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: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: KHAJURAHO GROUP OF TEMPLES/Dr. Manoj Kumar

Source: TEMPLE ARCHITECTURE AND SCULPTURE

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

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

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

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

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

Source: Lakshmana Temple, Khajuraho, Madya Pradesh, India

Source: Lakshmana Temple, Khajuraho, Madya Pradesh, India

Source: The Religious Imagery of Khajuraho

Source: The Religious Imagery of Khajuraho

Source: KHAJURAHO GROUP OF MONUMENTS / Group 12

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]

Source: PARAMETRIZING INDIAN KARNATA-DRAVIDA TEMPLE USING GEOMETRY

Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE

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

Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE

Source: VASTU PURUSHA MANDALA- A HUMAN ECOLOGICAL FRAMEWORK FOR DESIGNING LIVING ENVIRONMENTS

Source: Space and Cosmology in the Hindu Temple

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Source: Role of Fractal Geometry in Indian Hindu Temple Architecture

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE.

Source: HINDU TEMPLE: MODELS OF A FRACTAL UNIVERSE. 

Types of Vastu Purush Mandala

Source: Indian Architectural Theory: Contemporary Uses of Vastu Vidya

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

Types of Ceilings (Vitana) of Temples

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

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

Questions on Relationships

(Kshetra)

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

Square and Circle in a Group

Shiva temples (Squares) and Shakti Temples (Circular)

(Group of Squares and Group of Circles)

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

Temple Architecture Canonical Texts

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

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

UNESCO

https://whc.unesco.org/en/list/240/

Khajuraho Group of Monuments

WIKIPEDIA

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

https://doi.org/10.2307/602657
https://www.jstor.org/stable/602657

https://www.academia.edu/783948/Mandala_and_Practice_in_Nagara_Architecture_in_North_India

Geometry and Measure in Indian Temple Plans: Rectangular Temples

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

https://doi.org/10.2307/3249613
https://www.jstor.org/stable/3249613

https://www.academia.edu/783957/Geometry_and_measure_in_Indian_temple_plans_rectangular_temples

Measurement and proportion in Hindu temple architecture

Michael W Meister
1985, Interdisciplinary science reviews

https://www.academia.edu/783956/Measurement_and_proportion_in_Hindu_temple_architecture

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

ASI

Temple Architecture Khajuraho and Brihadeshwar

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

https://www.researchgate.net/publication/360928465_Temple_Architecture_Khajuraho_and_Brihadeshwar

KHAJURAHO GROUP OF TEMPLES

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

KHAJURAHO

Based on the text of Krishna Deva

Published by
The Director General
ARCHAEOLOGICAL SURVEY OF INDIA New Delhi, 2002


.

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

https://minds.wisconsin.edu/bitstream/handle/1793/36637/Gill,%20Chelsea%20-%20Temples%20Khajuraho%20Chandellas%20India.pdf;jsessionid=9BE7345C9D1C1A27B6513E02E295F9B8?sequence=1


Homage to Khajuraho

Mulk Raj Anand

Click to access 10136.pdf

KHAJURAHO GROUP OF MONUMENTS

Group-12
MAHENDRA KUMAR (M23) RAHUL KUMAR GUPTA (M31) JAMUNA PRASAD BAHUGUNA (M21)

READING LOVE IMAGERY ON THE INDIAN TEMPLE

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

priyaankaamathur007@yahoo.co.in

ESAMSKRITI

https://www.esamskriti.com/e/Culture/India-Travel-ad-Yatras/Khajuraho-Temples-a-Journey-from-Real-to-Surreal-1.aspx

https://www.esamskriti.com/a/Madhya-Pradesh/Kandariya-Mahadeva-Khajuraho.aspx

https://www.esamskriti.com/a/Madhya-Pradesh/Lakshmana-Temple-Khajuraho.aspx

https://www.esamskriti.com/e/Culture/India-Travel-Ad-Yatras/All-about-Khajuraho-Temples–1.aspx

https://www.esamskriti.com/e/Spirituality/Philosophy/Meaning-behind-Erotic-Sculptures-in-Khajuraho–1.aspx

KHAJURAHO – BREATHTAKING INSIGHT INTO TEMPLE ARCHITECTURE

 · 

ASPIRITBEDOUIN

Khajuraho photo essay

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

https://www.academia.edu/66939222/Khajuraho_photo_essay

Khajuraho In Perspective (Khajurāho in Perspective)

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

https://www.academia.edu/33809363/Khajuraho_In_Perspective_Khajurāho_in_Perspective_

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

Medhavi Davda.

https://www.ravenouslegs.com/blog/khajuraho-temples-symbolism-of-sacred-union-of-the-divine-feminine-energy-masculine-consciousness

“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
DOI:10.13189/cea.2022.100211

https://www.researchgate.net/publication/359089367_Investigating_Architectural_Patterns_of_Indian_Traditional_Hindu_Temples_through_Visual_Analysis_Framework


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

HEAVEN ON EARTH

TEMPLES, RITUAL, AND COSMIC SYMBOLISM IN THE ANCIENT WORLD

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

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

oi.uchicago.edu

https://oi.uchicago.edu/research/publications/ois/ois-9-heaven-earth-temples-ritual-and-cosmic-symbolism-ancient-world

Revisiting the Vāstupuruṣamaṇḍala in Hindu Temples, and Its Meanings

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

http://dx.doi.org/10.5659/AIKAR.2014.16.2.45

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

2011

https://escholarship.org/uc/item/83r393vc

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

INFINITE SEQUENCES IN THE CONSTRUCTIVE GEOMETRY OF 10TH CENTURY HINDU TEMPLE SUPERSTRUCTURES

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

https://espace.curtin.edu.au/bitstream/handle/20.500.11937/26171/202269_202269.pdf?sequence=2

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
DOI:10.2307/991561

https://www.researchgate.net/publication/271817065_On_the_Idea_of_the_Mandala_as_a_Governing_Device_in_Indian_Architectural_Tradition

Orissan Temple Architecture and Vastu Purusha Mandala for evolution process

December 2012

Partha Sarathi Mishra
Sri Sri University

https://www.researchgate.net/publication/275346850_Orissan_Temple_Architecture_and_Vastu_Purusha_Mandala_for_evolution_process

Shape and Geometry of Orissa Temple Architecture

Partha Sarathi Mishra

IIT Roorkee M Tech Thesis

2012

http://shodhbhagirathi.iitr.ac.in:8081/jspui/handle/123456789/2017?mode=full

Wikipedia Pages on

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

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

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

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

https://en.wikipedia.org/wiki/Panchayatana_(temple)

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

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

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.

https://oi.uchicago.edu/research/publications/ois/ois-9-heaven-earth-temples-ritual-and-cosmic-symbolism-ancient-world

“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.

https://www.academia.edu/69525989/Of_Palaces_and_Pagodas_Palatial_Symbolism_in_the_Buddhist_Architecture_of_Early_Medieval_China

Seeds and Mountains: The Cosmogony of Temples in South Asia

Michael W Meister

2013, HEAVEN ON EARTH TEMPLES, RITUAL, AND COSMIC SYMBOLISM IN THE ANCIENT WORLD

https://www.academia.edu/3658884/Seeds_and_Mountains_The_Cosmogony_of_Temples_in_South_Asia

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

https://www.academia.edu/43863568/IJERT_Role_of_Fractal_Geometry_in_Indian_Hindu_Temple_Architecture

https://www.ijert.org/research/role-of-fractal-geometry-in-indian-hindu-temple-architecture-IJERTV4IS050709.pdf

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

https://uwspace.uwaterloo.ca/bitstream/handle/10012/6356/Jayakrishnan_Kavitha.pdf?sequence=1

https://uwspace.uwaterloo.ca/handle/10012/6356?show=full

Building Science of Indian Temple Architecture

Shweta Vardia

shwetavardia@gmail.com

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,

243-258.

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

https://link.springer.com/article/10.1007/BF02153753#citeas

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

https://www.sciencedirect.com/science/article/abs/pii/S0360132307000273

https://www.semanticscholar.org/paper/Fractal-geometry-as-the-synthesis-of-Hindu-in-Rian-Park/719b6da37091121786525e4b99a667fb098abf49

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.

https://www.semanticscholar.org/paper/Symbolism-in-Hindu-Temple-Architecture-and-Fractal-Dutta-Adane/891db316ae9b06387a0e23ec4e2df649f43d2cd0

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.

https://www.semanticscholar.org/paper/Investigating-Architectural-Patterns-of-Indian-Singh-Das/76b27cc56ba003d899dbc8c0595f304f02e4bb28

“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 

sdatta@deakin.edu.au

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

https://www.semanticscholar.org/paper/Infinite-Sequences-in-the-Constructive-Geometry-Of-Datta/18ed5ec4e6e33a8ebdf5b5c1e6fdc8706b34fdcc

https://link.springer.com/article/10.1007/s00004-010-0038-0

Evolution and Interconnection: Geometry in Early Temple Architecture

DOI:10.1007/978-3-030-57907-4_11

Corpus ID: 238053244

https://www.semanticscholar.org/paper/Evolution-and-Interconnection%3A-Geometry-in-Early-Datta/c767a450532cf281e0bdd0cfac493343c2a2de07

PARAMETRIZING INDIAN KARNATA-DRAVIDA TEMPLE USING GEOMETRY

SRUSHTI GOUD

BMS School of Architecture, Yelahanka, Bangalore, India

goudsrushti@gmail.com

Click to access ascaad2016_042.pdf

A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE

PRATHAMESH GURME1,PROF. UDAY PATIL2

1UG SCHOLAR,2HEAD OF DEPARTMENT, DEPARTMENT OF CIVIL ENGINEERING BHARATI VIDHYAPEETH’S COLLEGE OF ENGINEERING , LAVALE , PUNE , INDIA

INTERNATIONAL JOURNAL FOR RESEARCH & DEVELOPMENT IN TECHNOLOGY

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

SHAPE AND GEOMETRY OF ORISSAN TEMPLE ARCHITECTURE

Authors: Mishra, Partha Sarathi

MS Thesis, IITR 2012

http://localhost:8081/xmlui/handle/123456789/2017

http://shodhbhagirathi.iitr.ac.in:8081/jspui/handle/123456789/2017


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

TEMPLE ARCHITECTURE AND SCULPTURE

NCERT

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

midhilachandra@gmail.com

Hindustan Institute of Technology and Sciences, Vijayawada, Andhra Pradesh

Dr. R. V. Nagarajan

rvnagarajan@hindustanuniv.ac.in

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

International Journal of Advance Research, Ideas and Innovations in Technology

2020

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

https://www.academia.edu/347639/Hindu_Temple_Fractals

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)
DOI:10.25215/0602.105

https://www.researchgate.net/publication/326258350_Pancha_Kosha_Theory_of_Personality

CRITICAL ANALYSIS OF PANCHAKOSHA THEORY OF YOGA PHILOSOPHY

Yagyik Mishra
SGT University

January 2019

World Journal of Pharmaceutical Research 8(13):413
DOI:10.20959/wjpr201913-16152

https://www.researchgate.net/publication/338936490_CRITICAL_ANALYSIS_OF_PANCHAKOSHA_THEORY_OF_YOGA_PHILOSOPHY

THE LEVELS OF HUMAN CONSCIOUSNESS AND CREATIVE FUNCTIONING: INSIGHTS FROM THE THEORY OF PANCHA KOSHA (FIVE SHEATHS OF CONSCIOUSNESS)

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

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

Panchakosha

Shiv Jakhar

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

NURTURING THE WHOLE BEING: AN INDIAN PERSPECTIVE

PANCHA KOSHA VIVEKA PRAKARANAM
(Discrimination of the Five Sheaths)

PANCHADASI Chapter 3

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

PANCHA KOSHA VIVEKA
“Differentiation of the Five Sheaths”

PANCHADASHEE – 03

Composed by Sri Swami Vidyaranyaji

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

Unit-5 Concept of Pancha Koshas, Concept of Panchmahabhutas

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

https://egyankosh.ac.in/handle/123456789/59786

COMMENTARY ON THE PANCHADASI

SWAMI KRISHNANANDA

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),

http://www.nexusjournal.com/N2002-Nathan.html

Hindu Temple and the Structure of Human Body

Sanskriti

Vastu-Vaastu

https://en.sthapatyaveda.net/vastuvaastu

Vaastu Purusha Mandalam

by Dr. V. Ganapati Sthapati

Book

Ayadi Calculations

by Dr. V. Ganapati Sthapati

Book

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.

VASTU SHASTRA PRINCIPLES APPLIED IN TEMPLE COMPLEX OF NEPAL

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
DOI:10.1007/978-94-007-3934-5_9717-2
In book: Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (pp.1-16)

https://www.researchgate.net/publication/304183079_Cosmologies_of_India

Indian Temple Architecture, 3 Vols

Ananya Gandotra, 2011

An Encyclopedia of Hindu Architecture

Vol VII

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”

Acharya,V.A,

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

https://www.ijser.org/researchpaper/Mandapa-Its-Proportion-as-a-tool-in-Understanding-Indian-Temple-Architecture.pdf

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

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

Source: Matrikas

Key Terms

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

Fractals, Circle, and Sphere Packing

Source: Circle Packing Explorations

Source: Circle Packing Explorations

Source: On a Diophantine Equation That Generates AllIntegral Apollonian Gaskets

Source:

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: Recursive Apollonian Gasket

Source: On the cover: Apollonian packing

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

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

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

Apollonius of Perga

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

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

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

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

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

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

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

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

Apollonian packings

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

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

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

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

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

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

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

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

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

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

Hyperbolic geometry

All these scorpions have the same hyperbolic size.

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

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

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

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

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

ab⊕cd=a+bc+dab⊕cd=a+bc+d

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

Möbius transformations

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

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

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

z↦az+bbz+dz↦az+bbz+d

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

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

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

Beyond the packing

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

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

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

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

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

To infinity

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

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

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

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

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

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

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

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

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

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

David Sheard

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

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

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

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

Source: Circle patterns in Gothic Architecture

My Related Posts

Indra’s Net: On Interconnectedness

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

A Calculus for Self Reference, Autopoiesis, and Indications

64 Yogini Hindu Temples Architecture

Fractal Geometry and Hindu Temple Architecture

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Cosmic Mirror Theory

Shape of the Universe

Geometry of Consciousness

Mind, Consciousness and Quantum Entanglement

Cantor Sets, Sierpinski Carpets, Menger Sponges

Interconnected Pythagorean Triples using Central Squares Theory

Key Sources of Research

ON A DIOPHANTINE EQUATION THAT GENERATES ALL APOLLONIAN GASKETS

JERZY KOCIK

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

Click to access Diophantine.pdf

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

Circle Packing Explorations.

Francesco De Comite ́
Laboratoire d’Informatique Fondamentale de Lille

University of Sciences and Technology of Lille , France

Francesco.De-Comite@univ-lille1.fr

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

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

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

Circle patterns in Gothic Architecture

Tiffany C. Inglis and Craig S. Kaplan
Published 2012

David R. Cheriton School of Computer Science

University of Waterloo

piffany@gmail.com

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

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

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

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

Frontiers of Architectural Research

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

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

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

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

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

Nelly Shafik Ramzy

Sinai University

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

QUADRALECTIC ARCHITECTURE – A Panoramic Review

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

https://quadralectics.wordpress.com

An Introduction to the Apollonian Fractal

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

Click to access apollony.pdf

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

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

Apollonian gaskets and circle inversion fractals

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

Farr RS, Griffiths E.

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

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

On the cover: Apollonian packing

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

David Sheard
17 April 2020

On the cover: Apollonian packing

Recursive Apollonian Gasket

Photostream on Flickr

by FDeComite

2011

Recursive Apollonian Gasket

Quadrilaterals & Triangles

Malin Christersson
2019 

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

Apollonian circle packings: Dynamics and Number theory

Hee Oh

Yale University
ICWM, 2014

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

L.M.G. FEIJS

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

Spatial Statistics of Apollonian Gaskets, 

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

Experimental Mathematics, 28:3, 263-270,

 DOI: 10.1080/10586458.2017.1385037

Revisiting Apollonian gaskets

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

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

Matrikas

Wikipedia

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

Indra’s Pearls: The Vision of Felix Klein.

David Mumford, Caroline Series, and David Wright. 

Cambridge University Press,Cambridge, 2002.

Geometry in Art and Architecture

Paul Calter

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

The Circular Church Plan

Quadralectic Architecture