Square and Circle of Hindu Temple Architecture

Square and Circle of Hindu Temple Architecture

Source: Khajuraho Group of Monuments/Wikipedia

Khajuraho temples use the 8×8 (64) Vastupurusamandala Manduka grid layout plan (left) found in Hindu temples. Above the temple’s brahma padas is a Shikhara (Vimana or Spire) that rises symmetrically above the central core, typically in a circles and turning-squares concentric layering design (right) that flows from one to the other as it rises towards the sky.[31][46]

Key Terms

Temple in Man

  • Panch Kosha Philosphy
  • Triguna Philosophy
  • Yoga Philosophy
  • Seven Chakras
  • Great Chain of Being
  • Higher/Lower Levels
  • Hierarchy Theory
  • Ascent of Men

Temple of Men

  • Beasts – Men – Devas – Gods
  • Ratha in Architecture
    • Triratha
    • Panchratha
    • Saptaratha
    • Navratha
  • Ayatan Plan Architecture
    • Ekayatan
    • Triayatan
    • Panchayatan
    • Saptayatan
    • Chhadyashikhar
    • Shikharanvit
    • Valabhichhandaj
    • Sandhar
    • Prasad
  • Nagara Architecture
    • Odisha
    • Chandella
    • Solanki Maru Gurjara
  • Mandapa in Architecture
    • Ardh Mandapa
    • Mandapa
    • Maha Mandapa
  • Vastu Purush Mandala
  • Ceilings in Architecture
  • Shikhar (Vimana, Spire)
  • Temple and the Tank
  • Temple and the Pond
  • Temple and the River
  • Temple and the Lake
  • Square and Circle
  • Earth and Heaven
  • As Above, So Below
  • As Below, So Above
  • Amalaka
  • Kalash
  • Garbh Graha
  • Outer to Inner
  • Lower to Higher
  • Purush Sukta
  • Shri Sukta
  • Antarala
  • Pada Devta
  • Lok Pals
  • Dwar Pals
  • Dik Pals
  • Ayadi Calculations
  • Toranas
  • Vitana (Ceiling)
  • Prakash and Vimarsh
  • Terrestial to Celestial

Nagara (North Indian) Hindu Temple Architecture









Source: Symbolism in Hindu Temple Architecture and Fractal Geometry – ‘Thought Behind Form’

Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India

Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India

Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India

Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India

Source: Lakshmana Temple, Khajuraho, Madya Pradesh, India

Source: Lakshmana Temple, Khajuraho, Madya Pradesh, India

Source: The Religious Imagery of Khajuraho

Source: The Religious Imagery of Khajuraho


Source: The Temples of Khajuraho in Central India

Ratha in Architecture

Source: https://en.wikipedia.org/wiki/Ratha_(architecture)

Square and Circle within a Temple

  • Ground Plan (Square) and Ceiling (Circle)
  • Using Circles to Generate Squares for Ground Plan
  • Pillars with Square Base and Circular Top

Vastu Purusha Mandala

Source: Hindu temple architecture/Wikipedia

Source: Hindu temple architecture/Wikipedia

Source: Hindu temple architecture/Wikipedia

Source: Khajuraho Group of Monuments/Wikipedia

Architecture of the temples

The layout plan of Kandariya Mahadeva Temple. It uses the 64 pada grid design. Smaller Khajuraho temples use the 9, 16, 36 or 49 grid mandala plan.[42]

Khajuraho temples, almost all Hindu temple designs, follow a grid geometrical design called vastu-purusha-mandala.[43] This design plan has three important components – Mandala means circle, Purusha is universal essence at the core of Hindu tradition, while Vastu means the dwelling structure.[44]

The design lays out a Hindu temple in a symmetrical, concentrically layered, self-repeating structure around the core of the temple called garbhagriya, where the abstract principle Purusha and the primary deity of the temple dwell. The shikhara, or spire, of the temple rises above the garbhagriya. This symmetry and structure in design is derived from central beliefs, myths, cardinality and mathematical principles.[45]

The circle of mandala circumscribe the square. The square is considered divine for its perfection and as a symbolic product of knowledge and human thought, while circle is considered earthly, human and observed in everyday life (moon, sun, horizon, water drop, rainbow). Each supports the other.[31] The square is divided into perfect 64 sub-squares called padas.[43]

Most Khajuraho temples deploy the 8×8 (64) padas grid Manduka Vastupurushamandala, with pitha mandala the square grid incorporated in the design of the spires.[42] The primary deity or lingas are located in the grid’s Brahma padas.

Khajuraho temples use the 8×8 (64) Vastupurusamandala Manduka grid layout plan (left) found in Hindu temples. Above the temple’s brahma padas is a Shikhara (Vimana or Spire) that rises symmetrically above the central core, typically in a circles and turning-squares concentric layering design (right) that flows from one to the other as it rises towards the sky.[31][46]

The architecture is symbolic and reflects the central Hindu beliefs through its form, structure, and arrangement of its parts.[47] The mandapas, as well as the arts, are arranged in the Khajuraho temples in a symmetric repeating patterns, even though each image or sculpture is distinctive in its own way. The relative placement of the images are not random but together they express ideas, just like connected words form sentences and paragraphs to compose ideas.[48] This fractal pattern that is common in Hindu temples.[49] Various statues and panels have inscriptions. Many of the inscriptions on the temple walls are poems with double meanings, something that the complex structure of Sanskrit allows in creative compositions.[29]

All Khajuraho temples, except one, face sunrise, and the entrance for the devotee is this east side.

An illustration of Khajuraho temple Spires (Shikhara, Vimana) built using concentric circle and rotating-squares principle. Four spires (left) are shown above, while the inside view of one Shikara ceiling (right) shows the symmetric layout.

Above the vastu-purusha-mandala of each temple is a superstructure with a dome called Shikhara (or Vimana, Spire).[44] Variations in spire design come from variation in degrees turned for the squares. The temple Shikhara, in some literature, is linked to mount Kailash or Meru, the mythical abode of the gods.[31]

In each temple, the central space typically is surrounded by an ambulatory for the pilgrim to walk around and ritually circumambulate the Purusa and the main deity.[31] The pillars, walls, and ceilings around the space, as well as outside have highly ornate carvings or images of the four just and necessary pursuits of life – kama, artha, dharma, and moksa. This clockwise walk around is called pradakshina.[44]

Larger Khajuraho temples also have pillared halls called mandapa. One near the entrance, on the east side, serves as the waiting room for pilgrims and devotees. The mandapas are also arranged by principles of symmetry, grids, and mathematical precision. This use of same underlying architectural principle is common in Hindu temples found all over India.[50] Each Khajuraho temple is distinctly carved yet also repeating the central common principles in almost all Hindu temples, one which Susan Lewandowski refers to as “an organism of repeating cells”.[51]



Layout of a Hindu temple pursues a geometrical design known as vastu-purusha-mandala, the name of which is derived from the three vital components of the design namely Vastu meaning Vaas or a place of dwelling; Purusha, meaning the Universal principle; and Mandala meaning circle. Vastupurushamandala is a mystical diagram referred in Sanskrit as a Yantra. The symmetrical and self-repeating model of a Hindu temple demonstrated in the design is derived from the primary convictions, traditions, myths, fundamentality and mathematical standards. According to Vastupurushamandala, the most sacred and typical template for a Hindu temple is the 8×8 (64) grid Manduka Hindu Temple Floor Plan also referred as Bhekapada and Ajira. The layout displays a vivid saffron centre with intersecting diagonals which according to Hindu philosophy symbolises the Purusha. The axis of the Mandir is created with the aid of the four fundamentally significant directions and thus, a perfect square is created around the axis within the available space. This square which is circumscribed by the Mandala circle and divided into perfect square grids is held sacred. On the other hand, the circle is regarded as human and worldly that can be perceived or noticed in daily life such as the Sun, Moon, rainbow, horizon or water drops. Both the square and the circle support each other. The model is usually seen in large temples while an 81 sub-square grid is observed in ceremonial temple superstructures. Each square within the main square referred as „Pada‟ symbolise a specific element that can be in the form of a deity, an apsara or a spirit. The primary or the innermost square/s of the 64 grid model called Brahma Padas is dedicated to Brahman. The Garbhagruha or centre of the house situated in the Brahma Padas houses the main deity. The outer concentric layer to Brahma Padas is the Devika Padas signifying facets of Devas or Gods which is again surrounded by the next layer, the Manusha Padas, with the ambulatory. The devotees circumambulate clockwise to perform Parikrama in the Manusha Padas with Devika Padas in the inner side and the Paishachika Padas, symbolising facets of Asuras and evils, on the outer side forming the last concentric square. The three outer Padas in larger temples generally adorn inspirational paintings, carvings and images with the wall reliefs and images of different temples depicting legends from different Hindu Epics and Vedic stories. Illustrations of artha, kama, dharma and moksha can be found in the embellished carvings and images adorning the walls, ceiling and pillars of the temples.



Source: Space and Cosmology in the Hindu Temple

Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho

Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho

Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho

According to Hindu philosophy, the main goal of man’s life is to achieve the ultimate liberation from the illusionary world where he suffers from his endless rebirth. Krishna says in Bhagavad Gita,9 if man worships, devotes and meditates to the manifested form of the wholeness of infinity, and sees the truth of its manifestation; he will surpass the confusion of the never-ending cycles of rebirth in this physical world, and finally will assimilate with the God. 10 Hindu calls it moksha.11 It brings the ultimate peace and harmony in man’s life. But how one, from the physical world, can practice to realize this ultimate truth of the creation? Brihatsamhita12 and Sthapatyaveda13 give the solution as the temple which should act as the microcosm of the cosmos [9]. It should be the bridge for the man of physical world to the God of divine world [10].

To connect the physical world with the divine world and to reflect the truth of creation, the layout of cosmos was copied graphically in the foundation of temple. Here, a mythical incident was interwoven where a formless being covered the sky and was, immediately, arrested down to the earth by the creator Brahma and other gods. This supernatural fact was depicted graphically by vastu purusha mandala, where vastu refers to physical environment, purusha refers to energy, power or cosmic being, and mandala is the diagram or chart. Central portion of the mandala represents the place of Brahma and other portions symbolize the other gods according to their capability. By laying down this metaphysical diagram in the foundation, various supernatural forces are captured beneath the temple whereas its centre is the source of cosmic energies. The basic shape of the vastu purusha mandala is a square which represents the earth [11]. Its four sides depict the four cardinal directions. It also symbolizes the order, the completeness of endless life and the perfectness of life and death [10]. According to Hindu philosophy, our mundane life is controlled by the number four—four castes, four stages of life, four great epochs or mahayugas, four heads of Brahma, the four Vedas etc. [12].

There are various types of vastu purusha mandala, which are nothing but the squares grids, produced from the basic shape; namely, a square which is known as sakala mandala. Each smaller square within the grid is called one pada. The number of pada may vary from 1, 4, 9, 16, 25 and so on 1024, where it follows the geometric progression of 1, 2, 3, 4, 5,…..,32 of common ratio 2. The mandala having even numbers of pada in its grid known as yugma squares mandala whereas the mandala, having odd numbers of pada known as ayugma squares mandala. Vastu purusha mandala is also known as different distinctive names according to the numbers of pada within the grid. The mandala having 1,4,9,16,25 and 36 numbers of pada within the grid are known as sakala mandala, pechaka mandala, pitah mandala, mahapitah mandala, manduka chandita mandala and para- masayika mandala, respectively14.

Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho

Source: Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho

Source: Maṇḍala in Architecture: Symbolism and Significance for Contemporary Design Education in India

Source: Maṇḍala in Architecture: Symbolism and Significance for Contemporary Design Education in India

Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings

Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings

Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings

Source: Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings

Source: Role of Fractal Geometry in Indian Hindu Temple Architecture














Types of Vastu Purush Mandala

Source: Indian Architectural Theory: Contemporary Uses of Vastu Vidya

  • Sakala (1 x 1) = 1 Square
  • Pechaka (2 x 2) = 4 Squares
  • Pitha (3 x 3) = 9 Squares
  • Mahapitha (4 x 4) = 16 Squares
  • Upapitha (5 x 5) = 25 Squares
  • Ugrapitha (6 x 6) = 36 Squares
  • Sthandila (7 x 7) = 49 Squares
  • Manduka (Chandita) (8 x 8) = 64 Squares
  • Paramashayika (9 x 9) = 81 Squares

Types of Ceilings (Vitana) of Temples

Source: A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India

  • Samatala vitāna decorated with padmaśila
  • Ksipta vitāna of Nābhicchanda order decorated with kola courses
  • Ksipta vitāna of Nābhicchanda order decorated with gajatalu courses
  • Ksipta vitāna of Nābhicchanda order in a set of diagonally arranged squares decorated with kola courses
  • Ksipta vitāna of Nābhicchanda order in a set of nine arranged squares decorated with kola courses
  • Ksipta vitāna of Nābhicchanda order in a set of eight circles intersected by a big circle in the center
  • Sama-ksipta vitāna decorated with figural groups in the boxes
  • Karotaka of Sabhmarga order
  • Ksipta vitāna of Nābhicchanda order made of three diminishing squares decorated with kola courses

Questions on Relationships


  • What is a relationship between the Temple and the Tank (Kunda)?
  • What is a relationship between the Temple and the Pond (Talab)?
  • What is a relationship between the Temple and the River (Nadi)?
  • What is a relationship between the Temple and the Lake (Jheel)?

Square and Circle in a Group

Shiva temples (Squares) and Shakti Temples (Circular)

(Group of Squares and Group of Circles)

  • Bateswer Group of Temples + Mitawali Chausath Yogini Temple (Circular)
  • Khajuraho Group of Temples + Chausath Yogini Temple (Rectangular)
  • Bhuvneswar Group of Temples + Hirapur Chausath Yogini Temple (Circular)

Temple Architecture Canonical Texts

Source: Rediscovering the Hindu Temple: The Sacred Architecture and Urbanism of India

My Related Posts

Fractal Geometry and Hindu Temple Architecture

Fractal and Multifractal Structures in Cosmology

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

Chausath (64) Yogini Hindu Temples Architecture

Sapta Matrikas (Seven Mothers) and Cosmology

Dasa (Ten) Maha Vidyas

Key Sources of Research

Khajuraho Group of Monuments



Khajuraho Group of Monuments



Maṇḍala and Practice in Nāgara Architecture in North India

Michael W. Meister
Journal of the American Oriental Society
Vol. 99, No. 2 (Apr. – Jun., 1979), pp. 204-219 (16 pages)



Geometry and Measure in Indian Temple Plans: Rectangular Temples

Michael W. Meister
Artibus Asiae
Vol. 44, No. 4 (1983), pp. 266-296 (31 pages)



Measurement and proportion in Hindu temple architecture

Michael W Meister
1985, Interdisciplinary science reviews


The Hindu Temple, Vol 1

Stella Kramrisch,

Motilal Banarsidass, ISBN 978-81-208-0222-3

The Temples of Khajuraho in Central India

Krishna Deva

Ancient India, 15


Temple Architecture Khajuraho and Brihadeshwar

May 2022
Project: Indian Art and Architecture
Authors: Aparna Joshi



Dr. Manoj Kumar
Deptt Of AIHC & Archaeology
Indira Gandhi National Tribal University

Click to access IGNTU-eContent-418924529482-MA-AIHC-4-Dr.ManojKumar(AIHC&Arch)-HistoryofAncientIndianArchitecture-II-2.pdf


Based on the text of Krishna Deva

Published by
The Director General


A Comparative Analysis of the Temples of Khajuraho and the Ruling Chandellas of India

Chelsea Gill

May 2008
A Senior Thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Archaeological Studies University of Wisconsin-La Crosse


Homage to Khajuraho

Mulk Raj Anand

Click to access 10136.pdf




Vidya Dehejia

Click to access Dehejia-Reading-love-imagery-1998.pdf

Sculptural Representation on the Lakshmana Temple of Khajuraho in the Light of Prabodhachandrodaya

Devangana Desai

The Religious Imagery of Khajuraho

Devangana Desai

Click to access khajuraho_desai.pdf

Lakshmana Temple, Khajuraho, Madya Pradesh, India

Khajuraho Temples a Journey from Real to Surreal

By Dr Priyaankaa Mathur

April 7 2021











Khajuraho photo essay

Dr. Uday Dokras
2021, Indo Nordic Author’s Collective


Khajuraho In Perspective (Khajurāho in Perspective)

Kalyan Kumar Chakravarty
1994, Khajurāho in Perspective published by Commissioner, Archaeology and Museums, Madhya Pradesh


Khajuraho Temples: symbolism of Sacred Union of the Divine – Feminine Energy & Masculine Consciousness

Medhavi Davda.


“Sacred space and symbolic form at Lakshmana Temple, Khajuraho (India),”

Dr. Cristin McKnight Sethi,

in Smarthistory, May 15, 2016, accessed March 10, 2023, https://smarthistory.org/lakshmana-temple/.

Investigating Architectural Patterns of Indian Traditional Hindu Temples through Visual Analysis Framework

March 2022

Aditya Kumar Singh
Maulana Azad National Institute of Technology, Bhopal

Vinay Mohan Das
Maulana Azad National Institute of Technology, Bhopal

Yogesh Kumar Garg
Maulana Azad National Institute of Technology, Bhopal

Mohammad Arif Kamal
King Fahd University of Petroleum and Minerals

Civil Engineering and Architecture 10(2):513-530


Maṇḍala in Architecture: Symbolism and Significance for Contemporary Design Education in India

Navin Piplani Ansal University India
Tejwant Singh Brar Ansal University India

IAFOR Journal of Education: Studies in Education

Volume 8 – Issue 4 – 2020

The Infinite Space
Manifestation of bindu and mandala in architecture

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

Jaffer Adam Ayub Khan

B.Arch (Madras)., M.Sc.Arch (Bartlett)., RIBA., RAIA

School of Architecture and Design College of Design and Social Context RMIT University

August 2017



edited by
with contributions by
Claus Ambos, John Baines, Gary Beckman, Matthew Canepa, Davíd Carrasco, Elizabeth Frood, Uri Gabbay, Susanne Görke, Ömür Harmanşah, Julia A. B. Hegewald, Clemente Marconi, Michael W. Meister, Tracy Miller, Richard Neer, Deena Ragavan, Betsey A. Robinson, Yorke M. Rowan, and Karl Taube

Papers from the Oriental Institute Seminar Heaven on Earth
Held at the Oriental Institute of the University of Chicago 2–3 March 2012



Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings

Young Jae Kim
Ph.D. in Architectural History and Theory (University of Pennsylvania, USA)


Research – Application of Bindu and Mandala as a model for Cultural and Sacred Architecture

Jaffer Khan
May 27, 2017

Listen how the wise one begins construction of a house for Visnu: Chapters 1-14 of the Hayasirsa Pancaratra

Raddock, Elisabeth Eva



Click to access qt83r393vc.pdf

Evolution and Interconnection: Geometry in Early Temple Architecture

Author : Sambit Datta

Published in: Digital Techniques for Heritage Presentation and Preservation


Dr. Sambit Datta
Senior Lecturer, Deakin University sdatta@deakin.edu.au


On the Idea of the Mandala as a Governing Device in Indian Architectural Tradition

Sonit Bafna
Georgia Institute of Technology

March 2000

The Journal of the Society of Architectural Historians 59(1):26-49


Orissan Temple Architecture and Vastu Purusha Mandala for evolution process

December 2012

Partha Sarathi Mishra
Sri Sri University


Shape and Geometry of Orissa Temple Architecture

Partha Sarathi Mishra

IIT Roorkee M Tech Thesis



Wikipedia Pages on








Temple Architecture-Devalaya Vastu –Part One (1 of 9)

sreenivasarao’s blogs

“Naturalizing Buddhist Cosmology in the Temple Architecture of China: The Case of the Yicihui Pillar.”

Tracy Miller

In Heaven on Earth: Temples, Ritual, and Cosmic Symbolism in the Ancient World

(The University of Chicago Oriental Institute Seminars, No. 9), edited by Deena Ragavan, 17-39. Chicago: University of Chicago Oriental Institute, 2013.


“Of Palaces and Pagodas: Palatial Symbolism in the Buddhist Architecture of Early Medieval China.” 

Tracy Miller

Frontiers of History in China 10.2 (2015): 222-263.


Seeds and Mountains: The Cosmogony of Temples in South Asia

Michael W Meister



Role of Fractal Geometry in Indian Hindu Temple Architecture

Dhrubajyoti Sardar
M.Arch Scholar
Architecture & Planning Department, IIT Roorkee

Roorkee, Uttarakhand, India

S. Y. Kulkarni
Professor & Former Head Architecture & Planning Department, IIT Roorkee Roorkee, Uttarakhand, India

International Journal of Engineering Research & Technology (IJERT) 

ISSN: 2278-0181 Vol. 4 Issue 05, May-2015



Physical Fractals: Self Similarity and Square-Integratibility

Akhlesh Lakhtakia

Penn State

Speculations in Science and Technology 18, 153-156, 1995

Dancing Architecture: The parallel evolution of Bharatanātyam and South Indian Architecture

Kavitha Jayakrishnan

University of Waterloo
Master of Architecture Thesis 2011



Building Science of Indian Temple Architecture

Shweta Vardia


2008 MS Thesis

Universidade do Minho, Portugal

Shapes, Patterns and Meanings in Indian Temple Architecture

Tanisha Dutta*, V. S. Adane

Department of Architecture and Planning, Visvesvaraya National Institute of Technology Nagpur, India *Corresponding author: ar.tanisha.dd@gmail.com

Received July 17, 2018; Revised August 20, 2018; Accepted November 05, 2018

American Journal of Civil Engineering and Architecture, 2018, Vol. 6, No. 5, 206-215

Available online at http://pubs.sciepub.com/ajcea/6/5/6 ©Science and Education Publishing DOI:10.12691/ajcea-6-5-6

Click to access ajcea-6-5-6.pdf

Hindu Temple: Models of a Fractal Universe. 

Trivedi, K. (1993).

International Seminar on Mayonic Science and Technology,


The Visual Computer 5, 243–258 (1989). https://doi.org/10.1007/BF02153753


Click to access Hindu_Temple_Models.pdf

Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho

IasefMd Riana Jin-HoParka HyungUk Ahna DongkukChangb

aDepartment of Architecture, Inha University, South Korea

bDepartment of Architecture, Chosun University, South Korea

Received 4 May 2006, Revised 21 July 2006, Accepted 15 January 2007, Available online 23 April 2007.

Building and Environment
Volume 42, Issue 12, December 2007, Pages 4093-4107

Click to access 2007_02.pdf



Symbolism in Hindu Temple
Architecture through Fractal Geoemtry- ‘Thought Behind Form’.

Dutta, T., & V.S.Adane. (2014).

International Journal of Science and Research (IJSR), 489-497.


Temples of Odisha- the Geometry of Plan Form

Rinku Parashar

Assistant Professor Department of Architecture Engineering NIT, Raipur, 492010, India

Dr Abir Bandyopadhyay

Professor & Head Department of Architecture Engineering NIT, Raipur, 492010, India

IJIRST –International Journal for Innovative Research in Science & Technology| Volume 2 | Issue 10 | March 2016 ISSN (online): 2349-6010

Click to access IJIRSTV2I10036.pdf

Shape and geometrical study of fractal cosmology through Orissan Temple architecture

Partha Sarathi Mishra

january 2013

“Investigating Architectural Patterns of Indian Traditional Hindu Temples through Visual Analysis Framework,”

Aditya Kumar Singh, Vinay Mohan Das, Yogesh Kumar Garg, Mohammad Arif Kamal ,

Civil Engineering and Architecture, Vol. 10, No. 2, pp. 513-530, 2022.

DOI: 10.13189/cea.2022.100211.


“Infinite Sequences in the Constructive Geometry Of Tenth-Century Hindu Temple Superstructures”,

Datta, S.,

School of Architecture and Building Deakin University 1, Gheringhap Street Geelong VIC 3219 AUSTRALIA 


Nexus Network Journal – Vol.12, No. 3, 2010 471 – 483

DOI 10.1007/s00004-010-0038-0; published online 15 September 2010
Kim Williams Books, Turin



Evolution and Interconnection: Geometry in Early Temple Architecture


Corpus ID: 238053244




BMS School of Architecture, Yelahanka, Bangalore, India


Click to access ascaad2016_042.pdf





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 2015


Authors: Mishra, Partha Sarathi

MS Thesis, IITR 2012



Vastu Purusha Mandala – A human ecological framework for designing living environments.

Venugopal, Jayadevi

In Jetty, C, Chandra, B, Bhashyam, A, & Prabhakara, R (Eds.) Proceedings of International Conference on Advances in Architecture and Civil Engineering (AARCV 2012), Volume 2.
Bonfring, India, pp. 870-877.

Rediscovering the Hindu Temple:
The Sacred Architecture and Urbanism of India

Vinayak Bharne and Krupali Krusche

Book, ISBN (10): 1-4438-4137-4, ISBN (13): 978-1-4438-4137-5



Space and Cosmology in the Hindu Temple

Subhash Kak

Presented at Vaastu Kaushal: International Symposium on Science and Technology in Ancient Indian Monuments, New Delhi, November 16-17, 2002.

Gender and space in temple architecture

D. Midhila


Hindustan Institute of Technology and Sciences, Vijayawada, Andhra Pradesh

Dr. R. V. Nagarajan


Hindustan Institute of Technology and Sciences, Padur, Chennai, Tamilnadu

International Journal of Advance Research, Ideas and Innovations in Technology


The Role of Five Elements of Nature In Temple Architecture 

Ar. Snigdha Chaudhary

International Journal of Scientific & Engineering Research Volume 8, Issue 7, July-2017 1149 ISSN 2229-5518

Trends in Fractal Dimension in Laxman and Kandariya Mahadev Temples, Khajuraho

Tanisha Dutta1,* and Vinayak S. Adane2

1Phd Research Scholar, Department of Architecture and Planning, Visvesvaraya National Institute of Technology, Nagpur- 440010, India.

2Professor, Department of Architecture and Planning, Visvesvaraya National Institute of Technology, Nagpur- 440010, India. (*Corresponding author)

nternational Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 3 (2018) pp. 1728-1741

© Research India Publications. http://www.ripublication.com

Hindu Temple Fractals

William J Jackson


Indian Architectural Theory: Contemporary Uses of Vastu Vidya

Vibhuti Chakrabarti

Oxford University Press (1999)

ISBN 0195650417

Pancha Kosha Theory of Personality

Biswajit Satpathy
Sambalpur University

July 2018

The International Journal of Indian Psychology 6(2)



Yagyik Mishra
SGT University

January 2019

World Journal of Pharmaceutical Research 8(13):413



Maharaj K. Raina, Ph.D.
Woodland, CA

The Journal of Transpersonal Psychology, 2016, Vol. 48, No. 2


Shiv Jakhar

International Journal of Yogic, Human Movement and Sports Sciences 2019; 4(1): 1382-1384


(Discrimination of the Five Sheaths)


Development of Panch Kosha

Human Resource Management

Indian Perspectives on Human Quality Development

Pathashala, MHRD, INDIA

Click to access 1513923600M6Q1DevelopmentofPanchkosha.pdf

The Pancha Koshas: Keys To Unveil Our True Self.

Aiswarya. H. N & Sowmya C. Nair

International Ayurvedic Medical Journal {online} 2019 {cited November, 2019}

Available from: http://www.iamj.in/posts/images/upload/2095_2100.pdf

Click to access 2095_2100.pdf

“Differentiation of the Five Sheaths”


Composed by Sri Swami Vidyaranyaji

on the 32 Lectures delivered by Swami Advayanandaji, Acharya at the Chinmaya International Foundation, Veliyanad, Kerala. from April 28th 2017 – November 7th, 2017

Unit-5 Concept of Pancha Koshas, Concept of Panchmahabhutas

Issue Date: 2020
Publisher: Indira Gandhi National Open University, New Delhi
URI: http://egyankosh.ac.in//handle/123456789/59786
Appears in Collections: Block-2 Yogic Concepts of Health




The Divine Life Society Sivananda Ashram, Rishikesh, India
Website: swami-krishnananda.org

“Vastu Purusha Mandala”, 

Vini Nathan,

Nexus Network Journal, vol. 4, no. 3 (Summer 2002),


Hindu Temple and the Structure of Human Body




Vaastu Purusha Mandalam

by Dr. V. Ganapati Sthapati


Ayadi Calculations

by Dr. V. Ganapati Sthapati


The Hindu Temple: An Introduction to its Meaning and Forms.

Michell, G. (1977, 1988).

Chicago, London: University of Chicago Press.

The temple architecture of india .

Hardy, A. (2007).

England: John Wiley & Sons Ltd.

Encyclopaedia of Indian temple architecture, South India, Lower Drāviḍadēśa 200 BC – AD 1324.

Meister M. W. (Ed.) (1999).

American Institute of Indian Studies and Manohar Publishers & Distributors. ISBN 81-7304-298-5.

Canonical Texts

The oldest and most complete canonical group of ancient texts includes

  • Mayamata
  • Mānasāra
  • Samarāṇgana Sūtradhāra
  • Rajavallabha
  • Vishvakarma Praksha

Architecture of Mānasāra.

Acharya, P. K. (2010).

New Bharatiya Book Corporation. ISBN 978-81-8315-133-7.

Architecture of Mānasāra: illustrations of architectural and sculptural objects.

Acharya P. K. (2010).

New Bharatiya Book Corporation.

Samarāṅgaṇa Sūtradharā of Bhojadeva: an ancient treatise on architecture.

Sharma, S. K. (2012). 

Parimal Publications. ISBN 978-81-7110-302-7.

The square and the circle of the Indian arts.

Vatsyayan, K. (1997). 

Abhinav Publications.

Vastu-Purusha mandala.

Vatsyayan, K. (1986).

In C. Kagan (Ed.), Vistāra – The architecture of Indiacatalogue of the exhibition, The Festival of India. https://architexturez.net/doc/az-cf-123753

Indian temple architecture: Form and transformation.

Hardy, A. (1995). 

Indira Gandhi National Centre for the Arts and Abhinav Publications.

ISBN 81-7017-312-4

Mayamatam: the treatise of housing, architecture and iconography.

Dagen, B. (1994). 

Indira Gandhi National Centre for the Arts. ISBN: 81-208-1226-3.

Indian architectural theory and practice: Contemporary uses of Vastu Vidya

Chakrabarti, V. (1998). 

Curzon. ISBN: 0-7007-1113-9.


Ms. TABASSUM SIDDIQUI Assistant Professor

Kantipur International College, Kathmandu, Nepal

International Journal For Technological Research In Engineering
Volume 8, Issue 3, November-2020

Cosmologies of India

John McKim Malville
University of Colorado Boulder

August 2014
In book: Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (pp.1-16)


Indian Temple Architecture, 3 Vols

Ananya Gandotra, 2011

An Encyclopedia of Hindu Architecture


Prasanna Kumar Acharya

Manasara Series, Volume I to VII

1946, Oxford Univ. Press

Shiva’s Waterfront Temples Reimaging the Sacred Architecture of India’s Deccan Region

Subhasini Kaligotla

PHD Thesis

Columbia University, 2015

”Indian Temple Architecture: form and Spaces”


Research Paper, Department of Architecture & Planning, I.I.T .Roorkee, Roorkee, India, 1991.

Understanding the Architecture of Hindu Temple: A Philosophical Interpretation

A. Bandyopadhyay

IJAEE, Vol 13, No 12, 2019

Temple Architecture

A Brief Overview and Its Symbolism

Purushottama Bilimoria

Temple Architecture and its Symbolism

Hindu Temple – A Systematic Study

Dr B Suresha

JETIR, December 2022, Vol 9, Issue 12

Significance of Mathematics and Geometry – Formation of Temples in India: The golden Era of Evolution

Santoshi Mishra and Sirisha David

ijeru, 2021, 01, 01, 050-055

The Iconography of Hindu Temple: Idea and Image

Sunil Kumar Patnaik

Transcendence from the matter to the mind

Understanding Spatial Experiance of a Temple Through its Architecture

Thesis, 2018 – 2019

Sneha B Krishnan

Ansal University, India

Maṇḍala in Architecture: Symbolism and Significance for Contemporary Design Education in India

Navin Piplani Ansal University India

Tejwant Singh Brar Ansal University India

IAFOR Journal of Education: Studies in Education

Volume 8 – Issue 4 – 2020

Mandapa: Its Proportion as a tool in Understanding Indian Temple Architecture

Ragima N Ramachandran

International Journal of Scientific & Engineering Research Volume 10, Issue 7, July-2019 2104 ISSN 2229-5518


Geometry of Human Form: Art and Science of Charles Henry

Geometry of Human Form: Art and Science of Charles Henry

Key Terms

  • Sacred Geometry
  • Sphere Packing
  • Light
  • Geometry
  • Optics
  • Pyramids
  • Vesica Piscis
  • Limestone
  • Close-packed reflective spheres
  • Golden Mean
  • Pi
  • Computer Imaging
  • Photography
  • Animation
  • Graphic Art
  • Sculpture

Meeting Prof. Charles Henry

I met Prof. Charles Henry in August of 2010. He was resident of Richmand, Virginia where I also live.

We met for lunch at one of the restaurant. He was kind to bring a CDROM with images and animations of his work for me.

We talked about Fractals and Packed Spheres.

He had a book with him on Pyramids by Christopher Dunn – The Giza Power Plant.

I kept in touch with him for next few days, We exchanged few emails.

Geometry of Human Form

Source: http://www.people.vcu.edu/~chenry/



New Discoveries

Linking The Great Pyramid to the Human Form


Copyright 1997 – 2000   CHARLES R. HENRY

All Rights Reserved

Professor, Department of Sculpture

Virginia Commonwealth University

Richmond, Virginia


This site is best viewed on Microsoft Internet Explorer 4.0 or higher

with screen set to 1024 X 768 pixels, 24 bit …16 million colors.

Set … View/Text Size … to Meduim

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For more than twenty years, I have been studying the image generating properties of reflective spheres stacked in 52 degree angle pyramids. The 52 (51.827) degree angle slope of the sides of The Great Pyramid in Cairo, Egypt embodies the Golden Mean which is the ratio that is used in Nature to generate growth patterns in space. Sacred Geometry studies such primal systems which reveal the unity of the cosmos by representing the relationships between numbers geometrically. The Vesica Piscis is one of the most fundamental geometrical forms of this ancient discipline and it reveals the relationship between the The Great Pyramid and the 2 dimensional expansion of a circle of one unit radius R as shown in Figure 1. This relationship is more completely described in The New View Over Atlantis by John Michell published by Thames and Hudson.

Figure1.gif (5952 bytes)

Figure 1         Vesica Piscis in 2 Dimensions


In the early 1970s, I became very interested in the three dimensional representation of this geometry and I visualized this as a three dimensional pyramid inside two intersecting spheres shown in Figure 2.

figure2.jpg (14868 bytes)

Figure 2          Vesica Piscis in 3 Dimensions


In an effort to visualize these 3D relationships in yet another way, I stacked reflective Christmas Tree balls in an inverted pyramid shell. I discovered that the patterns of multiple reflections created on the interior surfaces of reflective spheres stacked at this angle produce images that relate to the human form as photographed and shown in Figure 3 and Figure 4.


figure3.jpg (16443 bytes)Figure 3


figure4.jpg (27044 bytes)Figure 4

I made many pyramidal configurations of reflective spheres with different light sources and I photographed the patterns on the interior surfaces from many points of view. Another is shown here in Figure 5.

figure5.jpg (18367 bytes)Figure 5



Then in 1977, I discovered one stacking structure and viewpoint that produced a very convincing image of an archetypal human face. This structure of 10 spheres (2 5-ball pyramids) forming a cluster is shown in Figure 6.

figure6.jpg (8183 bytes)Figure 6



The face image is shown in the lower third of the pattern on one sphere inside this 10 sphere cluster and is shown in Figure 7.

figure7.jpg (34593 bytes)Figure 7   When I made this photograph, the structure was enclosed in a mirrored pyramid.



Later, I realized that the most natural structure for enclosing would be another 10 sphere, 2 pyramid structure that would totally enclose a smaller but similar cluster.  I worked out the math to find that by multiplying the inner sphere’s diameters by Pi gives the dimension for the outer sphere’s diameters as shown in Figure 8.

figure8.jpg (14227 bytes)Figure 8        One sphere is removed from the outer cluster to reveal the inner cluster. However, the inner cluster must be upside down with respect to the outer cluster to fit inside.


The expansion by Pi reinforced my suspicion that this 10 sphere cluster is a fundamental unit that is linked to the properties of three dimensional space.

Close-packed reflective spheres clustered in this concentric shell structure produce an optical distribution network that links the Golden Mean and Pi. The Golden Mean is expressed in the 52 degree angle pyramid structure and Pi is expressed in the ratio of the diameter to the circumference of each sphere of course; but it is also expressed in the ratio of the sizes of spheres in the 10 spheres within 10 spheres concentric shell structure that I discovered. This concentric shell structure could continue to expand with many shells and still retain the same ratio between shells. It was not until 1991 that I was able to build and photograph the images inside a ten-within-ten (2 shell) structure. I used 10 – 6″ diameter spheres and 10 – 19″ diameter hemispheres. The structure is shown here in Figure 9.

figure9.jpg (19664 bytes)Figure 9



Figure 10 and Figure 11 are photographs of the interior of the 10 spheres within 10 spheres cluster shown in Figure 9.

figure10.jpg (17924 bytes)Figure 10

figure11.jpg (13114 bytes)Figure 11

The photographs that I made from this 10-within-10 sphere structure revealed the inherent limitations of photography for this work. The final images were not what I was seeing with my eyes while directly viewing the interior of the structure. But this approach did reveal a more complete face form and I also realized that I would need at least 10 more spheres (about 60″ in diameter at $1500 each) to complete the enclosure and remove the remaining gaps in the images. Also, the lighting system was limited to the exterior and it was very difficult to control the positioning, color and brightness. So, in 1992, I purchased a computer to model these structures with ray-tracing software which enabled me to investigate more thoroughly the relationship between this cluster geometry and the archetypal images generated therein. During this tour into cyberspace, I could take a camera into the sacred chamber central to the concentric shells of reflective spheres which are simulated by a computer program (Real 3D by Realsoft Oy, Finland) that most accurately renders the effects of real-world light sources and records the patterns generated by multiple reflections on metal surfaces. With this method of investigation, I am able to more conveniently control the many variables which led to these discoveries and conclusions:

1.) At least 40 spheres (4-10 ball shells) are necessary to enclose the central area and fill in the gaps in the images.

2.) Most of the lights should be point sources placed at the points of contact between the innermost spheres of the structure; although additional symmetrically-paired, point-lights are necessary in the central area.

3.) Some of the innermost spheres are reduced in size and they can float within certain areas in the central space. Figure 12 shows typical positions and sizes.

4.) The camera position and field of view as shown in Figure 12 produces the most convincing image of the human form.

fig12.jpg (12536 bytes)Figure 12

5.) The image of this artificial anthropoid that is produced in these structures can be animated when the positions and sizes of certain spheres are modulated as shown in the three animations listed below. 

animation 1            .8 MB … estimated download time at 56k … 3.5 min.

animation 2           2.4 MB …  estimated download time at 56k … 10 min.


fig13.jpg (94727 bytes)

Figure 13       Human Form From Sacred Geometry

The computer image in Figure 13 was made from the camera position and field of view shown in Figure 12.


Figure 14   is from the same camera position (shown in Figure 12) but the lens set at a very wide angle.

fig14.jpg (110837 bytes)Figure 14



Figure 15 is from the same camera position (shown in Figure 12) with the camera lens set to zoom in.

fig15.jpg (92100 bytes)Figure 15  



Figure 16   is a stereo image and it shows the interior of the cluster in 3D. Stare through the images with your eyes focused at a distant point and the two images will turn into three images and the center image will appear in 3D.

fig16.jpg (42554 bytes)Figure 16    

In 1996, I produced an animation Sacred Spaces (6 minutes) which has been screened in many national and international film festivals and it has won some awards (see resume). I have also produced Flesh Tones (5.5 Minutes) another animation completed in 1997 and I have produced many high resolution images for prints/slides (some examples are shown in Figures 13 through 36) which I have presented with the video animations at lecture presentations. These images are stills from the animations and they demonstrate the variety of image generating that is inherent in this system. I have concentrated on exhibiting the videotape documentation of my work thus far because it is the most portable presentation format that describes this research most completely.

fig17.jpg (24384 bytes)Figure 17

fig18.jpg (27816 bytes)Figure 18

fig19.jpg (34590 bytes)Figure 19

fig20.jpg (27735 bytes)Figure 20

fig21.jpg (44628 bytes)Figure 21


This interdisciplinary research has taken me into many related areas of study.   Geometry, Optics, Ancient History of Art and Religion, Computer Imaging, Photography, Animation, Graphic Art and of course Sculpture are the major connecting disciplines that have contributed to this work.

I feel that I have rediscovered some of what was a highly developed understanding of Mankind’s relationship to the Universe and this knowledge was utilized and documented in the geometry of ancient structures. Sacred Geometry, the study of the unity of the cosmos, demonstrates relationships between Number and Space and the Human Form. It was used in the construction of ancient glyphs and monuments thereby preserving the knowledge of these principles of Natural Law for future generations.   This construction of reflective spheres may embody the technology that produced the animated images of the deities in the temples of antiquity. The Tree of Life which is a graphic representation of the interaction between cosmic forces is shown in Figure 22. It is found in many ancient texts of the Kabbalah.

fig22.jpg (11450 bytes)Figure 22.

I realized that The Tree of Life graphic can also represent the 10 sphere cluster made with 2 5-ball pyramids as shown in Figure 23.

fig23.jpg (7127 bytes)Figure 23

The construction of this structure of clustered metal reflective spheres (offering bowls) is well within the capability of many ancient cultures and with the addition of a few glass lenses, these images could be projected onto walls or into smoke. Perhaps there is some Truth behind the smoke and mirrors of Ancient Religion … perhaps it is geometry … Sacred Geometry.

This research which is documented in four computer animations Sacred Spaces, Flesh Tones, Our Mothers and Sacred Spaces 2, in color computer prints, and in color slides has given me new insight into the motives that may have inspired the construction of The Great Pyramid.

1998 – 2000 update

The images shown in Figures 24 – 35 were made with 50 spheres and 144 point-light sources.  Each of these images was made with unique brightness, color and value settings for various groupings of lights. 

fig24.jpg (49005 bytes)Figure24

fig25.jpg (61843 bytes)Figure25

fig26.jpg (45712 bytes)Figure26

fig27.jpg (30806 bytes)Figure27

fig28h.jpg (95562 bytes)Figure28 horizontal

fig28v.jpg (34955 bytes)Figure28 vertical

fig29.jpg (59673 bytes)Figure29

fig30.jpg (42660 bytes)Figure30

fig31.jpg (53032 bytes)Figure31

fig32.jpg (36716 bytes)Figure32

InFigure32,   the camera is aimed at the sphere opposite the face shown inFigure 31.Figure 12 shows the camera position for Figure 31.  The image in Figure 32 was formed when the camera position was rotated 180 degrees around the vertical axis  shown in Figurre 12    and  zoomed in.   The face image (a child?) in Figure 32 is much smaller  than the face (mother?) in  Figure 31.  

fig33h.jpg (72697 bytes)Figure33horizontal

fig33v.jpg (81648 bytes)Figure33vertical

figure34.jpg (54241 bytes)

Figure 34


Figure35This is a stereo 3D image that requires shutter glasses to view and your monitor must be set to interlace mode.



fig36.jpg (170331 bytes)

Figure 36            Stereo Image for Cross-Eyed Viewing

With your monitor at arm’s length away, focus on a point 6 inches in front of your nose (put your index finger 6 inches in front of your nose and focus on it). You will see a third image in 3D between the two images on the monitor (at the tip of your finger). Shift your attention from your finger to this third/middle image which will appear in 3D.



The cross-eyed viewing method is perhaps the most effective way to put the viewer inside the cluster to see the human image as it would exist in 3D from the cameras position as seen in Figure 12. There are many more identifiable images in this clustering geometry viewed from this position and from other camera positions and even more images with other color settings for the point lights.

fig36l.jpg (158045 bytes)Figure 36 Large

fig36zi.jpg (39939 bytes)Figure 36 Zoom-in  This is the image on the forehead of the face in Figure 36 Large.                     

fig36zob.jpg (162376 bytes)Figure 36 Wide-angle Large

fig36zod.jpg (148518 bytes)Figure 36 Very wide-angle Large

fig37.jpg (53779 bytes)

Figure 37          Rods connecting centers of nearest neighbors in 3 shells



I’m now very curious about the relationship between Sacred Geometry and Sacred Music and the Human Form. Number relates to all that science measures by virtue of the way that 3D Space is defined. Number is also used to measure Time … and, as Pythagoras observed, Music is a manifestation of Number in Time. The distribution of sounds i.e. amplitude and frequency, may well find an idealized model in the 10 within 10 sphere, space-filling, close-packing geometric system. The representation of this geometry with sticks or strings or rods as shown in Figure 37 (in which the centers of nearest neighbor spheres in three shells are joined) may represent the ideal space-filling matrix of linear oscillating elements. It may also be used to define spatially distributed, hierarchical, cellular arrays.



The camera positions on the symmetric planes within the cluster produce a bilateral symmetry that we identify with animal and human form. Naturally, we would expect this symmetry in the idealized images of higher life forms. The multiplicity of idealized beings in this cluster of reflective spheres suggests the presence of The Company of the Gods as described in many ancient Egyptian texts. It seems that the face images occur on a vertical plane linking the centers of spheres. There seem to be faces facing faces and faces within faces throughout this reflective environment. My guess is that the sacred part of what I have discovered is a result of the way the deity put the higher life forms in three dimensions.

This cluster geometry may have other properties that would be useful for spatial organization. The nesting of 10 reflective spheres within 10 reflective spheres geometry produces a distribution system that could be used for processing of optical information between the interior to the exterior of the structure. Because of the spatial distribution of the reflective spheres in two concentric shells of ten each, optical information must be reflected and diverged in order to enter or exit the system with the exception of a few radially arrayed directions. The system becomes a more selective filter of optical information as more shells are added to the structure. This inside-to-outside transformation/translation should have many practical applications in pattern recognition tasks. For example, any point source of coherent light (laser light) anywhere outside the two-shell cluster will produce a unique light distribution pattern on the inside as viewed from the center area of the cluster. This pattern could be recorded in a holographic medium and the exterior point source could then be reconstructed using conventional holographic means.

Is Number (Geometry and Time) the link between Art, Science and Religious experience? The language of number is perhaps the most convincing form of expression between humans and between humans and the Gods. We think and imagine in visual forms. Einstein constructed his mathematics based on mental images. He said that he would first try to visualize a space/time image and then mathematize it. We use mental images to construct possible scenarios of the future so that we don’t have to live out each one in “reality”. Words and pictures and mathematical formulas are ways to document, test, realize, and communicate these visions. Although there is seldom a need to mathematize images, we sense that it would be possible. We know that we could count and number the grains of sand on the beach. The geometry of our visions is what makes them real to us and it allows us to mentally work on them and to integrate them convincingly into our life here in 3D.

I’m not quite sure how the physical human form fits into the grand scheme of things but it does seem to be a result of the space-filling, spherical, close-packing geometric system that I’ve discovered and it is indeed “Sacred Geometry” by virtue of the definition God gave to three dimensions. I don’t think this geometric system is the matrix for all life systems but I think it can serve as a model for the interactions between the various dimensional realities in which we are immersed. It may also guide us in our attempt to develop new sciences and technologies that utilize the forces that operate in the regions that we now call consider paranormal.

According to contemporary Superstring Theory as described by Dr. Michio Kaku in his many recent writings, the mathematics that most appropriately describes the forces of nature requires an expression in ten dimensions. Einstein tried to describe the forces of nature in the mathematics of 3D and Time and found that the formulas were not broad enough to include all of the forces. We can perceive 3D and Time. The other 6D in Superstrings are hidden from our normal senses due to their incredibly small size according to Dr. Kaku. Our instincts inform us that there are more than 3 Dimensions and Time in the universe and the possiblity that something else exists mathematically beyond our perceptual horizon drives my curiosities about our possible links to these worlds. The new science of parapsychology has discovered many ways that humans can perceive by means beyond the physical senses and it has found that there are some people that are more capable of extrasensory perception than others. It may well be that our only contacts with this duality of nature are through numbers and mathematics on one hand or through dreams, remote viewing, telepathy, Ouija boards and tarot cards on the other. This would certainly confirm the existence of a God with a sense of humor




I am interested in any information relating this technology to ancient religious traditions.  Any references that you send will certainly be appreciated and I will certainly credit any references in future publications.

A more detailed version of this research is now available on CD-ROM and it includes the 10 minute animation Sacred Spaces 2 in streaming format.

I am also making available unlimited editions of selected images on this website which will be printed in very high resolution on archival paper with archival inks.

For more specific information about the availability, formats, sizes and pricing of the prints and CD relating to this work you may contact me at:


Related Material




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This page does not reflect the official position of Virginia Commonwealth University

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This site was last updated on 07/23/02

My Related Posts

Platonic and Archimedean Solids

The Great Chain of Being

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

Sapta Matrikas (Seven Mothers) and Cosmology

Chausath (64) Yogini Hindu Temples Architecture

Dasa (Ten) Maha Vidyas

On Holons and Holarchy

Fractal and Multifractal Structures in Cosmology

Fractal Geometry and Hindu Temple Architecture

Interconnected Pythagorean Triples using Central Squares Theory

Key Sources of Research

Charles Henry

VCU, Richmond, VA


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.


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

Das Mahavidyas – The 10 Tantric Goddesses of Wisdom

Written By Swami Ayyappa Giri

Kriya Tantra Institute


Goddess Worship in Hinduism: The Ten Wisdom Goddesses of Shaktism

Aishwarya Javalgekar



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Sarbeswar Satpathy
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Sarbeswar Satpathy
Punthi Pustak, 1991


Dasa Mahavidya (Tantrasara)  

Yogiraj Yashpal Bharti

Book, Hindi


Author(s): Vishnu Datta Rakesh

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



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  • Publisher – Choukhamba Krishnadas Academy
  • Language – Hindi and Sanskrit

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Sai Venkatesh Balasubramanian
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Fractal and Multifractal Structures in Cosmology

Fractal and Multifractal Structures in Cosmology

Key Terms

  • Spectral triple
  • Non commutative geometry
  • K-homology
  • Sierpinski Gasket
  • Cosmic Web
  • Cosmic Geometry
  • Menger Sponge
  • Apollonian Gasket
  • Apollonian Circles
  • Apollonian Spheres
  • Sphere Packing
  • Circle Packing
  • Ford Circles
  • Apollonian Strip
  • Farey Sequence
  • Fractals
  • Multifractals
  • Swiss Cheese Cosmology
  • Standard cosmological framework
  • Hubble diagram
  • Spatially homogeneous and isotropic spacetime
  • Swiss-cheese cosmological model
  • Strongly inhomogeneous on small scales
  • Einstein field equations
  • Cosmological constant
  • Large scale structure of Universe
  • Cosmology: dark matter – black hole physics
  • Einstein-Straus Swiss-cheese model
  • Swiss-Cheese model based on an Einstein- de Sitter cosmology
  • Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology
  • Homogeneous and isotropic: Friedmann universe
  • Homogeneous but not isotropic: Bianchi IX mixmaster models
  • Isotropic but not homogeneous? ⇒ Swiss Cheese Models
  • Noncommutative Cosmology
  • Square and Circle
  • Cube and Sphere
  • Incommensurability
  • Symmetry Breaking
  • Fractal Bubbles Model
  • Dodecahedron Model
  • Cosmic Topology
  • Algebraic Geometry

Inhomogeneous models of universe : an alternative to dark energy

Source: https://www.observatoiredeparis.psl.eu/inhomogeneous-models-of-universe-an-alternative.html

The problems of the homogeneous cosmological models

The equations of general relativity are such involved that few exact analytical solutions usable in cosmology or astrophysics are known. This is one of the reasons which contributed to the past and current success of the simplest model, the homogeneous model. Its main drawback is that 95% of the content of the Universe is unexplained. To make this so-called “concordance” model compatible with the cosmological data, more than 20% of dark matter and 75% of dark energy have been injected in it. However, the nature and properties of these new components are unknown in physics and they have been observed neither in laboratory experiments nor in the Universe. When the first observations of the light curves of remote type Ia supernovae were made, more than ten years ago, and when their interpretation in an homogeneous framework gave rise to the notion of dark energy, other proposals were made to explain these observations. The effect of the inhomogeneities was one of them.

The first inhomogeneous models

The first inhomogeneous models used to solve this problem were spherically symmetrical. Some among them were able to reproduce several cosmological data sets as well as or even better than the “concordance” model, without any need for dark energy. But, of course, they could only consider radial inhomogeneities.

Swiss-cheese models 

New more realistic Swiss-cheese-type models have been used afterwards. But their “holes” were also spherically symmetric which made them rather unphysical and prevented them to solve efficiently the dark energy problem. The models which appear the best adapted to solve this problem are Swiss-cheeses whose “holes” have no symmetry at all. An exact solution of general relativity able to model such “holes” exists, but it is much more difficult to implement because of its complexity. It is currently developed by a team composed of a researcher from the LUTH, two researchers from the N. Copernicus Astronomical Center (Warsaw) and one researcher from Cape Town University.

Source: Review: The Cosmic Web

By the early 1980s, it was clear to astronomers that galaxies were not randomly distributed across the universe but instead, in many cases, concentrated in clusters. One model developed by American astronomers saw these clusters as high-density “meatballs” isolated in a low-density universe. An alternative model developed by Soviet astronomers offered a “Swiss cheese” approach, with galaxies tracing out an intricate honeycomb pattern, with the voids, rather than galaxy clusters, being isolated. 

Around the same time, cosmologists developed the inflation model of the immediate aftermath of the Big Bang. Gott argued that neither the meatball nor the Swiss cheese models for galaxy clusters were consistent with inflation, and offered an alternative “sponge” model, where there is a complex interconnection of high- and low-density regions: the titular “cosmic web.” (This was inspired in part, he writes, by high school research he did on topology that earned him second prize in the Westinghouse Science Talent Search.)

New galaxy surveys created maps of structures that, at first, appeared to support the Swiss cheese model. But, over time, those structures grew ever larger and more intricate, and came to support Gott’s sponge model. Computer simulations, which also became more sophisticated, were able to produce models of galaxy structures using the sponge model that were similar to what astronomers were seeing. These giant structures that exist in the universe today, he writes, are the fossil remnants of random quantum fluctuations from the first instant after the Big Bang.

Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Source: Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Menger Sponge Space-Time

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

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

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

Apollonian Packing

Source: Lie sphere geometry in lattice cosmology

Source: Lie sphere geometry in lattice cosmology

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

Cantor Sets, Sierpinski Carpets, Menger Sponges

Geometry of Consciousness

Interconnected Pythagorean Triples using Central Squares Theory

Fractal Geometry and Hindu Temple Architecture

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Indra’s Net: On Interconnectedness

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Spectral Action Models of Gravity and Packed Swiss Cheese Cosmologies

Matilde Marcolli

MAT1314HS Winter 2019,

University of Toronto T 12-2 and W 12 BA6180

Click to access IntroNCGToronto11.pdf

Click to access SwissCheeseSlidesSMClass.pdf


Spectral action models of gravity on packed swiss cheese cosmology

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Classical and Quantum Gravity, Volume 33, Number 11
DOI 10.1088/0264-9381/33/11/115018

Multifractal Analysis of Packed Swiss Cheese Cosmologies

J. R. Mureika
W. M. Keck Science Center, The Claremont Colleges, Claremont, CA 91711 USA
Email: jmureika@jsd.claremont.edu
C. C. Dyer
Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 1A7 Canada

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Chapter 8: Fractal and Multifractal Structures in Cosmology

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https://doi.org/10.1142/10335 | February 2018


Discovery of Cosmic Fractals

https://doi.org/10.1142/4896 | 

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The Cosmic Web: Mysterious Architecture of the Universe

J Richard Gott
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Is the universe a sponge?

24 Nov 2016

Taken from the November 2016 issue of Physics World

Martin Bucher is a theoretical cosmologist

at the Université Paris 7/CNRS, Paris, France,

e-mail bucher@apc.univ-paris7.fr

( https://physicsworld.com/a/is-the-universe-a-sponge/ )

Light refraction in the Swiss-cheese model

Gyula Bene 1

Adelinda Csapó 1

  1. Institute of Physics, Eötvös University, Pázmány P. s. 1/A, H-1117, Budapest, Hungary
  2. F IJAIMS 2012; 10 (4) : 821-835;
  3. DOI: 10.2478/s11534-012-0069-0


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A team of researchers says the Milky Way resides in one of the observable universe’s darkest regions, but some experts aren’t so sure.

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A presentation for the CosmoStat CosmoClub

Sept. 10, 2015

Click to access AP_Thesis_Presentation_CEA.pdf

Backreaction in Cosmology

S. Schander1,2 and T. Thiemann2*
1Perimeter Institute, Waterloo, ON, Canada
2Institute for Quantum Gravity, FAU Erlangen – Nürnberg, Erlangen, Germany

Front. Astron. Space Sci., 23 July 2021

Volume 8 – 2021 |



There’s Now More Evidence That We’re All Living in a Giant Cosmic Void

07 June 2017



Interpretation of the Hubble diagram in a nonhomogeneous universe

Pierre Fleury,1,2,* He ́le`ne Dupuy,1,2,3,† and Jean-Philippe Uzan1,2,‡

1Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Universite ́ Pierre et Marie Curie, 98 bis bd Arago, 75014 Paris, France

2Sorbonne Universite ́s, Institut Lagrange de Paris, 98 bis bd Arago, 75014 Paris, France

3Institut de Physique The ́orique, CEA, IPhT, URA 2306 CNRS, F-91191 Gif-sur-Yvette, France

(Received 15 March 2013; published 24 June 2013)

PHYSICAL REVIEW D 87, 123526 (2013)

DOI: 10.1103/PhysRevD.87.123526

Cosmological black holes as seeds of voids in the galaxy distribution

S. Capozziello1,4, M. Funaro2, and C. Stornaiolo3,4

  1. Dipartimento di Fisica “E.R. Caianiello”, Universita` di Salerno, via S. Allende, 84081 Baronissi (Salerno), Italy
  2. Dipartimento di Matematica e Informatica, Universita` di Salerno, via Ponte Don Melillo, 84084 Fisciano (Salerno), Italy
  3. Dipartimento di Scienze Fisiche, Universita` di Napoli, Complesso Universitario di Monte S. Angelo, via Cinthia, EdificioN – 80126 Napoli, Italy
  4. Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, via Cinthia,Edificio G, 80126 Napoli, Italy

Receveid 5 February 2004 / Accepted 11 March 2004

Astronomy & Astrophysics

A&A 420, 847–851 (2004)
DOI: 10.1051/0004-6361:20041166

Review: The Cosmic Web

by Jeff Foust
Monday, May 9, 2016

The Cosmic Web: Mysterious Architecture of the Universe
by J. Richard Gott
Princeton Univ. Press, 2016
hardcover, 272 pp., illus.
ISBN 978-0-691-15726-9


Earth is at the center of a 1,000-light-year-wide ‘Swiss cheese’ bubble carved out by supernovas

By Harry Baker published January 12, 2022


Geometry and arithmetic of crystallographic sphere packings

Alex Kontorovich a,b,1 and Kei Nakamura a

a Department of Mathematics, Rutgers University, New Brunswick, NJ 08854

b School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540

December 26, 2018 116 (2) 436-441



The sequence of radii of the Apollonian packing


David W. Boyd

mathematics of computation volume 39,number 159
July 1982, pages 249-254

DOI: 10.1090/S0025-5718-1982-0658230-7


Ford Circles & Farey Series

Published February 9, 2023 

Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

Ronald L Graham
Jeffrey C. Lagarias at University of Michigan
Colin Mallows at Avaya Inc.
Allan R. Wilks at AT&T
Catherine H. Yan at Texas A&M University

Discrete & Computational Geometry 35(1):1-36
September 2006



Apollonian arrangements of spheres in d-dimensional space

H Moraal

Journal of Physics A: Mathematical and General, Volume 27, Number 23
Citation H Moraal 1994 J. Phys. A: Math. Gen. 27 7785
DOI 10.1088/0305-4470/27/23/021


Self-similar space-filling sphere packings in three and four dimensions

D. V. Stager 1 , ∗ and H. J. Herrmann1, 2 , †

1Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
2Departamento de F ́ısica, Universidade Federal do Cear ́a, 60451-970 Fortaleza, Ceara ́, Brazil

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


Scaling Hypothesis of Spatial Search on Fractal Lattice Using Quantum Walk

Rei Sato,∗ Tetsuro Nikuni, and Shohei Watabe 

Department of Physics, Faculty of Science Division I,

Tokyo University of Science, Shinjuku, Tokyo,162-8601, Japan



Yang, Yun & FENG, YUTING & Yu, Yanhua. (2017).

August 2017

Fractals 25(05):1750040


The generalization of Sierpinski carpet and Sierpinski triangle in n-dimensional space∗

Yun Yang, Yanhua Yu† 

Department of Mathematics, Northeastern University,

Shenyang, Liaoning, P. R. China, 110004


On the fractal Dimension of the Visible Universe

J.-P. Eckmann, E. Jarvenpaa, M. Javenpaa, I. Procaccia  


Fractal Geometries: Scaling of Intrinsic Volumes

Philipp Schoenhoefer

November 2014
Institut fur Theoretische Physik I

Friedrich-Alexander-Universitat Erlangen-Nurnberg

Is the Universe Actually a Fractal?

Ethan Siegel

Forbes, Jan 6, 2021,


Fractal cosmology

Jonathan J. Dickau
Alternative Cosmology Group, 8 Lynn Road, Poughkeepsie, NY, United States
Available online 12 November 2008.

Chaos, Solitons & Fractals
Volume 41, Issue 4, 30 August 2009, Pages 2103-2105


Observational constraints on the fractal cosmology

Mahnaz Asghari *1,2 and Ahmad Sheykhi †1,2
1Department of Physics, College of Sciences, Shiraz University, Shiraz 71454, Iran
2Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran


Quantum field theory, gravity and cosmology in a fractal universe. 

Gian luca Calcagni.

Journal of High Energy Physics, 03:120, 2010.

Fractal universe and quantum gravity. 

Gian luca Calcagni.

Phys. Rev. Lett., 104:251301, Jun 2010.

Dark energy in multifractional spacetimes. 

Gian luca Calcagni and Antonio De Felice.

Phys. Rev. D, 102:103529, Nov 2020.

Multifractional theories: An updated review. 

Gian luca Calcagni.

Modern Physics Letters A, 36(14):2140006, 2021.

Multi-scale gravity and cosmology. 

Gianluca Calcagni.

Journal of Cosmology and Astroparticle Physics, 2013(12):041–041, Dec 2013.

Multiscale spacetimes from first principles. 

Gianluca Calcagni.

Phys. Rev. D, 95:064057, Mar 2017.

Is there a pattern to the universe?

By Paul Sutter

 published May 06, 2021


Fractal cosmology in an open universe

M. Joyce 1,2,3, P. W. Anderson 4, M. Montuori 1,3, L. Pietronero 1,3 and F. Sylos Labini 3,5
Europhysics Letters, Volume 50, Number 3
Citation M. Joyce et al 2000 EPL 50 416
DOI 10.1209/epl/i2000-00285-3


On fractal dimension of the universe

Ali Eftekhari *

Electrochemical Research Center, P.O. Box 19395-5139, Tehran, Iran

Barrow Holographic dark energy in fractal cosmology

Abdulla Al Mamon, 

Ambuj Kumar Mishra 

Umesh Kumar Sharma


International Journal of Geometric Methods in Modern Physics

Vol. 19, No. 14, 2250231 (2022)


A Walk on the Fractal Universe

Publication Date: 2019
Publication Name: IOSR Journals


Fantappie’s group as an extension of special relativity on e(∞) Cantorian space-time

G.Iovane, P.Giordano

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata,

Universit´a di Salerno, Italy.


Dipartimento di Matematica e Informatica

Universit´a di Salerno, Italy.


The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity

Morag I. Scrimgeour1,2⋆, Tamara Davis3, Chris Blake4, J. Berian James5 et al

Mon. Not. R. Astron. Soc, (2002)


Robert L. Oldershaw

12 Emily Lane

Amherst, MA 01002



P.V. Grujic

Institute of Physics, P.O. Box 57, 11080 Belgrade, Yugoslavia

(Received: June 25, 2001)

Serb. Astron. J. 􏰀 163 (2001), 21 – 34

Click to access 021-034.pdf

The concept of fractal cosmos: III. Present state

Petar Grujic

Publication Date: 2011
Publication Name: Serbian Astronomical Journal


On the fractal structure of the universe

Petar Grujic


Conceptual Problems of Fractal Cosmology

Yurij V. Baryshev

Astronomical Institute of the Saint-Petersburg University, 198904, St.-Petersburg, Russia. E-mail: yuba@astro.spbu.ru

Fractals and the Large-Scale Structure in the Universe

Part 2: Is the Cosmological Principle Valid?

A K Mittal and T R Seshadri

Holographic, new agegraphic, and ghost dark energy models in fractal cosmology

AuthorsK. KaramiMubasher JamilS. GhaffariK. Fahimi, and R. Myrzakulov

Publication: Canadian Journal of Physics

26 June 2013



Fractal Universe and Quantum Gravity

Gianluca Calcagni

Phys. Rev. Lett. 104, 251301 – Published 24 June 2010


“Revisiting Fractional Cosmology” 

Micolta-Riascos, Bayron, Alfredo D. Millano, Genly Leon, Cristián Erices, and Andronikos Paliathanasis. 2023.

Fractal and Fractional 7, no. 2: 149.



The fractal and holographic universe

My Quantum Life, 2018

From Stanford University: “The fractal universe” Part 3

Sciencesprings, 2018

Towards a Fractal Universe 

Jaume Gine

Departament de Matematica Universitat de Lleida
Av. Jaume II, 69. 25001 Lleida, Spain


Adv. Studies Theor. Phys., Vol. 6, 2012, no. 10, 485 – 496

Click to access gineASTP9-12-2012.pdf

Chaotic Fractals at the Root of Relativistic Quantum Physics and Cosmology

L. Marek-Crnjac1, M. S. El Naschie2, Ji-Huan He3

1Technical School Center of Maribor, Maribor, Slovenia
2Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt

3National Engineering Laboratory for Modern Silk, College of Textile, Clothing Engineering Soochow University, Suzhou, China

Email: leila.marek@guest.arnes.si, Chaossf@aol.com, hejihuan@suda.edu.cn

Received November 27, 2012; revised January 5, 2013; accepted January 15, 2013

International Journal of Modern Nonlinear Theory and Application, 2013, 2, 78-88 http://dx.doi.org/10.4236/ijmnta.2013.21A010&nbsp;

Menger Universal Spaces

Introduction to Fractal Geometry and Chaos

Matilde Marcolli

Cal Tech 2020

MAT1845HS Winter 2020, University of Toronto M 5-6 and T 10-12 BA6180

Click to access FractalsUToronto7.pdf



Jose ́ Gaite

Instituto Universitario de Microgravedad, Escuela Te ́cnica Superior de Ingenieros Aeronauticos, Universidad Politecnica de Madrid, Madrid, Spain; jose.gaite@upm.es
Received 2006 June 15; accepted 2006 December 2

The Astrophysical Journal, 658:11Y24, 2007 March 20


Multifractals and El Naschie E-infinity Cantorian space–time

G. Iovane a,*, M. Chinnici b, F.S. Tortoriello c

a DIIMA, University of Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy
b DMA, University of Naples ‘‘Federico II’’, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy

c DIFARMA, University of Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy

Chaos, Solitons and Fractals xxx (2007) xxx–xxx

“Fractal holography: a geometric re-interpretation of cosmological large scale structure,”

Mureika, J. R.,

Journal of Cosmology and Astroparticle Physics 2007, (2007).

Halos and voids in a multifractal model of the dark matter distribution

José Gaite

AIP Conference Proceedings 878, 17 (2006);


Overview of fractals and multifractals

Yale University Math Dept

Is the universe actually a fractal?

On larger and larger scales, many of the same structures we see at small ones repeat themselves. Do we live in a fractal Universe?

 DECEMBER 28, 2021


Swiss Cheese and a Cheesy CMB

Wessel Valkenburg

LAPTH∗, Universit ́e de Savoie and CNRS BP110, F-74941 Annecy-le-Vieux Cedex France
E-mail: wessel.valkenburg@lapp.in2p3.fr


‘Cosmic void’ theory suggests our Universe is like Swiss cheese and we live in one of its holes

Further weight has been added to the theory that says our galactic neighbourhood is inside a void that happens to be seven times larger than the Universe’s average


07.06.2017 11:54 AM


Fractality in cosmic topology models with spectral action gravity


Coupling of gravity to matter, spectral action and cosmic topology

Branimir Ćaćić, M. Marcolli, K. Teh
Published 27 June 2011

DOI:10.4171/JNCG/162Corpus ID: 15213870


“Constraining the Swiss-Cheese IR-Fixed Point Cosmology with Cosmic Expansion” 

Mitra, Ayan, Vasilios Zarikas, Alfio Bonanno, Michael Good, and Ertan Güdekli. 2021.

Universe 7, no. 8: 263. https://doi.org/10.3390/universe7080263


Spectral action gravity and cosmological models

Action spectrale, gravitation et modèles cosmologiques

Matilde Marcolli

Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA

Available online 29 May 2017, Version of Record 6 June 2017.


Comptes Rendus Physique
Volume 18, Issues 3–4, March–April 2017, Pages 226-234


On Modelling a Relativistic Hierarchical (Fractal) Cosmology by Tolman’s Spacetime. I. Theory

Marcelo B Ribeiro



On modeling a relativistic hierarchical (fractal) cosmology by Tolman’s spacetime. II – Analysis of the Einstein-de Sitter model

Marcelo B Ribeiro
1992, The Astrophysical Journal


Cosmological observables in a Swiss-cheese universe

Edward Kolb

Valerio Marra

Sabino Matarrese

Antonio Riotto

2007, Physical Review D



On light propagation in Swiss-Cheese cosmologies

Sebastian Szybka

Published 2010


Dark Energy and Inhomogeneous Cosmologies

Juliana Kwan, Matthew J. Francis, Geraint F. Lewis
Institute of Astronomy, School of Physics, The University of Sydney jkwan@physics.usyd.edu.au

100 years of mathematical cosmology: Models, theories, and problems, Part A

Spiros Cotsakis and  A. P. Yefremov

Published:14 March 2022



100 years of mathematical cosmology: Models, theories and problems, Part B

Spiros Cotsakis and Alexander P. Yefremov
Published:04 July 2022




Lie sphere geometry in lattice cosmology

Michael Fennen1 and Domenico Giulini1,2
1 Center for Applied Space Technology and Microgravity, University of Bremen, Bremen, Germany
2 Institute for Theoretical Physics, Leibniz University of Hannover, Hannover, Germany
E-mail: giulini@itp.uni-hannover.de
Received 23 September 2019, revised 18 December 2019

Accepted for publication 10 January 2020
Published 18 February 2020

Class. Quantum Grav. 37 (2020) 065007 (30pp) 



Fractal universe and cosmic acceleration in a Lemaitre-Tolman-Bondi scenario

Leonardo Cosmai, Giuseppe Fanizza, Francesco Sylos Labini, Luciano Pietronero, Luigi Tedesco


Mohamed El Naschie’s Revision of Albert Einstein’s E = m0c2: A Definite Resolution of the Mystery of the Missing Dark Energy of the Cosmos

J. H. He1, L. Marek-Crnjac2*

1Nantong Textile Institute, National Engineering Laboratory for Modern Silk, College of Textile and Clothing, Soochow University, Suzhou, China

2Technical School Center of Maribor, Maribor, Slovenia

Email: hejihuan@suda.edu.cn, *leila.marek@guest.arnes.si

Received November 11, 2012; revised December 16, 2012; accepted December 25, 2012

International Journal of Modern Nonlinear Theory and Application
Vol.2 No.1(2013), Article ID:28827,5 pages



‘Swiss cheese’ universe challenges dark energy

Space 31 August 2007

By Anil Ananthaswamy


Swiss-cheese cosmologies with variable G and Λ from the renormalization group

Fotios K. Anagnostopoulos, Alfio Bonanno, Ayan Mitra, and Vasilios Zarikas

Phys. Rev. D 105, 083532 – Published 27 April 2022

Swiss-Cheese Inhomogeneous Cosmology and the Dark Energy Problem

Tirthabir Biswas(McGill U. and Penn State U.), Alessio Notari(McGill U.)
Feb, 2007
35 pages
Published in: JCAP 06 (2008) 021
e-Print: astro-ph/0702555 [astro-ph]
DOI: 10.1088/1475-7516/2008/06/021

The possibility of a stable flat dark energy-dominated Swiss-cheese brane-world universe

N. Ahmed, K. Bamba, Farid Salama
Published 23 April 2019



“Structures in the Universe by Exact Methods : Formation, Evolution, Interactions’’

Krzysztof Bolejko, Andrzej Krasinski, Charles Hellaby et Marie-Noêlle Célérier

Cambridge Monographs on Mathematical Physics, Cambridge University Press (2010)

Quantum Oppenheimer-Snyder and Swiss Cheese models

Jerzy Lewandowski, Yongge Ma, Jinsong Yang, Cong Zhang


‘ CMB seen through random Swiss Cheese ‘

Lavinto , M & Räsänen , S 2015 ,

Journal of Cosmology and Astroparticle Physics , no. 10 , 057 . https://doi.org/10.1088/1475-7516/2015/10/057


Fractals and multifractals in the description of the cosmic structure

V.J. Martínez
Departament de Matemàtica Aplicada i Astronomia Universitat de València Burjassot, E-46100 València, Spain
Available online 31 October 2002.

Vistas in Astronomy
Volume 33, Part 3, 1990, Pages 337-356



Fractal cosmology



The Fractal Geometry of the Cosmic Web and Its Formation

Jose Gaite

Applied Physics Dept., ETSIAE, Univ. Politécnica de Madrid, E-28040 Madrid, Spain

Advances in Astronomy

Volume 2019 | Article ID 6587138 | https://doi.org/10.1155/2019/6587138



Apollonian Packings

Related Publications

Platonic and Archimedean Solids

Platonic and Archimedean Solids

Source: The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions

Source: The History of Mathematics From the Egyptians to Archimedes

Source: Platonic Solids, or, the power of counting

Key Terms

  • Prapanch (Five Fold)
  • Panch (Five)
  • 5 Platonic Solids
  • 14 Archimedean Solids
  • Catalan Solids
  • Regular Convex Polyhedra
  • Semi Regular Convex Polyhedra
  • Kepler-Poinsot Polyhedra
  • 4D Polytopes
  • Five Elements
  • 5 Kosh (Sheaths)
  • 14 Lok (Levels, Realms)
  • 7 Upper Worlds
  • 7 Under Worlds
  • 7 Chakras
  • 5 Continents
  • 7 Seas
  • Hierarchy Theory
  • Mount Meru
  • Nested Platonic Solids
  • Soccer Ball Geometry
  • Uniform Polyhedra
  • Johnson Solids
  • Goldberg Polyhedra
  • Albrechet Durer
  • Leonardo da Vinci
  • Johannes Kepler
  • Fra Luca Bartolomeo de Pacioli (ca.1447–1517)
  • Buckminster Fuller
  • Fullerenes
  • Virus Geometry
  • Symmetry
  • Polygons
  • Max Brückner
  • H.S.M. Coxeter
  • George W. Hart

Five Platonic Solids

  • Five Platonic Solids
    • Tetrahedron
    • Octahedron
    • Cube
    • Icosahedron
    • Dodecahedron
  • Five Elements
    • Fire – Tetrahedron
    • Air – Octahedron
    • Earth – Cube
    • Space – Dodecahedron
    • Water – Icosahedron
  • Five Sense Organs
    • Tongue -Taste – Water
    • Eyes – Form – Fire
    • Ears – Sound – Space
    • Nose – Smell –  Earth
    • Skin – Touch – Air
  • Five Senses
    • Hearing
    • Sight
    • Touch
    • Taste
    • Smell

Source: Polyhedra: Plato, Archimedes, Euler

Source: Polyhedra: Plato, Archimedes, Euler

Source: Polyhedra: Plato, Archimedes, Euler

Source: Sacred Geometry and the Platonic Solids

Source: Sacred Geometry and the Platonic Solids

Source: The Platonic solids and fundamental tests of quantum mechanics

Source: The Platonic solids and fundamental tests of quantum mechanics

Source: The Platonic solids and fundamental tests of quantum mechanics

Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids

Source: Platonic Solids (Regular polytopes in 3D)

Source: Platonic Solids (Regular polytopes in 3D)

Source: Platonic Solids (Regular polytopes in 3D)

Source: Platonic Solids (Regular polytopes in 3D)

14 Archimedean Solids

  • Rhombicuboctahedron
  • Rhombicosidodecahedron
  • Cuboctahedron
  • Icosidodecahedron
  • Truncated Tetrahedron
  • Truncated Cube
  • Truncated Octahedron
  • Truncated Dodecahedron
  • Truncated Icosahedron
  • Truncated Cuboctahedron
  • Truncated Icosidodecahedron
  • Snub Cube
  • Snub Dodecahedron
  • Pseudorhombicuboctahedron ?

Source: Polyhedra: Plato, Archimedes, Euler

Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids

Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids

Source: The Stars Above Us:
Regular and Uniform Polytopes up to Four Dimensions

Catalan Solids

Source: Catalan Solids

Kepler-Poinsot Solid

Source: Kepler Poinsot Solids

Source: The Stars Above Us:
Regular and Uniform Polytopes up to Four Dimensions

Johnson Solids

Source: Johnson Solid

Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids

Source: Vertex Configurations: Platonic Solids, Archimedean Solids, And Johnson Solids

Source: Max Brücknerʼs Wunderkammer of Paper Polyhedra

Source: Max Brücknerʼs Wunderkammer of Paper Polyhedra

Platonic Solids and Plato’s Theory of Everything

Source: Platonic Solids and Plato’s Theory of Everything
The Socratic tradition was not particularly congenial to mathematics, as may be gathered from Socrates’ inability to convince himself that 1 plus 1 equals 2, but it seems that his student Plato gained an appreciation for mathematics after a series of conversations with his friend Archytas in 388 BC. One of the things that most caught Plato’s imagination was the existence and uniqueness of what are now called the five “Platonic solids”. It’s uncertain who first described all five of these shapes – it may have been the early Pythagoreans – but some sources (including Euclid) indicate that Theaetetus (another friend of Plato’s) wrote the first complete account of the five regular solids. Presumably this formed the basis of the constructions of the Platonic solids that constitute the concluding Book XIII of Euclid’s Elements. In any case, Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of (non-planar) points in space, and late in life he expounded a complete “theory of everything”, in the treatise called Timaeus, based explicitly on these five solids. Interestingly, almost 2000 years later, Johannes Kepler was similarly fascinated by these five shapes, and developed his own cosmology from them.
To achieve perfect symmetry between the vertices, it’s clear that each face of a regular polyhedron must be a regular polygon, and all the faces must be identical. So, Theaetetus first considered what solids could be constructed with only equilateral triangle faces. If only two triangles meet at a vertex, they must obviously be co-planar, so to make a solid we must have at least three triangles meeting at each vertex. Obviously when we have arranged three equilateral triangles in this way, their bases form another equilateral triangle, so we have a completely symmetrical solid figure with four faces, called the tetrahedron, illustrated below.
On the other hand, if we make four triangles meet at a vertex, we produce a square-bottomed pyramid, and we can obviously put two of these together, base to base, to give a completely symmetrical arrangement of eight triangular faces, called the octahedron, shown below.
Next, we can make five equilateral triangles meet at a point. It’s less obvious in this case, but if we continue this pattern, adding equilateral triangles so that five meet at each vertex, we arrive at a complete solid with 20 triangular faces. This is called the icosahedron, shown below.
Now, we might try putting six equilateral triangles together at a point, but the result is a planar arrangement of triangles, so it doesn’t give a finite solid. I suppose we could regard this as a Platonic solid with an infinite radius, which might have been useful in Plato’s cosmology, but it doesn’t seem to have been viewed this way. Perhaps this is not surprising, considering the well-known aversion of the ancient Greek mathematicians to the complete infinity. In any case, we clearly can’t construct any more perfectly symmetrical solids with equilateral triangle faces, so we must turn to other possible face shapes.
The next regular polygon shape is the square, and again we find that putting just two squares together does not yield a solid angle, so we need at least three squares to meet at each vertex. Putting three squares together we see that we can add three more to give the perfect solid with six faces, called the hexahedron (also known as the cube). This is shown below.
If we try to make four square faces meet at each vertex, we have another plane surface (giving another “infinite Platonic solid”), so clearly this is the only finite perfectly symmetrical solid with square faces.
Proceeding to pentagonal (five-sided) faces, we find that if we put together 12 pentagons so that three meet at each vertex, we arrive at the fifth Platonic solid, called the dodecahedron, illustrated below.
It isn’t self-evident that 12 identical regular pentagons would come together perfectly like this to form a closed solid, but it works, as Theaetetus proved and as Euclid demonstrates at the conclusion of The Elements. Of course, if we accept that the icosahedron works, then the dodecahedron automatically follows, because these two shapes are “duals” of each other. This means that the icosahedron has 20 faces and 12 vertices, whereas the dodecahedron has 12 faces and 20 vertices, and the angular positions of the face centers of one match up with the positions of the vertices of the other. Thus, once we have the icosahedron, we can just put a dot in the center of each face, connect the dots, and viola!, we have a dodecahedron. Similarly, the cube and the octahedron are duals of each other. Also, the tetrahedron is the dual of itself (so to speak).
Clearly it’s impossible for four (or more) pentagonal faces to meet at a vertex, because they subtend more than 360 degrees. For hexagonal (six-sided) faces, three hexagons meeting at a point constitute another “infinite solid”, i.e., a planar surface. It’s also obvious that no higher-order polygon can yield a solid, so the five solids already mentioned – tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron – are the only regular polyhedrons. Theaetetus not only proved that these solids exist, and that they are the only perfectly symmetrical solids, he also gave the actual ratios of the edge lengths E to the diameters D of the circumscribing spheres for each of these solids. This is summarized in Propositions 13 through 17 of Euclid’s Elements.
In Timaeus, Plato actually chose to constitute each of these solids from right triangles, which played the role of the “sub-atomic particles” in his theory of everything. In turn, these triangular particles consisted of the three legs (which we might liken to quarks), but these legs were ordinarily never separated. The right triangles that he chose as his basis particles were of two types. One is the “1,1,” isosceles triangle formed by cutting a square in half, and the other is the “1,2,” triangle formed by cutting an equilateral triangle in half. He used these to construct the faces of the first four solids, but oddly enough he didn’t just put two together, he used six “1,2, triangles to make a triangular face, and four “1,1,” triangles to make a square face, as shown below.
Of course, it’s not possible to build a pentagon from these two basic kinds of right triangles, and Plato doesn’t actually elaborate on how the faces of the dodecahedron are to be constructed, but from other sources we know that he thought each face should be composed of 30 right triangles, probably as shown on the right-hand figure above, so that the dodecahedron consisted of 360 triangles. The tetrahedron, octahedron, and icosahedron consisted of 24, 48, and 120 triangles (of the type 1,2,), respectively, and the hexahedron consisted of 24 triangles (of the type 1,1,).
Now, if the basic triangles were the subatomic particles, Plato regarded the solids as the “atoms” or corpuscles of the various forms of substance. In particular, he made the following identifications
The idea that all the constituents of nature consist of mixtures of a small number of “elements”, and in particular the selection of the four elements of earth, water, air, and fire, is attributed to an earlier Greek philosopher Empedocles of Agrigentum (495-435 BC). Empedocles believed that although these elements (which he called “the roots of all things”) could be mixed together in various proportions, the elements themselves were inviolable, and could never be changed. In contrast, one of the intriguing aspects of Plato’s theory was that he believed it was possible for the subatomic particles to split up and re-combine into other kinds of atoms. For example, he believed that a corpuscle of liquid, consisting of 120 “type 1” triangles, could be broken up into five corpuscles of plasma, or into two corpuscles of gas and one of plasma. Also, he believed that the “smaller” corpuscles could merge into larger corpuscles, so that (for example) two atoms of plasma could merge and form a single atom of gas. However, since the basic triangles making up “earth” (cubes) are dissimilar to those of the other forms of substance, he held that the triangles comprising cubes cannot be combined into any of the other shapes. If a particle of earth happened to be broken up into its constituent triangles, they will “drift about – whether the breaking up within fire itself, or within a mass of air or water – until its parts meet again somewhere, refit themselves together and become earth again”.
When Plato asserts that the [1,1,] triangles cannot combine into anything other than a cube, it’s conceivable that he was basing this on something more that just the geometric dissimilarity between this triangle and the [1,2,] triangle. He might also have had in mind some notion of the incommensurability of the magnitudes  and , not only with the unit 1, but with each other. Indeed the same Theaetetus who gave the first complete account of the five “Platonic” solids is also remembered for recognizing the general fact that the square root of any non-square integer is irrational, which is to say, incommensurable with the unit 1. It isn’t clear whether Theaetetus (or Plato) knew that two square roots such as  and  are also incommensurable with each other, but Karl Popper (in his anti-Plato polemic “The Free Society and its Enemies”) speculated that this might have been known, and that Plato’s choice of these two triangles as the basic components of his theory was an attempt to provide a basis (in the mathematical sense) for all possible numbers. In other words, Popper’s idea is that Plato tentatively thought the numbers 1, , and  are all mutually incommensurable, but that it might be possible to construct all other numbers, including , π, etc., as rational functions of these.
Of course, Book X of Euclid’s Elements (cf. Prop 42) dashes this hope, but it’s possible that the propositions recorded there were developed subsequent to Plato’s time. Popper also makes much of the numerical coincidence that + is approximately equal to π, and speculates that Plato might have thought these numbers were exactly equal, but this doesn’t seem credible to me. For one thing, it would give a means of squaring the circle, which would certainly have been mentioned if anyone had believed it. More importantly, the basic insight of Theaetetus was in recognizing the symmetry of all the infinitely many irrational square roots, and it just doesn’t seem likely that he (or Plato) would have been misled into supposing that just two of them (along with the unit 1) could form the basis for all the others. It’s a very unnatural idea, one that would not be likely to occur to a mathematician. (Still, an imaginative interpreter could probably discern correspondences between the four basis vectors of “The Platonic Field“, i.e., numbers of the form A + B+ C + D and Plato’s four elements, not to mention the components of Hamilton’s quaternions.)
It’s also interesting that Plato describes the “1,1,” triangle as the most “stable”, and the most likely to hold its shape, thus accounting for the inert and unchanging quality of the solid elements. He didn’t elaborate on his criterion for “stability”, although we can imagine that he had in mind the more nearly equal lengths of the edges, being closer to equilibrium. On the other hand, this would suggest that the equilateral triangle (which is the face of Plato’s “less stable” elements) was highly stable. Plato made no mention of the fact that the cube is actually the only unstable Platonic solid, in the sense of rigidity of its edge structure. In addition, the cube is the only Platonic solid that is not an equilibrium configuration for its vertices on the surface of a sphere with respect to an inverse-square repulsion. Nevertheless, the idea of stability of the sub-atomic structure of solid is somewhat akin to modern accounts of the stability of inert elements. 
We can also discern echoes of Plato’s descriptions in Isaac Newton’s corpuscular theory. Newton’s comments about the “sides” of light particles are very reminiscent of Plato’s language in Timaeus. It’s also interesting to compare some passages in Timaeus, such as
And so all these things were taken in hand, their natures being determined by necessity in the way we’ve described,  by the craftsman of the most perfect and excellent among things that come to be…
with phrases in Newton’s Principia, such as
…All the diversity of created things, each in its place and time, could only have arisen from the ideas and the will of a necessarily existing being… 
…all phenomena may depend on certain forces by which the particles of bodies…either are impelled toward one another and cohere in regular figures, or are repelled from one  another and recede…
…if anyone could work with perfect exactness, he would be the most perfect mechanic of all…
Plato explicitly addressed the role of necessity in the design of the universe (so well exemplified by the five and only five Platonic solids), much as Einstein always said that what really interested him was whether God had any choice in the creation of the world. But Plato was not naive. He wrote
Although [God] did make use of the relevant auxiliary causes, it was he himself who gave their fair design to all that comes to be. That is why we must distinguish two forms of cause, the divine and the necessary. First, the divine, for which we must search in all things if we are to gain a life of happiness to the extent that our nature allows, and second, the necessary, for which we must search for the sake of the divine. Our reason is that without the necessary, those other objects, about which we are serious, cannot on their own be discerned, and hence cannot be comprehended or partaken of in any other way.
The fifth element, i.e., the quintessence, according to Plato was identified with the dodecahedron. He says simply “God used this solid for the whole universe, embroidering figures on it”. So, I suppose it’s a good thing that the right triangles comprising this quintessence are incommensurate with those of the other four elements, since we certainly wouldn’t want the quintessence of the universe to start transmuting into the baser substances contained within itself!
Timaeus contains a very detailed discussion of virtually all aspects of physical existence, including biology, cosmology, geography, chemistry, physics, psychological perceptions, etc., all expressed in terms of these four basic elements and their transmutations from one into another by means of the constituent triangles being broken apart and re-assembled into other forms. Overall it’s a very interesting and impressive theory, and strikingly similar in its combinatorial (and numerological) aspects to some modern speculative “theories of everything”, as well as expressing ideas that have obvious counterparts in the modern theory of chemistry and the period table of elements, and so on.
Timaeus concludes
And so now we may say that our account of the universe has  reached its conclusion. This world of ours has received and  teems with living things, mortal and immortal. A visible  living thing containing visible things, and a perceptible God, the image of the intelligible Living Thing. Its grandness,  goodness, beauty and perfection are unexcelled. Our one  universe, indeed, the only one of its kind, has come to be.
The speculative details of Plato’s “account of the universe” are not very satisfactory from the modern point of view, but there’s no doubt that – at least in its scope and ambition as an attempt to represent “all that is” in terms of a small number of simple mathematical operations – Plato’s “theory of everything” left a lasting impression on Western science.
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Kepler’s Mysterium Cosmographicum

Source: https://en.wikipedia.org/wiki/Platonic_solid

Source: Kepler’s Polyhedra

Source: Kepler’s Polyhedra

Source: Kepler’s Polyhedra

Source: Platonic Solids, or, the power of counting

Six (6) 4D Polytopes

Source: 4d-polytopes described by Coxeter diagrams and quaternions

Discovery of the Platonic solids; tetrahedron, cube, octahedron, icosahedron and dodecahedron dates back to the people of Scotland lived 1000 years earlier than the ancient Greeks and the models curved on the stones are now kept in the Ashmolean Museum at Oxford [1]. Plato associated tetrahedron with fire, cube with earth, air with octahedron, and water with icosahedron. Archimedes discovered the semi-regular convex solids and several centuries later they were rediscovered by the renaissance mathematicians. By introducing prisms and anti-prisms as well as four regular non-convex polyhedra, Kepler completed the work in 1620. Nearly two centuries later, in 1865, Catalan constructed the dual solids of the Archimedean solids now known as Catalan solids [2]. Extensions of the platonic solids to 4D dimensions have been made in 1855 by L. Schlaffli [3] and their generalizations to higher dimensions in 1900 by T.Gosset [4]. Further important contributions are made by W. A. Wythoff [5] among many others and in particular by the contemporary mathematicians H.S.M. Coxeter [6] and J.H. Conway [7].

The 3D and 4D convex polytopes single out as compared to the polytopes in higher dimensions. The number of Platonic solids is five in 3D and there exist six regular polytopes in 4D contrary to the higher dimensional cases where there exist only three platonic polytopes which are the generalizations of tetrahedron, octahedron and cube to higher dimensions. The Platonic and Archimedean solids [8] as well as the Catalan solids [9] can be described with the rank-3 Coxeter groupsW(A3),W(B3) and W(H3).

The 4D polytopes are described by the rank-4 Coxeter groups W(A4 ), W(B4 ), W(H4 ) and the group (F4 ).

Source: The Stars Above Us:
Regular and Uniform Polytopes up to Four Dimensions

Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces

Source: Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces

Source: Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces


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Indra’s Net: On Interconnectedness

Shape of the Universe

Geometry of Consciousness

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

Cantor Sets, Sierpinski Carpets, Menger Sponges

Interconnected Pythagorean Triples using Central Squares Theory

Cosmic Mirror Theory

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Mind, Consciousness and Quantum Entanglement

Process Physics, Process Philosophy

Law of Dependent Origination

On Synchronicity

The Great Chain of Being

Hierarchy Theory in Biology, Ecology and Evolution

The Hidden Geometry of Trade Networks

Networks and Hierarchies

Multiplex Financial Networks

Shapes and Patterns in Nature

Key Sources of Research

Polyhedra V1.0

by Gian Marco Todesco

A Java applet for creating Nested Platonic Solids. I have not yet seen anything better than this applet.



The Topology and Combinatorics of Soccer Balls

When mathematicians think about soccer balls, the number of possible designs quickly multiplies

Dieter Kotschick

American Scientist, Volume 94, page 350

Platonic Solids

Paul Calter at Dartmouth


A Beginner’s Guide to Constructing the Universe

Schneider, Michael, 

Harper Perennial, 1994

A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller

Edmondson, Amy, 

Burkhauser Boston, 1987

5. Graph theory and platonic solids

Fine, Benjamin, Gaglione, Anthony, Moldenhauer, Anja, Rosenberger, Gerhard and Spellman, Dennis.

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Platonic Solids


Platonic Solids, or, the power of counting

Keith Jones

Pi Mu Epsilon Induction SUNY Oneonta April 2017

Click to access pimuepsilon2017.pdf

Platonic Solids and Plato’s Theory of Everything


Geometric, Algebraic and Topological Connections in the Historical Sphere of the Platonic Solids

A thesis submitted in partial fulfillment of the requirements for the degree of
Master of Science in Mathematics.

James Adam Smith

University of Nevada, Reno, 2012

Dodecahedrane-The chemical transliteration of Plato’s universe ( A Review )


Evans Chemical Laboratories,The Ohio State University, Columbus, Ohio 43210

Communicated by Daniel E. Koshlond, Jr., April 26, 1982

Proc.NatL Acad, Sci. USA Vol.79, p. 4495-4500, July 1982


4d-polytopes described by Coxeter diagrams and quaternions

Mehmet Koca 2011

J. Phys.: Conf. Ser. 284 012040


Article 40: Geometry – The Platonic Solids – Part 1 – Introduction

Cosmic Core


Cosmic Core

Sacred Geometry: The Geometry of Life, Matter, Consciousness, Space & Time

Cosmic Core


Egon Schulte

Click to access chap18.pdf

It’s a Long Way to the Stars
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Dimitrios C. Tzarouchis, Pasi Ylä-Oijala, Ari Sihvola

First published: 28 October 2017



Universe as Geometry : discovery of polyhedra

Published on Jun 20, 2016

Platonic Solids Revisited

Posted on September 2, 2012 by Suresh Emre

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Name: David Clark, Kelsey Kyllo, Kurt Maugerle, Yue Yuan Zhang

Course Number: Math 445

Professor: Julia Pevtsova

Date: 2013/06/03

Platonic Solids: The Language of the Universe

By David Mcconaghay
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The Platonic Solids: a Three-Dimensional Textbook

Martin Levin
604 Winona Court
Silver Spring, MD, 20902, USA

E-mail: mdlevin_public@msn.com

Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture


A 10-Dimensional Jewel

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CS Division, University of California, Berkeley

E-mail: sequin@cs.berkeley.edu

Platonic and Archimedean geometries in multicomponent elastic membranes

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Edited by L. Mahadevan, Harvard University, Cambridge, MA, and accepted by the Editorial Board February 8, 2011 (received for review August 30, 2010)

February 28, 2011
108 (11) 4292-4296


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Archimedean Polyhedra


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Daud Sutton 2002

Wooden Book

New York, NY. Walker and Company.

Plato’s Error and a Mean Field Formula for Convex Mosaics

Gabor Domokos
Budapest University of Technology and Economics

Zsolt Lángi

Budapest University of Technology and Economics

August 2019 Axiomathes 32(1)


Platonic Solids (Regular polytopes in 3D)

Written by Paul Bourke
December 1993




A Thesis
Taylor R. Brand
June 2012

Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses

Stan Schein stan.schein@gmail.com and James Maurice Gayed

Edited by Patrick Fowler, The University of Sheffield, Sheffield, United Kingdom, and accepted by the Editorial Board January 7, 2014 (received for review June 10, 2013)
February 10, 2014
111 (8) 2920-2925



The Platonic solids and fundamental tests of quantum mechanics

Armin Tavakoli and Nicolas Gisin
Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

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Robert L. Benedetto

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MathPath at Mount Holyoke College

Tuesday, July 15, 2014

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Allen Liu

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The Secrets of the Platonic Solids and Sacred Geometry

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Platonic Solids in All Dimensions

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Second Edition

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Symmetrical Analysis Techniques for Genetic Systems and
Bioinformatics: Advanced Patterns and Applications

Sergey Petoukhov
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Matthew He
Nova Southeastern University, USA

2010 by IGI Global

The mathematics of harmony : from Euclid to contemporary mathematics and computer science

by Alexey Stakhov ; assisted by Scott Olsen.


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Instructor: Associate Professor Dan Knopf
Email: danknopf@math.utexas.edu
Homepage: http://www.ma.utexas.edu/users/danknopf

Mathematics of Space Architectural Design, 2 edition

Author: Hoi Mun


Jole de Sanna

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V. L. Hansen∗

ICM 2002 · Vol. III · 1–3

Chapter 1

The Story of Numbers and Arithmetic from Ancient Times to the Beginning of the Second Millennium

Book Trilogy of Numbers and Arithmetic


A Secret of Ancient Geometry

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Brian J. McCartin

Applied Mathematics
Kettering University

Click to access mccartin-2.pdf

From Euclid to Riemann and Beyond∗ – How to describe the shape of the universe

Toshikazu Sunada

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Wiarton, ON, Canada, george@georgehart.com

Bridges 2019 Conference Proceedings

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Tomlow, J.,

Chapter in book

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Ivory shells and polyhedra


*Budapest University of Technology and Economics Budapest, Műegyetem rkp. 3., H-1521 Hungary tarnai@ep-mech.me.bme.hu

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The Polyhedron Formula and the Birth of Topology

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Michael Flicker

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Plato’s Mathematical Imagination

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and Their Interpretation



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12 November 2015

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Ümit Ziya SAVCI

 Cumhuriyet Sci. J., Vol.40-2(2019) 458-464 

Wikipedia pages for: 

Sapta Matrikas (Seven Mothers) and Cosmology

Sapta Matrikas (Seven Mothers) and Cosmology

Source: Matrikas / Wikipedia

Key Terms

  • Sapta Matrikas
  • Seven Mothers
  • Seven Sisters
  • Seven Cows
  • Apollonian Strip
  • Apollonian Gasket
  • Kartik
  • Pleiades Constellation
  • Harmonic
  • Enharmonic
  • Music Tuning Theory
  • In Tune
  • Out of Tune
  • Musical Scale
  • Seven Swar
  • Sa Re Ga Ma Pa Dha Ni
  • Seven Colors
  • Sapta Puri
  • Tripura
  • Tripurari
  • Virabhadra
  • Veenadhara
  • Ganesh Lord of Music
  • Nataraj Lord of Dance
  • Rudra Veena
  • Cymatics
  • Sound and Form
  • Naam Rupa
  • Bootstraped Creation
  • Seven Rays of Sun
  • Shiva
  • Indian Classical Dances
  • Indian Classical Music
  • Natyashastra of Bharatmuni
  • Participatory Geometry
  • MATRYOSHKA’S Dolls of Russia
  • Nested Set of Seven Dolls
  • Diophantine Equation
  • Ford Circles
  • Farey Sequence
  • Consonance
  • Dissonance
  • Unbounded Apollonian Gasket
  • Bounded Apollonian Gasket
  • Menger Sponge + Swiss Cheese Cosmology
  • Shape of the Universe
  • Cosmic geometry
  • Dark Matter and Dark Energy
  • Pythagorean Triples
  • Triad of Goddess ( Saraswati + Lakshmi + Parwati)
  • Triad of Gods ( Brahma, Vishnu, Mahesh)
  • Tri Loka
  • Sapta Rishi Mandala (Ursa Major)
  • Seven Sisters of Pleiades
  • Three Stars of Orion
  • Misr
  • Goddess Hathor (Cow Godess)
  • Nested Platonic Solids
  • Rig Veda
  • Maharishi Vishwamitra
  • Gayatri Mantra
  • Krishna, Balaram, Subhadra
  • Rama, Sita, Lakshman
  • Hanuman
  • Madan Mohan
  • Sacred Geography
  • Archeo Astronomy
  • Archeo Musicology
  • Circle and Square as Limit Sets
  • Southern Sky
  • Northern Sky
  • Western Sky
  • Eastern Sky

Cosmology and Triads

Interaction / Intersection of Menger Sponge and Apollonian Sphere in cosmic geometry produces pythagorean triples / triads.

Seven plus one spheres to triads of three.

Apollonian Gasket, Circle and Sphere Packing and Cosmic Geometry


A transfer operator method is proposed to calculate 𝑑𝐻, the Hausdorff dimension of the Apollonian gasket. Compared with previous operator-based methods, we make two improvements in this paper. We adopt an infinite set of contractive Möbius transformations (rather than a finite set of parabolic ones) to generate the Apollonian gasket. We also apply an efficient finite matrix approximation of an infinite sum of infinite-dimensional operators. By using this method, a high precision estimate of 𝑑𝐻 is obtained: 

𝑑𝐻=1.305 686 728 049 877 184 645 986 206 851 0….


The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2−19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both algorithms are given.


Source: What Type of Apollonian Circle Packing Will Appear?

Source: Self-similar space-filling sphere packings in three and four dimensions

Source: Self-similar space-filling sphere packings in three and four dimensions

Sapta Matrikas

  • Brahmi
  • Maheswari
  • Kumari 
  • Vaishnavi 
  • Varahi
  • Indrani 
  • Chamunda

Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern

In Varahapurana the Devi, Vaishnavi in account of the creation of the matrika, is doing asceticism on mount Mandara. At one point she losses her concentration. From her disgraced mind, several beautiful attendants were created. They later became Devi’s helpmates on the battlefield when she fights the demon. Although the Matrikas are described as lovely in this account, it is important to note that they are born when Devi losses control of her concentration. This suggests that the matrikas are essentially of uncontrolled natures. Born from lack of mental control, they lack control themselves. Varahapurana relates them to vices or inauspicious emotions; Brahmi of Mada, Maheswari of Krodha, Kumari of MohaVaishnavi of LobhaIndrani of Matsarya, Varahi of Asuya and Chamunda of Paisunya.

Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern

The follower of Tantrasara has an esoteric interpretation of the seven matrikas. According to them, Brahmi represents the primordial Nada, the energy in which even the first throb has not yet appeared. This is the manifest sound, the origin of all creation. It is the same substance or energy represented by the pranava. When Brahmi creates the universe, the power of Vaishnavi gives definite shape. The symmetry, beauty, organization and order in the universe are the work of Vaishnavi. Maheswari stands for the power that gives individuality to the created beings. She resides in the hearts of all and makes them play, as dolls mounted on a machine. Kumari represents the ever present force of aspiration of the evolving soul. She is ‘Guruguha’, the Guru in guha (the cave of the heart, the intellect). Varahi is the all-consuming power of assimilation and enjoyment. Because of her, all living beings get their food and enjoyments. Indrani symbolizes the terrible power that destroys all that opposes the cosmic law. Chamunda is the force of concentrated awareness, the spiritual awakening in the heart that devours that ceaseless activity of the immature mind and uplifts it to the highest level (Harshananda 1981.95-99).

Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern

Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern

Source: Matrikas/Siddha Pedia

Source: The Seven Ancient Mothers

The Pythagoreans considered the figure seven as the image and model of the divine order and harmony in nature. As the harmony of cosmic sound takes place on the space between the seven planets, the harmony of audible sound takes place on a smaller plane within the musical scale of the seven tones. Therefore, the syrinx of the nature god Pan consists of seven pipes, and the lyre of Apollo (the god of music) consists of seven strings. As the number seven is a union between the number three (the symbol of the divine triad) and of four (the symbol of the cosmic forces or elements), the number seven points out symbolically to the union of the divine with the universe.

Source: Internet

Source: Internet

Source: Internet

Source: Regional Variations in Mātṛkā Conventions

Source: Regional Variations in Mātṛkā Conventions

Source: Regional Variations in Mātṛkā Conventions

My Related Posts

On Holons and Holarchy

Maha Vakyas: Great Aphorisms in Vedanta

The Great Chain of Being

Cosmic Mirror Theory

Shape of the Universe

Law of Dependent Origination

Mind, Consciousness and Quantum Entanglement

Third and Higher Order Cybernetics

Indra’s Net: On Interconnectedness

Interconnected Pythagorean Triples using Central Squares Theory

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

Sounds True: Speech, Language, and Communication

Semiotics and Systems

Semiotic Boundaries

Semiotic Self and Dialogic Self

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

64 Yogini Hindu Temples Architecture

Cantor Sets, Sierpinski Carpets, Menger Sponges

Understanding Rasa: Yoga of Nine Emotions

Growth and Form in Nature: Power Laws and Fractals

Fractal Geometry and Hindu Temple Architecture

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Recursion, Incursion, and Hyper-incursion

Rituals | Recursion | Mantras | Meaning : Language and Recursion

A Calculus for Self Reference, Autopoiesis, and Indications

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

The Aesthetics of Charles Sanders Peirce

Key Sources of Research

A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets

Indubala I Satija1,a
Department of Physics, George Mason University, Fairfax, VA, 22030


The Hausdorf dimension of the Apollonian packing of circles

P B Thomas1 and D Dhar1

Journal of Physics A: Mathematical and GeneralVolume 27Number 7

Citation P B Thomas and D Dhar 1994 J. Phys. A: Math. Gen. 27 2257

DOI 10.1088/0305-4470/27/7/007


An introduction to the Apollonian fractal


Click to access apollony.pdf



Fractals VOL. 26, NO. 04





Click to access Li,H.pdf

Spectral action gravity and cosmological models

Action spectrale, gravitation et modèles cosmologiques

Testing quantum gravity with cosmology/Tester les théories de la gravitation quantique à l’aide de la cosmologie

Matilde Marcolli
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA

Comptes Rendus Physique
Volume 18, Issues 3–4, March–April 2017, Pages 226-234



Cosmological observables in a Swiss-cheese universe

Valerio Marra
Universidade Federal do Espírito Santo

Edward W. Kolb
University of Chicago

Sabino Matarrese
University of Padova

Antonio Riotto

September 2007

Physical review D: Particles and fields 76(12)




Mean-field approach to Random Apollonian Packing

Pierre Auclair∗
Cosmology, Universe and Relativity at Louvain (CURL), Institute of Mathematics and Physics, University of Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium (Dated: November 15, 2022)



Click to access SwissCheese.pdf

Appolonian Gaskets and Ford Circles


Estimate for the fractal dimension of the Apollonian gasket in dimensions

R. S. Farr*
Unilever R&D, Olivier van Noortlaan 120, AT3133 Vlaardingen, The Netherlands
and The London Institute for Mathematical Sciences, 22 S. Audley Street, Mayfair, London, United Kingdom

E. Griffiths

297 Sandy Bay Road, Sandy Bay, Tasmania, Australia

PHYSICAL REVIEW E 81, 061403 􏰀2010􏰁

DOI: 10.1103/PhysRevE.81.061403

Packing my circles


Recursive Apollonian Gaskets with Python Turtle

Lie sphere geometry in lattice cosmology

Michael Fennen and Domenico Giulini 2020

Class. Quantum Grav. 37 065007



A Study of the Sapta Matrikas’ Origins and Evolution: From the Perspectives of the Art and Literature of Western India Dating from B.C. 1400 to 500 A.D.

Rebecca Sholes

Mother Goddess in Central India

Om Prakash Misra
Agam Kala Prakashan, 1985

The Little Goddesses (mātrikās)

K. C. Aryan
Rekhā, 1980

The Iconography of the Saptamatrikas: Seven Hindu Goddesses of Spiritual Transformation

Katherine Anne Harper
E. Mellen Press, 1989

ISBNS 9780889460614, 0889460612
OCLC ocm19921123

Saptamātr̥kā Worship and Sculptures: An Iconological Interpretation of Conflicts and Resolutions in the Storied Brāhmanical Icons

Shivaji Panikkar
D.K. Printworld, 1997

Mātr̥kās, Mothers in Kuṣāṇa Art

Nilakanth Purushottam Joshi
Kanak Publications, 1986

Saptamātṛkas in Indian Religion and Art

Vaidhyanathan Raja Mani
Mittal Publications, 1995

The Badoh-Pathari Saptamātṛ Panel Inscription

Dániel Balogh


Publication Date:  2019

Publication Name:  Indo-Iranian Journal


From Mātṛ to Yoginī: Continuity and Transformation in the South Asian Cults of the Mother Goddesses

Shaman Hatley
2012, Transformations and Transfer of Tantra in Asia and Beyond, ed. by István Keul (Walter de Gruyter)


Yoginis & Mātṝkās: Ecstatic Divine Celebration

Stella Dupuis


Regional Variations in Mātṛkā Conventions

Michael W Meister

1986, Artibus Asiae


Early Pāṇḍya Siṃhavāhinī and Sapta Mātṛkā Sculptures in the Far South of India

R.K.K. Rajarajan



shankar pandiyan



Meghali Goswami, Dr.Ila Gupta, Dr.P.Jha

Indian Institute of Technology Roorkee, INDIA


Click to access Saptmatrika.pdf

From Mātṛgaṇa to Sapta Mātṛkās: Brahmanical Transformation of Autochthonous Goddesses

The Memoirs of the Institute for Advanced Studies on Asia, no. 116, the University of Tokyo, 2011: 566-92.

Jae-Eun Shin
Published 2011


Saptamatrikas in Kerala: Iconography and Distribution Pattern

Arya Nair V.S.

Department of History, University of Calicut, Kerala, India,

(Email: aryanairveeyes@gmail.com)

Heritage: Journal of Multidisciplinary Studies in Archaeology 4 (2016): 376-400

Click to access 24.pdf



The Matrkas (Mothers)

The Company of Men — Early Inscriptional Evidence for the Male Companions of Mother Goddesses

Dániel Balogh
Published 2018


Saptamatrka – Part One – Devi

sreenivasarao’s blogs





Dasha Mahavidya – Part One – Introduction

Blog by Sreenivasarao Subbanna


Mandalic Mothers Of Bhaktapur, Nepal – PART 1

by Laura Amazzone 

Sutra J, February, 2016


Mandalic Mothers Of Bhaktapur, Nepal – PART 2

by Laura Amazzone

March, 2016


Saptamatrikas – Legends, History, Iconography And Temples

Anuradha Goyal

October 12, 2020

Saptamatrikas in art : some depictions


Soma Ghosh


How the Ancient Mother Goddesses Elevate and Transform Consciousness

February 25, 2020 


The Saptamātṝkās

Iconography: Meaning and Myths of Icons – Assignment 3

Aditi Trivedi


Manish Jaishree



The Fractal Dimension of the Apollonian Sphere Packing

R. Peikert
1994, Fractals


IFS for apollonian gaskets

Roger L. Bagula


Apollonian gasket



Apollonian sphere packing



Statistical Regularity of Apollonian Gaskets, 

Xin Zhang,

International Mathematics Research Notices, Volume 2021, Issue 2, January 2021, Pages 1055–1095, https://doi.org/10.1093/imrn/rnz241

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Jeffrey Lagarias
2000, Dcg


Apollonian circle packings: geometry and group theory. II. Super-Apollonian

Jeffrey Lagarias
Published 2006


Apollonian circle packings: number theory

Jeffrey Lagarias

Ronald Graham
2003, Journal of Number Theory


Apollonian circle packings: Number theory II. Spherical and hyperbolic packings

Jeffrey Lagarias
2007, The Ramanujan Journal


Apollonian Circle Packings: Geometry and Group Theory — II. Super-Apollonian Group and Integral Packings

Colin Mallows
Published 2000


Apollonian circle packings: geometry and group theory III. Higher Dimensions

Colin Mallows
Published 2005


Geometric Sequences Of Discs In The Apollonian Packing

Dov Aharonov

Published 1998


Visualizing hyperbolic honeycombs

Roice Nelson & Henry Segerman


Journal of Mathematics and the Arts
Volume 11, 2017 – Issue 1


A tale of two fractals

A. A. Kirillov
Department of Mathematics, The University of Pennsylvania,

Philadelphia, PA 19104-6395

E-mail address: kirillov@math.upenn.edu

Webpage of Jerzy Kocik 


Apollonian gaskets: beautiful math can be simple

worlds of math & physics

Luca Moroni


Integral Apollonian Packings

Author(s): Peter Sarnak
Source: The American Mathematical Monthly, Vol. 118, No. 4 (April 2011), pp. 291-306

Published by: Mathematical Association of America



Fractals Vol. 02, No. 04, pp. 521-526 (1994)



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


Self-similar space-filling sphere packings in three and four dimensions *

D. V. St ̈ager 1 , ∗ and H. J. Herrmann 1, 2 , †
1 Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
2 Departamento de F ́ısica, Universidade Federal do Cear ́a, 60451-970 Fortaleza, Ceara ́, Brazil

What Type of Apollonian Circle Packing Will Appear?

Jan E. Holly

Department of Mathematics, Colby College, Waterville, ME 04901

The American Mathematical Monthly 128 (2021) 611–629.

On a Diophantine Equation That Generates All Integral Apollonian Gaskets

Jerzy Kocik

International Scholarly Research Notices

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


The early history of the cult of THE MOTHER GODDESS
in Northern Indian Hinduism with special reference


Thesis submitted for examination for the degree of

Chapter 2, Pages 193 to 241

Sapta Matrikas – The Seven Divine Mothers

Mothers, Lovers & Others: A study of the Chausathi Yogini Temple in Hirapur

Monalisa Behera




Siddha Pedia

The Seven Ancient Mothers

Martini Fisher





Ashish Sarangi

Sapta Matrikas: The Seven Divine Mothers

Posted by The Editor | Sep 17, 2015 


The Seven Mothers or Sapta Matrikas in Ancient Indian Texts


DR Saroj Rani

Earliest epigraphic evidence for Saptamatrikas discovered

It is also the oldest Sanskrit inscription found in South India

December 25, 2019 10:41 pm |


Non Euclidean Geometry

Malin C.


Ford circle



Clifford Algebras and Euclid’s Parameterization of Pythagorean Triples

Jerzy Kocik




Fractals Vol. 26, No. 04, 1850050 (2018)


Matrëshka. History of Russian Nesting Dolls


History of Wooden Matryoshka Nesting Dolls



Matryoshka: A History of Russian Nesting Doll with Asian Roots

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: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket