

Man blunder

https://manblunder.com/articles/dasa-mahavidya

Dus Mahavidyas

https://www.rudraksha-ratna.com/articles/dus-mahavidyas

Dasha Mahavidya – Part One – Part Four

sreenivasarao’s blogs

The Ten Great Cosmic Powers (Dasa Mahavidyas)

By S.Shankaranarayanan

November 2013

Book

samatabooks@gmail.com

https://www.esamskriti.com/e/Spirituality/Philosophy/The-Ten-Great-Cosmic-Powers-(Dasa-Mahavidyas)-1.aspx

Das Mahavidyas

Gita Press, Gorakhpur

Book, Hindi

https://archive.org/details/HindiBookDashaMahavidyaByGitaPress

Dasha Mahavidya and the 10 dimensions

December 15, 2019

Dasa Mahavidya

Orissa Review 2009

Click to access 48-50.pdf

Dasa Maha Vidya: The Ten Wisdom Goddesses

Sahasrakshi

Dasa Mahavidya

Vinita Rashinkar

2020

The Dasa Mahavidyas

Chamunda Swami ji

Das Mahavidya book set (दस महाविद्द्या)

  • Author – Goswami Prahlad Giri
  • Publisher – Choukhamba Krishnadas Academy
  • 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

Reaching GODHOOD

– SREEVIDYA –

Ten Goddesses / Dasamahavidya

Dasha(10) Mahavidya

(SriVidya aka BrahmaVidya aka ShuddhaVidya)

Learn Kriya Yoga

The Dasa(Ten) Mahavidyas

Hindu Online

Click to access dasamahavidya.pdf

Dasha Maha Vidya: Wisdom from the Ten Directions

Sai Venkatesh Balasubramanian
Bengaluru, India: saivenkateshbalasubramanian@gmail.com

Dus Mahavidyas
the Ten Forms of the Devi

Dolls of India

Dasa Mahavidya – The 10 aspects of Adi Parashakti

Dasha Mahavidya – The Ten Great Sources of Wisdom

Sabda Institute

Maha Vidya

Wikipedia

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

Cantor Sets, Sierpinski Carpets, Menger Sponges

Cantor Sets, Sierpinski Carpets, Menger Sponges

Key Terms

  • Cantor Sets
  • Sierpinski Carpets
  • Menger Sponges
  • Fractals
  • Dimension
  • Georg Cantor
  • Waclaw Sierpinski
  • Karl Menger
  • 3 Steps of Vishnu
  • Maha Maya
  • Space Time Geometry
  • Fractal Space Time
  • Cantorian Space Time
  • 9 x 9 square
  • 8 x 8 square
  • 18 and 81
  • 56 and 65 (64 + 1)
  • Chhapan Bhog
  • Chousath Kala
  • Chousath Yogini
  • Fractal Dimension
  • Noninteger Dimension
  • Euler’s Number
  • 64 Tetrahedron
  • Kal Bhairav
  • Kali
  • Geometry of Space Time
  • Truncated Icosahedron
  • Soccer Ball Geometry
  • Pentagon and Hexagon
  • 9 Planets
  • 27 Nakshatras
  • 14 Lokas
  • Hausdorff Dimension
  • Non Commutative Geometry

Cantor Sets, Sierpinski Carpets, Menger Sponges

  • The Menger Sponge is formed by removing cubes from the original cube in an iterative manner.
  • This fractal has dimension d = log 20 / log 3 = 2.7268 placing
    it a space between two and three dimensions.
  • The Sierpinski carpet has fractal dimension = log 8/ log 3 = 1.8928.
  • The fractal dimension of the ternary Cantor set is DH = ln(2)/ln(3) = 0.6309.

Source: A Short History of Fractal Dimension

Source: A Short History of Fractal Dimension

Source: A Short History of Fractal Dimension

Source: A Short History of Fractal Dimension

Source: A Short History of Fractal Dimension

Source: A Short History of Fractal Dimension

Source: Menger Universal Spaces: Introduction to Fractal Geometry and Chaos

Source: Menger Universal Spaces: Introduction to Fractal Geometry and Chaos

Source: Menger Universal Spaces: Introduction to Fractal Geometry and Chaos

Source: Fractals: The Menger Sponge

Source: THE UNIVERSALITY OF THE HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE

Source: THE UNIVERSALITY OF THE HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE

Source: THE UNIVERSALITY OF THE HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE

Source: Mathematical Impressions: The Surprising Menger Sponge Slice

Source: Mathematical Impressions: The Surprising Menger Sponge Slice

Source: Mathematical Impressions: The Surprising Menger Sponge Slice

Source: Mathematical Impressions: The Surprising Menger Sponge Slice

Source: Mathematical Impressions: The Surprising Menger Sponge Slice

Source: Mathematical Impressions: The Surprising Menger Sponge Slice

Space, Fractal Spacetime, Cantorian Spacetime in Cosmology

Prof. M. S. El Naschie has published many research papers on Fractal Spacetime, Cantorian Spacetime, Dark Energy, and structure and geometry of space.

Please see references.

Source: A Fractal Menger Sponge Space-Time Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy

My Related Posts

Growth and Form in Nature: Power Laws and Fractals

Fractal Geometry and Hindu Temple Architecture

Indra’s Net: On Interconnectedness

Shapes and Patterns in Nature

Shape of the Universe

Cosmic Mirror Theory

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Geometry of Consciousness

Maha Vakyas: Great Aphorisms in Vedanta

The Great Chain of Being

Interconnected Pythagorean Triples using Central Squares Theory

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

The Pillar of Celestial Fire

Key Sources for Research

Mathematical Impressions: The Surprising Menger Sponge Slice

Simons Foundation

Menger Universal Spaces

Introduction to Fractal Geometry and Chaos

Matilde Marcolli

MAT1845HS Winter 2020, University of Toronto

Click to access FractalsUToronto7.pdf

Fractals: The Menger Sponge

LCAO approximation for scaling properties of the Menger sponge fractal

Kazuaki Sakoda

Optics Express Vol. 14, Issue 23, pp. 11372-11384 (2006) • https://doi.org/10.1364/OE.14.011372

Squeezing Pi from a Menger Sponge

Ed Pegg, Wolfram Research 

https://community.wolfram.com/groups/-/m/t/822984

mrly fractals

Inspired by Carlo Mats Vincenti Mitchener
Graphics by Paul Bourke
December 2022

http://paulbourke.net/fractals/mrlymath/index.html

Marley Math: Cantor Sets, Sierpinski Carpets, Menger Sponges, And More

Carlo Mats Vincenti Mitchener

24 April 2022

Click to access mrlymath.pdf

A Fractal Menger Sponge Space-Time Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy

Mohamed S. El Naschie

Department of Physics, University of Alexandria, Egypt

Email: Chaossf@aol.com

International Journal of Modern Nonlinear Theory and Application
Vol. 2  No. 2 (2013) , Article ID: 32969 , 15 pages

 DOI:10.4236/ijmnta.2013.22014

https://www.scirp.org/html/1-2340074_32969.htm

The Mystery of the Menger Sponge

NY Times

2011

New Classes of Regular Symmetric Fractals.

Kak, Subhash (2021):

TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.14714094.v2 

THE GENERALIZATION OF SIERPINSKI CARPET AND MENGER SPONGE IN n-DIMENSIONAL SPACE

YUN YANG

YUTING FENG and 

YANHUA YU

https://doi.org/10.1142/S0218348X17500402

Fractals

Vol. 25, No. 05, 1750040 (2017)

https://www.worldscientific.com/doi/10.1142/S0218348X17500402

Hausdorff Measures and Hausdorff Dimensions of the Invariant Sets for Iterated Function Systems of Geometric Fractals

Md. Jahurul Islam1,*, Md. Shahidul Islam1, Md. Shafiqul Islam2

1Department of Mathematics, University of Dhaka, Bangladesh
2School of Mathematics and Computational Science, University of Prince Edward Island, Canada

Mathematics and Statistics 6(3): 25-33, 2018

DOI: 10.13189/ms.2018.060301

Sound absorption by Menger sponge fractal

Tetsuji Kawabea), Takatsuna Miyazakib), Daisuke Oka, Sin’ichiro Koyanagic), and Atsushi Hinokidani

The Journal of the Acoustical Society of America 125, 2830 (2009); https://doi.org/10.1121/1.3095807

https://asa.scitation.org/doi/full/10.1121/1.3095807

Fractal Dimensions in Circular and Spiral Phenomena.

Kak, Subhash (2022):

TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.21766706.v1 

Sierpinski Carpets as Julia sets for Imaginary 3-Circle Inversions

Daniel M. Look

Williams College

Is Spacetime Fractal and Quantum Coherent in the Golden Mean?

Mae-Wan Ho , Mohamed el Naschie & Giuseppe Vitiello

Global Journal of Science Frontier Research

Volume XV Issue I Version I Year 2015

Real Analysis: The Sierpinski Carpet and its Remarkable Area Explained

Analysis on the Sierpinski Carpet

M. T. Barlow

Centre de Recherches Mathematiques

CRM Proceedings and Lecture Notes

Homeomorphisms of the Sierpinski Carpet

Karuna S. Sangam
Bard College, ks4217@bard.edu

Senior Projects Spring 2018

Bard Undergraduate Senior Projects

A SIERPINSKI CARPET LIKE FRACTAL WITHOUT STANDARD SELF-SIMILAR ENERGY

SHIPING CAO AND HUA QIU

The construction of brownian motion on the Sierpinski carpet

MARTIN T. BARLOW 

RICHARD F. BASS

Annales de l’I. H. P., section B, tome 25, no 3 (1989), p. 225-257 

Click to access AIHPB_1989__25_3_225_0.pdf

Sierpinski

https://gofiguremath.org/fractals/sierpinski/

On the Hausdorff dimension of general Cantor sets,

A. F . Beardon,

Proc. Camb. Phil. Soc, 61 (1965), 679-694.

“Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donne”

W. Sierpiήski,

Comptes Rendus, 162 (1916), 629-642.

THE HAUSDORFF DIMENSION OF GENERAL SIERPINSKI CARPETS

CURT McMULLEN

Nagoya Math. J. Vol. 96 (1984), 1-9

Uniqueness of Brownian motion on Sierpinski carpets. 

Richard F. Bass, Takashi Kumagai, Martin T. Barlow, Alexander Teplyaev, 

J. Eur. Math. Soc. 12 (2010), no. 3, pp. 655–701

DOI 10.4171/JEMS/211

https://ems.press/journals/jems/articles/3456

GEODESICS IN THE SIERPINSKI CARPET AND MENGER SPONGE

ETHAN BERKOVE and DEREK SMITH

Fractals

Vol. 28, No. 07, 2050120 (2020)
https://doi.org/10.1142/S0218348X20501200

https://www.worldscientific.com/doi/10.1142/S0218348X20501200

The Theory of Cantorian Spacetime and High Energy Particle Physics (An Informal Review).

El Naschie, M.S. (2009)

Chaos, Solitons & Fractals, 41, 2635-2646. 
http://dx.doi.org/10.1016/j.chaos.2008.09.059

The Ultimate Unified Physico Mathematical Theory of Nature

M.S. El Naschie

Distinguished Professor, Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt.

Date of publication (dd/mm/yyyy): 22/08/2022

International Journal of Innovation in Science and Mathematics

Volume 10, Issue 4, ISSN (Online): 2347–9051

Click to access IJISM_974_FINAL.pdf

Analysis on Fractal Objects

Uta Freiberg
Technische Universität Chemnitz

December 2005

Meccanica 40(4):419-436
DOI:10.1007/s11012-005-2107-0

https://www.researchgate.net/publication/226551998_Analysis_on_Fractal_Objects

Exponentially Decaying Discrete Dynamical Systems

Yogesh Joshi
CIty University of New York – Kingsborough Community College

Denis Blackmore
New Jersey Institute of Technology

April 2012

Recent Patents on Space Technology 2(1):37-48
DOI:10.2174/1877611611202010037

https://www.researchgate.net/publication/267391812_Exponentially_Decaying_Discrete_Dynamical_Systems


THE UNIVERSALITY OF THE HYPERPOLAR IMAGES OF THE SIERPINSKI CARPET AND THE MENGER SPONGE

Glaser, F. (1997, Fall).

The Cal Poly Pomona Journal of Interdisciplinary Studies, 10, 9-18.

https://scholarworks.calstate.edu/downloads/gx41mk81q?locale=es

EVALUATING THE EXACT INFINITESIMAL VALUES OF AREA OF SIERPINSKIS CARPET AND VOLUME OF MENGERS SPONGE

Yaroslav D. Sergeyev

Dipartimento di Elettronica, Informatica e Sistemistica, Universita` della Calabria, Via P. Bucci, Cubo 42-C, 87030 Rende (CS) – Italy

http://wwwinfo.deis.unical.it/∼yaro yaro@si.deis.unical.it

Chaos, Solitons & Fractals
Volume 42, Issue 5, 15 December 2009, Pages 3042-3046

https://www.sciencedirect.com/science/article/abs/pii/S0960077909003877

Dynamics of fractals in Euclidean and measure spaces

Md. Shahidul Islam and Md. Jahurul Islam 2017

 J. Phys.: Conf. Ser. 890 012058

https://iopscience.iop.org/article/10.1088/1742-6596/890/1/012058/pdf

Cantor sets

March 2010

Ferdinand Chovanec
Armed Forces Academy of General Milan Rastislav Štefánik

https://www.researchgate.net/publication/228747023_Cantor_sets

Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets

by Rebecca Stettin

Ashland University

May 2017

Study of Variants of Cantor Sets Using Iterated Function System

Gen. Math. Notes, Vol. 23, No. 1, July 2014, pp. 45-58

https://www.semanticscholar.org/paper/Study-of-Variants-of-Cantor-Sets-Using-Iterated-Rani-Chugh/a110f82b7bf693815b53d97ad1ae8e74eb99c7c4

Click to access 6_GMN-4682-V23N1.247122947.pdf

Volume of the hyperbolic cantor sets

Habibulla Akhadkulov & Yunping Jiang (2020) 

Dynamical Systems, 35:2, 185-196, 

DOI: 10.1080/14689367.2019.1659754

https://www.tandfonline.com/doi/full/10.1080/14689367.2019.1659754

THE GENERALIZED CAPACITY OF CANTOR SETS

A. F. BEARDON

The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 301–304, https://doi.org/10.1093/qmath/19.1.301

https://academic.oup.com/qjmath/article-abstract/19/1/301/1570320?redirectedFrom=PDF

Box-Counting Dimension of the Cantor Set

https://users.math.yale.edu/public_html/People/frame/Fractals/FracAndDim/BoxDim/CantorBoxDim/CantorBoxDim.html

A Note on the History of the Cantor Set and Cantor Function

JULIAN F. FLERON 

SUNY at Albany

Albany, New York

Mathematics Magazine, Vol. 67, No. 2 (Apr., 1994), pp. 136-140

Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2690689 .

Cantor Set and Its Properties

Zhixing Guo
University of California, Santa Barbara

April 23, 2014

Click to access mathcs103_s2014_zhixing_presentation.pdf

The Cantor Set and the Cantor Function

TMA4225 – Foundations of Analysis

On the measure of arithmetic sums of Cantor sets

Boris Solomyak∗
Department of Mathematics, University of Washington, USA

Indagationes Mathematicae
Volume 8, Issue 1, 1997, Pages 133-141

https://www.sciencedirect.com/science/article/pii/S0019357797833575

Multi Dimensional Cantor Sets in Classical and Quantum Mechanics

M. S. EL NASCHIE

Chaos, Solitons & Fractals Vol. 2, No. 2. pp.211-220. 1992

An Exploration of the Cantor Set

Christopher Shaver
Rockhurst University, shaverc@hawks.rockhurst.edu

Rose-Hulman Undergraduate Mathematics Journal: Vol. 11 : Iss. 1 , Article 1.

2010

An Exact Mathematical Picture of Quantum Spacetime

Mohamed S. El Naschie

Department of Physics, University of Alexandria, Alexandria, Egypt Email: Chaossf@aol.com

Advances in Pure Mathematics, 2015, 5, 560-570

http://dx.doi.org/10.4236/apm.2015.59052

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=57895

On El Naschie’s Fractal-Cantorian Space-Time and Dark Energy—A Tutorial Review

Leila Marek-Crnjac

Department of Mathematics, Technical School Center of Maribor, Maribor, Slovenia

Natural Science, 2015, 7, 581-598

http://dx.doi.org/10.4236/ns.2015.713058

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=61825

High Energy Physics and Cosmology as Computation. 

El Naschie, M. (2016)

American Journal of Computational Mathematics6, 185-199.

doi: 10.4236/ajcm.2016.63020.

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=67938

On the Need for Fractal Logic in High Energy Quantum Physics. 

Naschie, M. , Olsen, S. , He, J. , Nada, S. , Marek-Crnjac, L. and Helal, A. (2012)

International Journal of Modern Nonlinear Theory and Application1, 84-92.

doi: 10.4236/ijmnta.2012.13012.

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=23087

Quantum Entanglement as a Consequence of a Cantorian Micro Spacetime Geometry

M. El Naschie

Journal of Quantum Information Science, Vol. 1 No. 2, 2011, pp. 50-53.

doi: 10.4236/jqis.2011.12007.

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=7623

The Three Page Guide to the Most Important Results of M. S. El Naschie’s Research in E-Infinity Quantum Physics and Cosmology

M. A. Helal, L. Marek-Crnjac and J. He,

Open Journal of Microphysics, Vol. 3 No. 4, 2013, pp. 141-145.

doi: 10.4236/ojm.2013.34020.

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=39841

An Exact Mathematical Picture of Quantum Spacetime. 

Naschie, M. (2015)

Advances in Pure Mathematics5, 560-570.

doi: 10.4236/apm.2015.59052.

https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/journal/paperinformation.aspx?paperid=57895

A guide to the mathematics of E-infinity Cantorian spacetime theory


El Naschie, M. S.

Chaos, Solitons and Fractals, Volume 25, Issue 5, p. 955-964.
Pub Date: September 2005

DOI: 10.1016/j.chaos.2004.12.033

https://www.sciencedirect.com/science/article/abs/pii/S0960077905000688?via%3Dihub

Dimensional symmetry breaking, information and fractal gravity in Cantorian space.

el Naschie MS.

Biosystems. 1998 Apr;46(1-2):41-6.

doi: 10.1016/s0303-2647(97)00079-8. PMID: 9648673.

https://pubmed.ncbi.nlm.nih.gov/9648673/

Time Symmetry Breaking, Duality and Cantorian Space-Time

M. S. EL NASCHIE DAMTP, Cambridge, UK

Chaos, Solitons & Fractals Vol 7. No. 4, pp. 499 – 518, 1996

Click to access 5.Time%20Symmetry%20Breaking,%20Duality.pdf

Renormalization Approach to the Dimension of Diffusion in Cantorian Space

M. S. EL NASCHIE

Appl. Math. Lett. Vol. 8, No. 1, pp. 59-63, 1995

On the Missing Link between Cosmology and Biology 

Mohamed S. El Naschie1, Scott Olsen2, M.A. Helal3, L. Marek-Crnjac4 and S. Nada

International Journal of Innovation in Science and Mathematics

Volume 6, Issue 1, ISSN (Online): 2347–9051

Click to access MissingLinkbetweenCosmologyandBiology.pdf

Information theory and dimensionality of space. 

Kak, S.

Sci Rep 10, 20733 (2020).

https://doi.org/10.1038/s41598-020-77855-9

https://www.researchgate.net/publication/346540015_Information_theory_and_dimensionality_of_space

https://www.nature.com/articles/s41598-020-77855-9

The Ontology of Space

Subhash Kak

2021, Chapman University

https://www.academia.edu/49175956/The_Ontology_of_Space

https://www.academia.edu/49175956/The_Measure_of_Space

Our e-dimensional universe

Subhash Kak
Nov 27, 2020

https://subhashkak.medium.com/our-e-dimensional-universe-febb3a20fa64

Black holes, disk structures, and cosmological implications in e-dimensional space


Kak, Subhash ; Kafatos, Menas

Physics Essays, vol. 35, issue 4, pp. 345-355
Pub Date: December 2022

DOI: 10.4006/0836-1398-35.4.345

https://www.ingentaconnect.com/content/pe/pe/2022/00000035/00000004/art00004;jsessionid=1ngq6d7uavci8.x-ic-live-03

Asymptotic freedom and noninteger dimensionality.

Kak S.


Scientific Reports, 09 Feb 2021, 11(1):3406
DOI: 10.1038/s41598-021-83002-9 PMID: 33564046 PMCID: PMC7873067

https://europepmc.org/article/pmc/pmc7873067

On the dimensionality of spacetime

Max Tegmark†
Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA

Received 10 February 1997

Class. Quantum Grav. 14 (1997) L69–L75

The Topology and Combinatorics of Soccer Balls

Dieter Kotschick

American Scientist, Volume 94

A Short History of Fractal Dimension

DECEMBER 26, 2020 by David D. NOLTE

Box counting fractal dimension of volumetric data

Written by Paul Bourke
April-May 2014

http://paulbourke.net/fractals/cubecount/