Seeds and Mountains: The Cosmogony of Temples in South Asia

Seeds and Mountains: The Cosmogony of Temples in South Asia

Key Terms

  • Viraj
  • Purusha
  • Hindu Temple
  • Architecture
  • Purush Sukta
  • Rg Veda
  • Creation Myth
  • Hinduism
  • Vedic Philosophy
  • Amalaka
  • Vastupurush Mandala
  • Amalaka Ekadashi
  • Gavaksa
  • Kumbha
  • Kalash
  • Shakti
  • Prakriti
  • Amla Fruit
  • Square and Circle
  • Circling the Square
  • Squaring the Circle
  • Heaven and Earth

Researchers

  • Adam Hardy
  • Stella Kramrisch
  • Alice Boner
  • Michael W Meister
  • George Michell
  • Ananda K. Coomaraswamy
  • Madhusudan A. Dhaky
  • Vasu Renganathan
  • Brown, Percy
  • Subhash Kak
  • Datta, Sambit
  • Sonit Bafna
  • Zimmer, Heinrich
  • Pramod Chandra
  • Krishna Deva

THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Source: THE TEMPLE AS PURUSA

Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

Source: Seeds and Mountains: The Cosmogony of Temples in South Asia

My Related Posts

You can search for these posts using Search Posts feature in the right sidebar.

  • Temples of the Indus: Studies in the Hindu Architecture of Ancient Pakistan
  • Art and Architecture of Gandhara Buddhism
  • An Infinity of Stupas: Design and Architecture of Chinese Buddhist Temples and Pagodas
  • Design Principles of Early Stone Pagodas in Ancient Korean Architecture
  • The Architecture and Sacred Temple Geometry of Japanese Buddhist Temples
  • Nichiren School of Buddhist Philosophy
  • Pure Land School of Buddhist Philosophy
  • Shingon (Esoteric) School of Buddhist Philosophy
  • Three Treatise School (Sanlun) of Chinese Buddhist Philosophy
  • Hua Yan Buddhism : Reflecting Mirrors of Reality
  • What is Yogacara Buddhism (Consciousness Only School)?
  • Dhyan, Chan, Son and Zen Buddhism: Journey from India to China, Korea and Japan
  • Chinese Tiantai and Japanese Tendai Buddhism
  • Intersubjectivity in Buddhism
  • Schools of Buddhist Philosophy 
  • Meditations on Emptiness and Fullness
  • Self and Other: Subjectivity and Intersubjectivity
  • Law of Dependent Origination 
  • The Fifth Corner of Four: Catuskoti in Buddhist Logic
  • Square and Circle of Hindu Temple Architecture
  • Indira’s Pearls: Apollonian Gasket, Circle and Sphere Packing
  • Cantor Sets, Sierpinski Carpets, Menger Sponges
  • Fractal Geometry and Hindu Temple Architecture
  • The Great Chain of Being
  • INTERCONNECTED PYTHAGOREAN TRIPLES USING CENTRAL SQUARES THEORY
  • The Pillar of Celestical Fire
  • Purush – The Cosmic Man
  • Platonic and Archimedean Solids
  • Fractal and Multifractal Structures in Cosmology

Key Sources of Research

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

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

Studies in Indian Temple Architecture

Michael W Meister

1976, Artibus Asiae

PARTS AND WHOLES: THE STORY OF THE GAVĀKSA

Adam Hardy

Vārāṭa Temples: The Lost Tradition In-Between 

Adam Hardy

EARLY ARCHITECTURE AND ITS TRANSFORMATIONS: NEW EVIDENCE FOR VENACULAR ORIGINS FOR THE INDIAN TEMPLE

Michael W Meister

Indian Architecture

(Buddhist and Hindu Periods)

Percy Brown

Encyclopedia of Indian Temple Architecture

North India

Foundations of North Indian Style

Encyclopedia of Indian Temple Architecture

South India

Lower Dravidadesa

Vāstupuruṣamaṇḍalas: Planning in the Image of Man

In: Maṇḍalas and Yantras in the Hindu Traditions
Author: Michael W. Meister

Type: Chapter
Pages: 251–270
DOI: https://doi.org/10.1163/9789004492370_012

https://brill.com/display/book/9789004492370/B9789004492370_s012.xml

Maṇḍalas and Yantras in the Hindu Traditions

Series:
Brill’s Indological Library, Volume: 18
Author: Gudrun Bühnemann

E-Book (PDF)
Availability: Published
ISBN: 978-90-04-49237-0
Publication: 15 Nov 2021

Hardback
Availability: Published
ISBN: 978-90-04-12902-3
Publication: 30 Jun 2003

Michael W. Meister, M. A. Dhaky and Krishna Deva (ed.): Encyclopaedia of Indian temple architecture: North India: Foundations of North Indian style c. 250 b.c.–a.d. 1100. I: xvii, 422 pp. 158 figures, 15 maps; II: viii, 778 plates.

New Delhi: American Institute of Indian Studies, and Princeton: Princeton University Press, 1988.

Published online by Cambridge University Press: 24 December 2009

George Michell

https://www.cambridge.org/core/journals/bulletin-of-the-school-of-oriental-and-african-studies/article/abs/michael-w-meister-m-a-dhaky-and-krishna-deva-ed-encyclopaedia-of-indian-temple-architecture-north-india-foundations-of-north-indian-style-c-250-bcad-1100-i-xvii-422-pp-158-figures-15-maps-ii-viii-778-plates-new-delhi-american-institute-of-indian-studies-and-princeton-princeton-university-press-1988/6051A1F5A8E6E9BDE4F13143E4C2BBDF

Michael W. Meister. Review of “The Temple Architecture of India” by Adam Hardy

CAA Reviews
April 2008
DOI:10.3202/caa.reviews.2008.40
Authors:
Michael Meister
University of Pennsylvania

https://www.researchgate.net/publication/240807621_Michael_W_Meister_Review_of_The_Temple_Architecture_of_India_by_Adam_Hardy

Early Indian Architecture and Art

Subhash Kak

Migration & Diffusion – An international journal, Vol.6/Nr.23, 2005, pages 6-27

Encyclopedia of Indian Temple Architecture, North India, Volume II, Part I: Foundations of North Indian Style.

(Two books: text and plates)

(ENCYCLOPAEDIA OF INDIAN TEMPLE ARCHITECTURE) First Edition

by Michael W. Meister (Editor), Madhusudan A. Dhaky (Editor)

Publisher ‏ : ‎ Princeton University Press; First Edition (October 21, 1989)
Language ‏ : ‎ English
Hardcover ‏ : ‎ 816 pages
ISBN-10 ‏ : ‎ 0691040532
ISBN-13 ‏ : ‎ 978-0691040530

Drāviḍa Temples in the Samarāṅgaṇasūtradhāra

Adam Hardy

Encyclopedia of Indian Temple Architecture – North and South India (Eight Volumes in 16 Books)

AUTHOR:MICHAEL W. MEISTERGEORGE MICHELLM. A. DHAKY
PUBLISHER:AMERICAN INSTITUTE OF INDIAN STUDIES
LANGUAGE:ENGLISH
EDITION:1986 – 2001

Hardy, “The Temple Architecture of India” review

Michael W Meister
Published 2008

https://www.academia.edu/5688611/Hardy_The_Temple_Architecture_of_India_review

“Geometry and measure in Indian temple plans: Rectangular temples.”


Meister, Michael W..

Artibus Asiae 44 (1983): 266. DOI: 10.2307/3249613

Measurement and proportion in Hindu temple architecture

Michael W Meister

1985, Interdisciplinary science reviews

Tradition and Transformation: Continuity and Ingenuity in the Temples of Karnataka

Adam Hardy
2001, The Journal of the Society of Architectural Historians

De- and Re-constructing the Indian Temple. 

Meister, M. W. (1990).

Art Journal49(4), 395–400. https://doi.org/10.1080/00043249.1990.10792723

https://architexturez.net/doc/az-cf-175759

https://www.tandfonline.com/doi/pdf/10.1080/00043249.1990.10792723

Re-creation and self-creation in temple design

Adam Hardy

Architectural Research Quarterly

Dependence and Freedom in the Theory and Practice of Indian Temple Architecture

Adam Hardy

2023, Embodied Dependencies and Freedoms: Artistic Communities and Patronage in Asia

2013 – Indian Temple Typologies

Adam Hardy

Kashmiri Temples: a Typological and Aedicular Analysis

Adam Hardy

2019, Indology’s Pulse: Arts in Context (Essays Presented to Doris Meth Srinivasan in Admiration of Her Scholarly Research) edited by Corinna Wessels-Mevissen and Gerd J. R. Mevissen

Mandala and Practice in Nagara Architecture in North India

Michael W Meister
Published 1979

Building science of Indian temple architecture

Shweta Vardia

Published 2013

Prāsāda as Palace: Kūṭina Origins of the Nāgara Temple

Michael W Meister
1988, Artibus Asiae

https://www.jstor.org/stable/3250039

On the Development of Indian Temple Architectural Morphology and the Origin of Superstructure

Vasu Renganathan
Published 2007

On the development of a morphology for a symbolic architecture: India

Michael W Meister

1986, RES: Anthropology and Aesthetics

Symbol and Surface: Masonic and Pillared Wall-Structures in North India

Michael W Meister

Artibus Asiae

A Note on the Superstructure of the Marhia Temple

Michael W Meister
1974, Artibus Asiae

Indo-Aryan’ Temples: Noodling Seventh-Century Nagara

Michael W Meister
JISOA ns vol. 27

Region, Style, Idiom, and Ritual in History

Michael W. Meister on the Study of Jain Art

John E. Cort

Chakshudana or Opening the Eyes
Seeing South Asian Art Anew
Edited by Pika Ghosh and Pushkar Sohoni
First published 2024
ISBN: 978- 1- 032- 20783- 4 (hbk)
ISBN: 978- 1- 032- 27121- 7 (pbk)
ISBN: 978- 1- 003- 29147- 3 (ebk)

DOI: 10.4324/ 9781003291473- 3

https://library.oapen.org/bitstream/handle/20.500.12657/76179/9781003291473_10.4324_9781003291473-3.pdf?sequence=1

Mountains and cities in Cambodia: Temple architecture and divine vision.

Meister, M.W.

Hindu Studies 4, 261–268 (2000). https://doi.org/10.1007/s11407-000-0009-2

https://link.springer.com/article/10.1007/s11407-000-0009-2

https://www.jstor.org/stable/20106740

References cited
  • Boisselier, Jean. 1997. The meaning of Angkor Thom. In Helen Ibbitson Jessup and Thierry Zephir, eds., Sculpture of Angkor and ancient Cambodia: Millennium of glory, 117–22. London: Thames and Hudson.Google Scholar 
  • Coomaraswamy, A. K. 1992. Early Indian architecture, I–IV. In Michael W. Meister, ed., Ananda K. Coomaraswamy: Essays in early Indian architecture, 3–69, 105–24. Delhi: Oxford University Press.Google Scholar 
  • Coomaraswamy, A. K. 1993 [1928–31]. Yakṣas: Essays in the water cosmology (ed. Paul Schroeder). Delhi: Oxford University Press.Google Scholar 
  • Dagens, Bruno. 1997. Angkor: ‘The city on the Ganges.’ Connaissance des arts [special exhibition issue] pp. 23–24.
  • Giteau, Madeleine. 1997. The profound sense of the sacred. Connaissance des arts [special exhibition issue] pp. 42–57.
  • Groslier, Bernard Philippe 1962. Indochina: Art in the melting-pot of races. London: Methuen.Google Scholar 
  • Jessup, Helen Ibbitson. 1997. Temple-mountains and the Devarāja cult. In Helen Ibbitson Jessup and Thierry Zephir, eds. 1997. Sculpture of Angkor and ancient Cambodia: Millennium of glory, 101–16. London: Thames and Hudson.Google Scholar 
  • Jessup, Helen Ibbitson and Thierry Zephir, eds. 1997. Sculpture of Angkor and ancient Cambodia: Millennium of glory. London: Thames and Hudson.Google Scholar 
  • Meister, Michael W. 1979. Maṇḍala and practice in Nāgara architecture in North India. Journal of the American Oriental Society 99, 2: 204–19.Article MathSciNet Google Scholar 
  • Meister, Michael W. 1984. Śiva’s forts in central India: Temples in Dakṣiṇa Kosala and their ‘daemonic’ plans. In Michael W. Meister, ed., Discourses on Śiva: Proceedings of a symposium on the nature of religious imagery, 119–42. Philadelphia: University of Pennsylvania Press.Google Scholar 
  • Meister, Michael W. 1986a. Measurement and proportion in Hindu temple architecture. Interdisciplinary science reviews 10, 3: 248–58.Google Scholar 
  • Meister, Michael W. 1986b. On the development of a morphology for a symbolic architecture: India. Res, anthropology, and aesthetics 12: 33–50.Google Scholar 
  • Meister, Michael W. 1989. Prāsāda as palace: Kūṭina origins of the nāgara temple. Artibus Asiae 49, 3–4: 254–80.Google Scholar 
  • Meister, Michael W. 1990. De- and re-constructing the Indian temple. Art journal 49, 4: 395–400.Article Google Scholar 
  • Meister, Michael W. 1991. The Hindu temple: Axis and access. In Kapila Vatsyayan, ed., Concepts of space, ancient and modern, 269–80. New Delhi: Abhinav.Google Scholar 
  • Meister, Michael W. 1992. Symbology and architectural practice in India. In Emily Lyle, ed., Sacred architecture in the traditions of India, China, Judaism, and Islam, 5–24. Edinburgh: Edinburgh University Press.Google Scholar 
  • Meister, Michael W. 1996. Reassessing the text. In Farooq Ameen, ed., Contemporary architecture and urban form: The South Asian paradigm, 88–100. Bombay: Marg.Google Scholar 
  • Meister, Michael W. and M. A. Dhaky, eds. 1983. Encyclopaedia of Indian temple architecture: South India, lower Drāviḍadeśa. Philadelphia: University of Pennsylvania Press.Google Scholar 
  • Meister, Michael W., M. A. Dhaky and Krishna Deva, eds. 1988. Encyclopaedia of Indian temple architecture: North India, foundations of North Indian style. Princeton: Princeton University Press.Google Scholar 
  • Menon, C. P. S. 1932. Early astronomy and cosmology: A reconstruction of the earliest cosmic system. London: G. Allen & Unwin.Google Scholar 
  • Muschamp, Herbert. 1997. The designs of a genius redesigning himself. The New York times 18 July: C1, 29.
  • Zimmer, Heinrich. 1983 [1955]. The art of Indian Asia. 2 vols. Princeton: Princeton University Press.Google Scholar 

Constructing Community: Tamil Merchant Temples in India and China, 850-1281

Lee, Risha

https://doi.org/10.7916/D8W95H8W

https://academiccommons.columbia.edu/doi/10.7916/D8W95H8W

This dissertation studies premodern temple architecture, freestanding sculpted stones, and Tamil language inscriptions patronized by south Indian merchants in south India and China. Between the ninth and thirteenth centuries, Indian Ocean trade was at its apex, connecting populations on European and Asian continents through complex interlocking networks. Southern India’s Tamil region, in particular, has been described as the fulcrum of the Indian Ocean circuit; however, knowledge of intra-Asian contact and exchange from this period has been derived mostly from Arabic and Chinese sources, which are abundant in comparison with the subcontinent’s dearth of written history. My project redresses this lacuna by investigating the material culture of Tamil merchants, and aims to recover their history through visual evidence, authored by individuals who left few written traces of their voyages across the Indian Ocean. 

The arguments of my dissertation are based primarily on unpublished and unstudied monuments and inscriptions, weaving together threads from multiple disciplines–art history, literature, epigraphy, and social theory–and from across cultures, the interconnected region of the eastern Indian Ocean and the South China Seas, spanning the Sanskritic, Tamil, Malay, and Sinocentric realms. My dissertation challenges traditional narratives of Indian art history that have long attributed the majority of monumental architecture to royal patrons, focusing instead on the artistic production of cosmopolitan merchants who navigated both elite and non-elite realms of society. I argue that by constructing monuments throughout the Indian Ocean trade circuit, merchants with ties to southern India’s Tamil region formulated a coherent group identity in the absence of a central authority. 

Similar impulses also are visible in merchants’ literary production, illustrated through several newly translated panegyric texts, which preface mercantile donations appearing on temple walls in the modern states of Tamil Nadu, Karnataka, and Andhra Pradesh. Moreover, my work analyzes the complex processes of translation visible in literary and material culture commissioned by merchants, resulting from inter-regional and intercultural encounters among artisans, patrons, and local communities. Rather than identifying a monolithic source for merchants’ artistic innovations, in each chapter I demonstrate the multiple ways in which merchants employed visual codes from different social realms (courtly, mercantile, and agrarian) to create their built environments. In Chapter Four, I provide a detailed reconstruction and historical chronology of a late thirteenth century temple in Quanzhou, coastal Fujian Province, and southeastern China, which both echoes and transforms architectural forms of contemporaneous temples in India’s Tamil region. 

Piecing together over 300 carvings discovered in the region in light of archaeological and art historical evidence, I develop a chronology of the temple’s history, and propose that Ming forces destroyed the temple scarcely a century after its creation. In Chapter Three, I interpret stone temples patronized by the largest south Indian merchant association, the Ainnurruvar, as being integral to their self-fashioning in India and abroad. While the temples do not project a merchant identity per se, I show that they employ an artistic vocabulary deeply entrenched in the visual language of the Tamil region. Chapter Two looks at other forms through which merchants created a shared mercantile culture, including literary expressions and freestanding sculptural stones. These texts demonstrate that merchants engaged in both elite and non-elite artistic production. Chapter One analyzes the distribution, content, and context of Tamil merchant sponsored inscriptions within the Indian Ocean circuit, focusing on the modern regions of Tamil Nadu, Karnataka, Kerala, and Andhra Pradesh. An appendix offers new translations of important Tamil language mercantile inscriptions discovered throughout south India.

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

Datta, Sambit.

Nexus Network Journal 12 (2010): 471-483.

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

https://espace.curtin.edu.au/bitstream/handle/20.500.11937/26171/202269_202269.pdf;jsessionid=F9680661D8F6240F552A9D80C0741853?sequence=2

“Evolution and Interconnection: Geometry in Early Temple Architecture.” 

Datta, Sambit.

Digital Techniques for Heritage Presentation and Preservation (2021): n. pag.

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

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
ISSN 2229-5518

The Construction Geometry of Early Javanese Temples

David Beynon, Deakin University

Sambit Datta, Curtin University

Papers presented to the 30th Annual Conference of the Society of Architectural Historians, Australia and New Zealand held on the Gold Coast, Queensland, Australia, July 2-5, 2013.
http://www.griffith.edu.au/conference/sahanz-2013/

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

Sonit Bafna,

Journal of the Society of Architectural Historians. 59, no. 1 (2000), 26-49.

https://online.ucpress.edu/jsah/article-abstract/59/1/26/59369/On-the-Idea-of-the-Mandala-as-a-Governing-Device?redirectedFrom=fulltext

Mountain Temples and Temple-Mountains: Masrur 

Michael W. Meister

Journal of the Society of Architectural Historians (2006) 65 (1): 26–49.

https://doi.org/10.2307/25068237

https://online.ucpress.edu/jsah/article-abstract/65/1/26/60132/Mountain-Temples-and-Temple-Mountains-Masrur?redirectedFrom=fulltext

Decoding a Hindu Temple

Toronto’s Bochasanwasi Akshar Purushottam Swaminarayan Sanstha (BAPS) Shri Swaminarayan Mandir and the Mandala as a Principle of Design

Krupali Uplekar Krusche
Volume 46, Number 2, 2021 World religions in Canada
Religions mondiales au Canada

URI: https://id.erudit.org/iderudit/1088489ar DOI: https://doi.org/10.7202/1088489ar

Aspects of Indian Art and Architecture

Centre for Historical Studies

M21409

Architecture of India

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

Space and Cosmology in the Hindu Temple

Subhash Kak
2002, Vaastu Kaushal: International Symposium on Science and Technology in Ancient Indian Monuments

Hindu Temple Architecture in India

Dr. Vinod Kumar

Sociology, Vallabh Government College, Mandi Himachal Pradesh 175001, India

Studies in Art and ArchitectureISSN 2958-1540

http://www.pioneerpublisher.com/SAA

Volume 3 Number 1 March 2024

https://www.pioneerpublisher.com/SAA/article/view/700




WORKSHOP IN INDIAN ARCHITECTURE

Art 514: Proseminar in Indian Art Fall 2000

Hist. of Art Dept., Jaffe Building 113, Weds. 3-5

Professor Michael W. Meister, Jaffe 308

http://www.arthistory.upenn.edu/fall00/514/syl.html

Archive: The University of Pennsylvania houses a photographic archive of Indian art and architecture (now over 100,000 photographs) as part of the W. Norman Brown South Asia Reference Room on the fifth floor west end of Van Pelt library. To gain access, contact the South Asia bibliographer, David Nelson, or his staff. This Archive should be an integral part of your work this semester.

Intention: This seminar will both introduce you in the remarkable variety of India’s architectural accomplishments and encourage you to discuss the broader issues of how architecture can be designed to express meaning. In the past I have sometimes asked students to divide into groups to work together to frame one area of India’s architecture. Categories have been: Early Indian architecture; South Indian architecture; North Indian architecture; early Islamic architecture in India.

This year, I propose to organize readings around a variety of approaches and methodologies: issues of construction, translation of architectural forms into new materials, architectural symbolism, typology and chronology, and praxis (the use and survival of buildings over time).

I will ask you to work collectively, but on different aspects or examples of the general area, reporting in class on the literature, issues, ideas, and substance appropriate to each.

Books: Four books have been ordered by the Penn Book Center (130 S. 34th St.):

Ananda K. Coomaraswamy, Essays in Early Indian Architecture, ed. Michael W. Meister, Oxford University Press, 1993. (This may now be out of print, but is available in the Fine Arts Library reserve.)

Richard H. Davis, Lives of Indian Images, Princeton, 1997 (Princeton University Press paperback).

James C. Harle, The Art and Architecture of the Indian Subcontinent (Pelican History of Art). New York, 1986.

George Michell, The Hindu Temple: An Introduction to Its Meaning and Forms, New York, 1977 (Chicago University Press paperback).

Other books of interest will be placed on Reserve in the Fischer Fine Arts Library and the South Asia Reading Room in Van Pelt, near the photo archive.

Course assignments: In addition to participation in class discussion, students will be asked to prepareshort reports on reading for presentation in class and to choose an area for research leading to a final presentation and paper.

________________________________

Brief Bibliography of General Surveys:

Batley, Claude. The Design Development of Indian Architecture, 3rd rev. enl. ed., London, 1973.

Brown, Percy. Indian Architecture, vol. 1. Buddhist and Hindu periods, vol. 2. Islamic period, 5th ed., Bombay, 1965-68.

Coomaraswamy, Ananda K. The Arts and Crafts of India and Ceylon. London, 1913.

Coomaraswamy, Ananda K. History of Indian and Indonesian Art. New York, 1927.

Fergusson, James. History of Indian and Eastern Architecture, London, 1876; rev. and ed. by James Burgess, 2 vol., London, 1910.

Herdeg, Klaus. Formal Structure in Indian Architecture, preface by Balkrishna Doshi, New York: Rizzoli, 1990 (1978).

Mayamata. An Indian Treatise on Housing, Architecture, and Iconography, trans. by Bruno Dagens, Delhi, 1985.

Pereira, José. Elements of Indian Architecture, Delhi, 1987.

Tadgell, Christopher. The History of Architecture in India: From the Dawn of Civilization to the End of the Raj, London: Architecture Design and Technology Press, 1990.

Volwahsen, Andreas. Living Architecture: Indian and Living Architecture: Islamic Indian, New York, 1969-70.

Architecture of India

RBSI

https://rarebooksocietyofindia.org/grid-layout.php?t=23

From Sikhara to Sekhari: Building from the Ground Up

Michael W. Meister

Temple Architecture and Imagery of south and southeast Asia

Temple Architecture and Imagery of South and Southeast Asia: Prasadanidhi:

Papers Presented to Professor M.A. Dhaky Hardcover – May 27, 2016

by Parul Pandya Dhar (Author), Gerd J.R. Mevissen (Author)

THE TEMPLE AS PURUSA

STELLA KRAMRISCH

(Plates 1-6)

STUDIES IN INDIAN TEMPLE ARCHITECTURE

Papers presented at a Seminar held in Varanasi, 1967

Edited with an introduction by

PRAMOD CHANDRA 

Professor in the Department of Art

The University of Chicago

Virāja

https://en.wikipedia.org/wiki/Virāja#:~:text=Viraja%20is%20born%20from%20Purusha,and%20Shiva(Lord%20Ayyappa).

Vedā: Puruṣa Śuktam

Chapter 6 – Description of the Virāṭ Puruṣa—Exposition of the Puruṣa Sūkta

https://www.wisdomlib.org/hinduism/book/the-bhagavata-purana/d/doc1122474.html

Viraj

https://static.hlt.bme.hu/semantics/external/pages/Rta/en.wikipedia.org/wiki/Viraj.html

Purusha

https://en.wikiquote.org/wiki/Purusha

Purusha Sukta

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

Chapter 6: The Purusha Sukta of the Veda

https://www.swami-krishnananda.org/religious.life/religious.life_06.html

Chapter 7: The Doctrine of Creation in the Purusha Sukta

https://www.swami-krishnananda.org/religious.life/religious.life_07.html

Creation Myths

XII—Purusha Sukta: an Aurobindonian Interpretation (A)

http://savitri.in/library/resources/sanatana-dharma/aug-09-2009

Purusha Sukta (Text, Transliteration, Translation and Commentary)

AUTHOR: S.K. RAMACHANDRA RAO
PUBLISHER: SRI AUROBINDO KAPALI SASTRY INSTITUTE OF VEDIC CULTURE
LANGUAGE: SANSKRIT TEXT WITH TRANSLITERATION AND ENGLISH TRANSLATION
EDITION: 2014
ISBN: 8179940462
PAGES: 96

“Cosmogony as Myth in the Vishnu Purāṇa.” 

Penner, Hans H.

History of Religions 5, no. 2 (1966): 283–99. http://www.jstor.org/stable/1062116.

Chapter 53 – Importance of studies of Vedas for Varnins; Origin of the universe, of speech and Vedas

https://www.swaminarayan.faith/scriptures/en/satsangi-jeevan/prakran-5/53

The Purusha sukta

by Kamesvara Aiyar, B. V., tr

Publication date 1898
Publisher Madras, G. A. Natesan

English Commentary on Purusha Sukta of Veda

by S K Ramachandra Rao

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

Fractal Geometry and Hindu Temple Architecture

Fractal Geometry and Hindu Temple Architecture

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

Proportion and measurements were the guiding tools for Indian temple construction starting from the 5th century onwards and it continuous even now. Through out the history proportion dominated as a tool, which determined the monuments both spatial arrangements as well as form. The ancient texts, therefore, insist on a high degree of precision in their measurements.

The standard text Mayamata mentions- ”Only if the temple is constructed correctly according to a mathematical system can it be expected to function in harmony with the universe. Only if the measurement of the temple is in every way perfect, there will be perfection in the universe as well.”

Source: A REVIEW STUDY ON ARCHITECTURE OF HINDU TEMPLE

Source: TEMPLE ARCHITECTURE AND SCULPTURE

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

Source: TEMPLE ARCHITECTURE AND SCULPTURE

Source: TEMPLE ARCHITECTURE AND SCULPTURE

Key Terms

  • Fractals
  • Cosmology
  • Temple Architecture
  • Fractal Dimension
  • Recursive
  • Algorithmically
  • Hindu Temples
  • Vastu Purush Mandala
  • Vastu Shastra
  • Shilpa Shastra
  • Nagara Style
  • Dravidian Style
  • Vesara Style
  • Kalinga Style
  • 64 Yogini Temple Style
  • Jain Temple Architecture
  • Buddhist Stupa Architecture
  • Cellular Automata
  • 3D Fractals
  • 2D Cellular Automata
  • Nine Cell Square
  • Nav Grah Yantra
  • Sierpinski Carpet
  • Box Counting Method
  • Biophilic Architecture
  • Symbolism
  • Square and Circle
  • Earth and Heaven
  • Squaring the Circle
  • Correspondence
  • Equivalence
  • Symmetry
  • As Above, So below
  • Cosmic Mirrors

Hindu Temples: Models of a Fractal Universe

Source: Hindu temples: Models of a fractal universe

Hindu philosophy views the cosmos to be holonomic and self-similar in nature. According to ancient architectural tradition, Hindu temples are symbols of models of the cosmos and their form represents the cosmos symbolically.

The procedures and methods used in the construction of Hindu temples bear a striking resemblance to the procedures of computer graphics, including discretization, fractalization and extensive use of recursive procedures, including self-similar iteration. The instructions given in ancient Vastu shastras (texts on architecture) work like general programmes to generate various types of temples.

The paper is an attempt to draw attention to the similarities between the procedures and resulting forms in computer graphics and Hindu temple architecture and to explain the relationship that exists between the form of the temple and the concepts of Hindu philosophy. It is proposed that Hindu temples may be viewed as three dimensional fractal models and that the use of fractal geometry procedures has a special symbolic meaning in the generation of the forms of Hindu temples.

Introduction to the Temple Architecture in the Indian Context

Source: Temples of Odisha- the Geometry of Plan Form

The evolution of temple architecture is marked by a strict adherence to the original ancient models, that were derived from sacred thought which persisted over many centuries. The commencement of the main style of Hindu temple architecture in India dates back to the Mauryan period i.e 3rd century BC, as evident from the archaeological excavation at Sanchi (Madhya Pradesh, temple no.40 and18) and Bairat (Rajastan), (DB Garnayak , 2007) . The Indian Silpasastras recognize three main types of temples known as the Nagara, Dravida and Vesara. Nagara temple belongs to the country from the Himalaya to the Vindhya, Vesara from the Vindhya to the Krishna and the Dravida from the Krishna to the Cape Comorin (DB Garnayak , 2007). An inscription in 1235 A.D in the mukhamandapa of the Amritesvara temple at Holal in Bellary district of Karnataka speaks of the fourth style i.e. Kalinga, in addition to the above three. The Kalinga style of Architecture is explained exclusively in the texts like Bhubana Pradip, Silpa Prakasa, Silpa Ratnakosha etc.

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

Source: Fractal Geometry as a source of innovative formations in interior design

Source: The fractal analysis of architecture: calibrating the box-counting method using scaling coefficient and grid disposition variables

Source: PARAMETRIZING INDIAN KARNATA-DRAVIDA TEMPLE USING GEOMETRY

Vastu purusha mandala

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: Exploring Ancient Architectural Designs with Cellular Automata

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

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: 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

FRACTAL DESIGN, ARCHITECTURE AND ART IN HUMAN HISTORY

Source: Working with Fractals

Fractals have permeated cultures spanning across many centuries and continents, classical art and vernacular architecture from the column capitals of ancient Greece, Egyptian, Aztec, Incan civilisations, the art of Ancient Mayans, Islamic and Hindu temples, Angkor Wat in Cambodia, the Eifel Tower in Paris, and the structures of Santiago Calatrava. Fractals are also evident in such well known works as those of Botticelli, Vincent van Gogh, and Jackson Pollock. Their visual properties were also explored by mathematicians when Benoit Mandelbrot published The Fractal Geometry of Nature (1982) in which he catalogued nature’s statistical fractals and discussed them using mathematical methods for their replication.

Fractals constitute a central component of human daily experience of the environment (Taylor & Spehar, 2016). While extensive research has documented the negative effects of environments that do not have
a complement of rich experiential aesthetic variety (Mehaffy & Salingaros, 2013), their proliferation in art and design has continued to grow and diversify, creating architecture, interiors and products designed for human needs (Taylor & Spehar, 2016). Over the past two decades, interdisciplinary teams have confirmed that the aesthetic qualities of nature’s fractal patterns can induce striking effects on health.1

PARAMETERS OF FRACTAL PREFERENCE

Source: Working with Fractals: For the Love of Fractals

The universal preference for ‘statistical’ fractals peaks at low to moderate degree of complexity, while universal preference for ‘exact’ fractals peaks at a higher complexity. The high level of symmetry in exact fractals enables greater tolerance for visual complexity compared to statistical fractals (Abboushi et al., 2019). Four factors influence complexity in exact fractals:

1. Fractal dimension (D)

2. Symmetry

3. Recursion

4. Number of elements introduced at each recursion

Fractal dimension. 

The Euclidean simplicity and symmetry of exact fractals increases tolerance and peak preference for medium-high complexity exact fractals (D= 1.5–1.7) (Abboushi et al., 2019). Medium- high exact fractals can enhance visual preference and mood, particularly in less complex Euclidean interior spaces (Abboushi et al., 2019; Taylor et al., 2018).

When complex fractal patterns are experienced within a low-complexity interior space, the visual preference can shift to those available higher D values (1.5 to 1.7, medium-high range), suggesting that a low complexity environment enables a tolerance and preference for higher complexity statistical fractals such as found in artworks or casted light patterns unique within that space (Abboushi et al., 2019). A good example of this scenario is museums with an abundance of geometrical rooms and white walls adorned with highly complex artworks that captivate.

Symmetry. 

Patterns with symmetry and geometry, such as common among exact fractals, can be visually appealing as they balance interest and comprehensibility. Mirror symmetry is generally considered one of the most predictive factors when judging whether a geometric pattern is ‘beautiful’. A lack of radial and mirror symmetry can be overcome by including more recursion and higher fractal dimensionality.

The orderliness of exact fractals allows a pattern to approach the maximum use of space at a particular dimension while retaining its elegance. Patterned tiles and carpet, wall coverings and textiles, artefacts and ornaments found in many cultures (Eglash, 2002) are evidence of this spatial orderliness and symmetry.

Recursion. 

Fractals generated by a finite subdivision rule bear a striking resemblance to both nature and human ornament. In mathematics, the finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. In a sense, subdivision rules are generalisations of regular exact fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals (Cannon, et al., 2001).

Source: The application of complexity theory and Fractals in architecture, urban planning and design

Source: The application of complexity theory and Fractals in architecture, urban planning and design

My Related Posts

Shapes and Patterns in Nature

Shape of the Universe

Cosmic Mirror Theory

Interconnected Pythagorean Triples using Central Squares Theory

Indra’s Net: On Interconnectedness

The Great Chain of Being

Maha Vakyas: Great Aphorisms in Vedanta

Growth and Form in Nature: Power Laws and Fractals

Geometry of Consciousness

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Mind, Consciousness and Quantum Entanglement

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

From Systems to Complex Systems

The Pillar of Celestial Fire

Key Sources of Research:

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

The Hindu Temple as a Model of Fractal Cosmology – Forecasting Architecture with Recursive Instruction

Data is Nature

Monday, 6 April 2015

http://www.dataisnature.com/?p=2138

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

Building Science of Indian Temple Architecture

Shweta Vardia

shwetavardia@gmail.com

2008 MS Thesis

Universidade do Minho, Portugal

The Fractal Structure of Hindu Temples

Fractal Enlightenment

Fractal Geometry And Self-Similarity In Architecture: An Overview Across The Centuries

Nicoletta Sala
Academy o f Architecture o f Mendrisio, University o f Italian Switzerland Largo Bernasconi CH- 6850 Mendrisio
Switzerland
E-mail: nsala @ arch.unisLch

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

Fractal geometry and architecture: some interesting connections

N. Sala

Accademia di Architettura, Università della Svizzera italiana, Mendrisio, Switzerland

Eco-Architecture: Harmonisation between Architecture and Nature 163

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

Fractal Geometry as a source of innovative formations in interior design 

Omniah Bahaa Ibrahim

Teaching Assistant, at Faculty of Applied Arts, Interior Design & Furniture Department, Helwan University, Giza, Egypt

DOI: 10.21608/jdsaa.2021.42275.1075

Fractal Geometry and Architecture Design: Case Study Review

Xiaoshu Lu1,2, Derek Clements-Croome3, Martti Viljanen1

1Department of Civil and Structural Engineering, School of Engineering, Aalto University, PO Box 12100, FIN-02150, Espoo, Finland
E-mail: xiaoshu@cc.hut.fi
2Finnish Institute of Occupational Health, Finland
3School of Construction Management and Engineering, Whiteknights, University of Reading, PO Box 219, Reading RG6 6AW, UK

Click to access 2_CMSIM_2012_Lu_clements_Croome_viljanen_2_311-322.pdf

A review of the fractal geometry in structural elements

Aman Upadhayay, Dr. Savita Maru
Department of Civil Engineering, Ujjain Engineering College, India

International Journal of Advanced Engineering Research and Science (IJAERS)
Peer-Reviewed Journal
ISSN: 2349-6495(P) | 2456-1908(O)

Vol-8, Issue-7; Jul, 2021
Journal Home Page Available: https://ijaers.com/&nbsp;

Article DOI: https://dx.doi.org/10.22161/ijaers.87.3

The fractal analysis of architecture: calibrating the box-counting method using scaling coefficient and grid disposition variables

Michael J Ostwald

School of Architecture and Built Environment, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia;

email: Michael.Ostwald@newcastle.edu.au
Received 15 July 2011; in revised form 21 March 2012

Environment and Planning B: Planning and Design 2013, volume 40, pages 644 – 663 

doi:10.1068/b38124

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

African Fractals

MODERN COMPUTING AND INDIGENOUS DESIGN

RON EGLASH

Book

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

Exploring Ancient Architectural Designs with Cellular Automata

Hokky Situngkir
[hokky.situngkir@surya.ac.id]
Dept. Computational Sociology, Bandung Fe Institute Center for Complexity Studies in Surya University

BFI Working Paper Series WP-9-2010

WORKING WITH FRACTALS

A RESOURCE FOR PRACTITIONERS OF BIOPHILIC DESIGN

A PROJECT OF THE EUROPEAN ‘COST RESTORE ACTION’ 

PREPARED BY RITA TROMBIN

The application of complexity theory and Fractals

in architecture, urban planning and design

Click to access finalCh4.pdf

Chapters

http://www.toofanhaghani.com

The influence of traditional Indian architecture in Balkrishna Doshi’s IIM Complex at Bangalore: A comparative analysis using fractal dimensions and lacunarity

,Mario Lodeweik LIONAR,  (Ph.D. Program of Architecture, Institute of Natural Sciences, Bursa Uludağ University, Bursa, Turkey) 

Özgür Mehmet EDİZ (Department of Architecture, Faculty of Architecture, Bursa Uludağ University, Bursa, Turkey) 

A|Z ITU Mimarlık Fakültesi Dergisi 

DOI: 10.5505/itujfa.2021.80388

https://search.trdizin.gov.tr/yayin/detay/503974/

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

PRINCIPLES OF FRACTAL GEOMETRY AND APPLICATIONS IN ARCHITECTURE AND CIVIL ENGINEERING

Anton Vrdoljak, M.Sc.
Faculty of Civil Engineering, University of Mostar, anton.vrdoljak@gf.sum.ba Kristina Miletić, B.Sc.(Math.)
Faculty of Civil Engineering, University of Mostar, kristina.miletic@gf.sum.ba

Number 17, June 2019.

https://hrcak.srce.hr/file/324620

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

The Shape of Cities: Geometry, Morphology, Complexity and Form

Chapter in book Fractal Cities

WHOLENESS, VISUAL COMPLEXITY AND MATERIALITY:

A Comparative Analysis Using Fractal Dimension Analysis And Mirror Of The Self-Test In The Case Of Material Imitations.

author | FILIP KINNERT supervisor | doc. PhDr. MARTIN HORÁČEK Ph.D.


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

Borobudur was Built Algorithmically

Hokky Situngkir

[hs@compsoc.bandungfe.net]

Dept. Computational Sociology Bandung Fe Institute

TEMPLE ARCHITECTURE AND SCULPTURE

NCERT

Fractal Cities: A Geometry of Form and Function,

Batty, M., Longly, P., 1994, 

Academic Press, San Diego.

Book

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

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 Intriguing Temples of the 64 Yoginis

2022