Article: Mathematics of the universe

Contributed by Nalini Balbir

Jain cosmology observes mathematical principles found in calculations and geometry formulae. Mathematics is an intellectual field to which Jain theorists have made valuable contributions. More importantly, it is a way of thinking about and mentally organising details of the universe and time in a coherent manner. Classifying space and time into measurable units and seeing recurring patterns in these units, categories and structure of units is a feature of the Jain approach to cosmology.

The unique Jain notion of the universe has deep religious significance. Understanding the whole concept and the different elements that form it, including the structure, cycles of time and actions over time, is vital to spiritual progress. Crucial religious ideas – such as the physical soul going through the cycle of birth, being reborn in different parts of the three worlds, in order to gain omniscience and then liberation – are tied into understanding traditional cosmology. For this reason, passing on accurate knowledge of Jain cosmology has been an important part of scholarly work for centuries.

Communicating information about the universe is based on texts, including scriptures and popular stories, but images of the universe have also been important teaching aids since the earliest times. Since Jain cosmology is remarkably intricate, with recurring patterns, such descriptions and artwork can seem confusingly exhaustive and repetitive.

Units of time and space

The Jain universe is thought of in terms of dimensions and quantities of units. These ideas are discussed at length in philosophical and sacred literature, starting with definitions of time and space units. Jain thinkers have produced a vast vocabulary to describe and understand units of time and space, going from the smallest to the largest, beyond what can be imagined.

Components of Jain cosmology are classified in one of the following ways:

  • numerable – saṃkhyeya
  • innumerable – asaṃkhyeya
  • infinite – ananta.

The smallest unit of physical matter is the atom. Infinite combinations of atoms make up the smallest unit of measurement that can be counted. This is called the 'extremely fine'.

Similarly, the Jains have gone into great detail in analysing the extremely large or highly numerous (Plofker in Granoff 2009: 65ff.).

Dimensions and quantities

This detail of a manuscript page gives information about the distances between the suns and moons in the Jain triple world. Numbers and mathematics underlie the symmetry and repetition that are noticeable in traditional Jain cosmology.

Distances separating the suns and moons
Image by British Library © CC0 1.0 (Creative Commons Public Domain)

Cosmological texts detail the quantities and proportions of parts of the universe. There are many examples of the vital part mathematics plays in traditional Jain concepts of the universe.

One is the location and number of certain items. For instance, in the middle world of humans, the Jyotiṣka gods – the astral bodies such as planets – are distributed as follows:

  • Kālodadhi – 42 suns, 42 moons
  • Half-Puṣkara – 72 suns, 72 moons.

Each moon has the following retinue of:

  • planets – 88
  • constellations – 28
  • stars – 6,697,500,000,000,000,000,000.

A second example is the dimensions of the various continents and oceans that make up the Two and A Half Continents.

Sizes of the parts of the Two and A Half Continents

Name

Size in yojanas

Jambū-dvīpa

100,000

Lavaṇa-samudra

200,000 x 2 (east and west) = 400,000

Dhātakīkhaṇḍa

400,000 x 2 = 800,000

Kālodadhi

800,000 x 2 = 1,600,000

Half-Puṣkara

800,000 x 2 = 1,600,000

Aḍhāī-dvīpa

4,500,000 yojanas in total

Repetition and symmetry

This manuscript painting of the temples and trees on the three terraces of Mount Meru emphasises the symmetry and repetition that are hallmarks of Jain cosmology. Mount Meru is the cosmic axis, centre of the three worlds of the Jain universe

Temples and trees on Mount Meru
Image by British Library © CC0 1.0 (Creative Commons Public Domain)

The mathematical characteristics of recurrence and symmetry are very noticeable features of Jain cosmology. The universe is 'a self-replicating composite' (Granoff 2009: 56). These qualities are especially obvious in the intricate pictures of the universe, which may make them puzzlingly complex.

There are plenty of illustrations of the universe’s repetitive, symmetrical nature, chiefly in the three worlds. Examples abound in the middle world, particularly in the Two and A Half Continents, where humans live.

Firstly, the 45 continents of the middle world are all modelled on the central continent of Jambū-dvīpa.

Next, each of the 90 continents and oceans is bigger by a factor of two than the one inside it.

Thirdly, in the Two and a Half Continents there are a total of five Mount Merus, the cosmic axis, distributed as follows:

  • one at the centre of Jambū-dvīpa
  • two in the second continent of Dhātakīkhaṇḍa, one per half
  • two in the inner part of the third continent of Puṣkara, one in each half.

In addition, Jambū-dvīpa has many features that are noticeably identical, symmetrical or both. For instance, the landmass itself demonstrates recurring, often symmetrical, proportions of mountains, regions, lakes, rivers and so on.

Prominent examples include the regions of Airāvata, Bharata and Mahā-videha. The northern region of Airāvata is a mirror image of Bharata, the southern region. Jambū-dvīpa’s ten capital cities are equally divided between Bharata and Airāvata. They sit between identical branches of the river that flows across each region and are all called Ayodhyā. The central region of Mahā-videha is comprised of 32 provinces laid out in two matching sets of eight rectangles. These are either side of each of the twin rivers that flow east and west across the region. Each pair of sets therefore makes up 16 rectangles in the east and 16 in the west.

The next example is the social organisation of the deities in the three worlds. The gods and goddesses have a hierarchical society very similar to traditional Indian cultures, in which the ideal form of government is monarchy.

Finally, in the Lands of Actions – karma-bhūmi – around half of Mount Meru, time moves in cycles. These cycles of descending and ascending periods of time are endless. Although not of equal length, they are characterised by repetitiveness and predictability.

Images

  • Distances separating the suns and moons This detail of a manuscript page gives information about the distances between the suns and moons in the Jain triple world. Numbers and mathematics underlie the symmetry and repetition that are noticeable in traditional Jain cosmology. Cosmological texts go into immense detail about all measurements of the universe, ranging from the sub-microscopic to the dizzingly huge.. Image by British Library © CC0 1.0 (Creative Commons Public Domain)
  • Temples and trees on Mount Meru This manuscript painting of the temples and trees on the three terraces of Mount Meru emphasises the symmetry and repetition that are hallmarks of Jain cosmology. Mount Meru is the cosmic axis, centre of the three worlds of the Jain universe.. Image by British Library © CC0 1.0 (Creative Commons Public Domain)

Further Reading

Commentary on Tattvārtha Sūtra of Vācaka Umāsvāti
Pandit Sukhlalji
translated by K. K. Dixit
L. D. series; volume 44
L. D. Institute of Indology; Ahmedabad, Gujarat, India; 1974

Full details

Steps to Liberation: 2500 Years of Jain Art and Religion
Jan van Alphen
Etnografisch Museum Antwerpen; Antwerp, Belgium; 2000

Full details

‘Une peinture cosmologique jaina déposée au Musée Guimet: texte et traduction’
Nalini Balbir
Bulletin d’Études Indiennes
volume 24–25
Association Française pour les Études Indiennes; Paris, France; 2006 to 2007

Full details

‘Le monde médian: une peinture cosmologique jaina sur tissu déposée au Musée Guimet’
Nalini Balbir
Arts Asiatiques
volume 64
École Française d’Extrême-Orient; Paris, France; 2009

Full details

Elements of Jaina Geography: The Jambūdvīpasaṃgrahaṇī of Haribhadra Sūri
Haribhadra
translated and edited by Frank van den Bossche
Motilal Banarsidass; New Delhi, India; 2007

Full details

Jain Cosmology
Colette Caillat
and Ravi Kumar
translated by R. Norman
Bookwise (India) Pct. Ltd; New Delhi, India; 2004

Full details

Essays on Jaina Art
Anand K. Coomaraswamy
edited by Richard J. Cohen
Indira Gandhi National Centre for the Arts and Manohar; New Delhi, India; 2003

Full details

Victorious Ones: Jain Images of Perfection
Phyllis Granoff
Mapin Publishing Pvt. Ltd and Rubin Museum of Art, New York; Ahmedabad, Gujarat, India and New York, USA; 2009

Full details

‘Oceans, Islands and Sacred Mountains: Representations of Cosmic Geography in Jaina Art and Architecture’
Julia A. B. Hegewald
Cosmos: the Journal of the Traditional Cosmological Society
volume 16
Traditional Cosmological Society; 2000

Full details

‘Meru, Samavasaraṇa and Siṃhāsana: The Recurrence of Three-Tiered Structures in Jaina Cosmology, Mythology and Ritual’
Julia A. B. Hegewald
Kalhar (White Water-Lily): Studies in Art, Iconography and Archaeology of India and Bangladesh – Professor Enamul Haque Felicitation Volume
edited by Gerd R. Mevissen, Gourishwar Bhattacharya, Mallar Mitra and Sutapa Sinha
Kaveri Books; New Delhi, India; 2007

Full details

Jaina Temple Architecture in India: The Development of a Distinct Language in Space and Ritual
Julia A. B. Hegewald
Monographien zur Indischen Archäologie, Kunst und Philologie series; volume 19
Stiftung Ernst Waldschmidt, G+H Verlag; Berlin, Germany; 2009

Full details

Cosmology Old & New: Being a modern commentary on the fifth chapter of Tattvārthādhigama Sūtra
G. R. Jain
Jñānapīṭha Mūrtidevī granthamālā: English series; volume 5
Bhāratīya Jñānapīṭh Publication; New Delhi, India; 1991

Full details

Jainendra Siddhānta Kośa
Jinendra Varṇi
volume 38
Bhāratīya Jñānapītha Publication; New Delhi, India; 2003

Full details

‘Lokākāśa and Lokadhātu: A Comparison of Jain and Buddhist Cosmology’
Padmanabh S. Jaini
The Clever Adulteress and Other Stories: A Treasury of Jain Literature
edited by Phyllis Granoff
Mosaic Press; Oakville, Ontario, Canada; New York, USA; London, UK; 1990

Full details

Triṣaṣṭiśalākapuruṣacaritra: Lives of the Sixty-three Illustrious Persons
Hemacandra
translated by Helen M. Johnson
Gaekwad's Oriental series; volume 2
Oriental Institute; Baroda, Gujarat, India; 1937

Full details

Die Kosmographie der Inder: nach den Quellen dargestellt
Willibald Kirfel
Georg Olms; Hildesheim, Lower Saxony, Germany; 1967

Full details

The Scientific Foundations of Jainism
K. V. Mardia
edited by Dayanand Bhargava
Lala Sunder Lal Jain Research series; volume 5
Motilal Banarsidass; New Delhi, India; 1996

Full details

Die Erlösungslehre der Jaina: Legenden, Parabeln, Erzählungen
translated and edited by Adelheid Mette
Insel Verlag; Berlin, Germany; 2010

Full details

The Peaceful Liberators
Pratapaditya Pal
Los Angeles County Museum of Art and Thames and Hudson; Los Angeles, California, New York USA and London, UK; 1994

Full details

Illustrated Jambudveep Prajnapti Sūtra
edited by Pravarttak Shri Mamar Muniji Maharaj, Amar Muni and Srichand Surana 'Saras'
Illustrated Agam series; volume 20
Padma Prakashan and Shree Diwakar Prakashan; New Delhi, India; 2006

Full details

The Doctrine of the Jainas: Described after the Old Sources
Walther Schubring
translated by Wolfgang Bühlen
edited by Satya Ranjan Banerjee
Lala Sunder Lal Jain Research series; volume 15
Motilal Banarsidass; New Delhi, India; 2000

Full details

That Which Is: Tattvārtha Sūtra
Umāsvāti / Umāsvāmi
translated by Nathmal Tatia
Sacred Literature series
International Sacred Literature Trust in association with Harper Collins; London, UK; 1994

Full details

Treasures of Jaina Bhandāras
Umakant Premanand Shah
L. D. series; volume 69
L. D. Institute of Indology; Ahmedabad, Gujarat, India; 1978

Full details

Tiloyapannatti: Teaching on the Three Worlds
Yativṛṣabha
translated by Pandit Balchandra Shastri
edited by A. N. Upadhye and Hiralal Jain
Jaina Saṃskṛti Saṃrakshaka Sangha series
Jīvarāja Jaina Granthamālā; Solapur, Maharashtra, India; 2008

Full details

Glossary

Aḍhāī-dvīpa

The Hindi phrase for 'Two and A Half Continents' describes the only part of the universe where human beings live in the Middle World of Jain cosmology. It is made up of the central continent, Jambū-dvīpa, the second continent, Dhātakīkhaṇḍa, and Lavaṇa-samudra, the circular ocean that separates them. Kālodadhi is the ring of ocean around Dhātakīkhaṇḍa, dividing it from the 'half' continent, which is the inner part of the Puṣkara continent.

Ākāśa

Space – one of the five non-material substances that is non-sentient in Jain belief. These five substances make up the universe along with the sentient substance, called jīvastikaya.

Bharata

One of the Lands of Action or Karma-bhūmi in the first continent, Jambū-dvīpa, in the Middle World where humans live. Bharata is also the name of the eldest son of the first Jina, Ṛṣabha, who succeeded his father as king.

Cosmology

A belief system about the universe that covers its origin, structure and parts, and natural laws and characteristics such as space, time, causality and freedom.

Deity

A god or divine figure, often with physical powers beyond those of a human and with superhuman abilities.

Dhātakīkhaṇḍa

The second continent in the Middle World of Jain cosmology. Dhātakīkhaṇḍa forms part of the Two and A Half Continents where human beings live.

Jain

Follower of the 24 Jinas or an adjective describing Jain teachings or practices. The term 'Jaina' is also used although 'Jain' is more common.

Jambū-dvīpa

The innermost island-continent in the Middle World, in Jain cosmology. It is divided into seven continents separated by six mountain ranges. It takes its name - 'Rose-Apple Continent' - from a rock formation that resembles a rose-apple tree, which is found on Mount Meru in the centre of the island.

Jyotiṣka

The third class of gods, who are the astral or luminous bodies, such as the sun, moons, planets and stars. They live in the middle of the three worlds.

Kāla

Time. One of the five insentient non-material substances that make up the universe along with the sentient substance, called jīvastikaya.

Lavaṇa-samudra

The Lavaṇa-samudra or 'Salt Ocean' in Sanskrit is the first ocean in the Two and A Half Continents of the Middle World in Jain cosmology. It encircles the central continent, Jambū-dvīpa.

Loka

The universe in Jain cosmology, composed of the upper, middle and lower worlds. Human beings can live only in part of the Middle World.

Madhya-loka

There are three worlds in traditional Jain cosmology. The middle world is where human beings and animals live, and sits between the upper and the lower worlds.

Mahā-videha

In Jain cosmology, one of the Lands of Action or karma-bhūmi in the first continent, Jambū-dvīpa, in the middle world of humans. Mahā-videha consists of 32 provinces between the Niṣadha and the Nīla mountain ranges. Thanks to the repetitive nature of Jain cosmology, there are also two Mahā-videhas on each of the continents of Dhātakīkhaṇḍa and Puṣkara-dvīpa.

Mount Meru

The cosmic axis of the Jain universe. Located in the middle of Jambū-dvīpa, the innermost continent of Jain cosmology, Mount Meru consists of three forested terraces, each smaller than the one below. When a Jina is born, the gods visit the earth, take him away and wash him in the standard birth ritual on the mountain. Jain temples often have a tower symbolising Mount Meru. Mount Meru is also the centre of the universe in traditional Buddhist and Hindu belief.

Pudgala

Matter. One of the five insentient material substances of dravya that make up the universe along with the sentient substance, jivastikaya.

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Related Manuscripts

Related Manuscript Images

  • Chart of measurements

    Chart of measurements

    With commentary by Pārśva-candra. British Library. Add. 26374. Ratnaśekhara. 1769

  • Chart of measurements

    Chart of measurements

    With commentary by Pārśva-candra. British Library. Add. 26374. Ratnaśekhara. 1769

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