Geometry in IndiaA brief history

Lecture 01

Ananda Dasgupta

Indian Institute of Science Education and Research Kolkata

Spring Semester 2015

Ananda Dasgupta Geometry in India

The beginning

Traditionally, the history of Indian mathematics has beensupposed to start with the Sulba-sutras (800-200 BCE).

They deal with a set of rules for construction of ritual pyres.

The oldest slokas have been dated at 1000 BCE by F. Stahlon the basis of some of them appearing in the earliestYajurveda Samhita - the Maitrayani.

The absence of any prior records has prompted speculationthat mathematics in India originated from West Asia.

However, discoveries in the Indus valley region has revealedthat Indians had a sophisticated idea of geometry at least1000 years before.

It is not clear whether there was some continuity of ideasbetween the geometric notions of the Indus valley civilizationand the Sulba-sutras.

Ananda Dasgupta Geometry in India

The beginning

Traditionally, the history of Indian mathematics has beensupposed to start with the Sulba-sutras (800-200 BCE).

They deal with a set of rules for construction of ritual pyres.

The oldest slokas have been dated at 1000 BCE by F. Stahlon the basis of some of them appearing in the earliestYajurveda Samhita - the Maitrayani.

The absence of any prior records has prompted speculationthat mathematics in India originated from West Asia.

However, discoveries in the Indus valley region has revealedthat Indians had a sophisticated idea of geometry at least1000 years before.

It is not clear whether there was some continuity of ideasbetween the geometric notions of the Indus valley civilizationand the Sulba-sutras.

Ananda Dasgupta Geometry in India

The beginning

Traditionally, the history of Indian mathematics has beensupposed to start with the Sulba-sutras (800-200 BCE).

They deal with a set of rules for construction of ritual pyres.

The oldest slokas have been dated at 1000 BCE by F. Stahlon the basis of some of them appearing in the earliestYajurveda Samhita - the Maitrayani.

The absence of any prior records has prompted speculationthat mathematics in India originated from West Asia.

However, discoveries in the Indus valley region has revealedthat Indians had a sophisticated idea of geometry at least1000 years before.

It is not clear whether there was some continuity of ideasbetween the geometric notions of the Indus valley civilizationand the Sulba-sutras.

Ananda Dasgupta Geometry in India

The beginning

Traditionally, the history of Indian mathematics has beensupposed to start with the Sulba-sutras (800-200 BCE).

They deal with a set of rules for construction of ritual pyres.

The oldest slokas have been dated at 1000 BCE by F. Stahlon the basis of some of them appearing in the earliestYajurveda Samhita - the Maitrayani.

The absence of any prior records has prompted speculationthat mathematics in India originated from West Asia.

However, discoveries in the Indus valley region has revealedthat Indians had a sophisticated idea of geometry at least1000 years before.

It is not clear whether there was some continuity of ideasbetween the geometric notions of the Indus valley civilizationand the Sulba-sutras.

Ananda Dasgupta Geometry in India

The beginning

Traditionally, the history of Indian mathematics has beensupposed to start with the Sulba-sutras (800-200 BCE).

They deal with a set of rules for construction of ritual pyres.

The oldest slokas have been dated at 1000 BCE by F. Stahlon the basis of some of them appearing in the earliestYajurveda Samhita - the Maitrayani.

The absence of any prior records has prompted speculationthat mathematics in India originated from West Asia.

However, discoveries in the Indus valley region has revealedthat Indians had a sophisticated idea of geometry at least1000 years before.

It is not clear whether there was some continuity of ideasbetween the geometric notions of the Indus valley civilizationand the Sulba-sutras.

Ananda Dasgupta Geometry in India

The beginning

Traditionally, the history of Indian mathematics has beensupposed to start with the Sulba-sutras (800-200 BCE).

They deal with a set of rules for construction of ritual pyres.

The oldest slokas have been dated at 1000 BCE by F. Stahlon the basis of some of them appearing in the earliestYajurveda Samhita - the Maitrayani.

The absence of any prior records has prompted speculationthat mathematics in India originated from West Asia.

However, discoveries in the Indus valley region has revealedthat Indians had a sophisticated idea of geometry at least1000 years before.

It is not clear whether there was some continuity of ideasbetween the geometric notions of the Indus valley civilizationand the Sulba-sutras.

Ananda Dasgupta Geometry in India

Part I

Indus Valley Civilization

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

The Indus Valley Civilization,also called the MatureHarappan Civilization wasspread over what is now Pakistanand North-western India.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

It covered about a million squarekilometers.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

It was contemporaneous withOld Kingdom Egypt, the EarlyDynastic period of Sumeria andthe ancient Akkadian civilization.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

It was contemporaneous withOld Kingdom Egypt, the EarlyDynastic period of Sumeria andthe ancient Akkadian civilization.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

It was contemporaneous withOld Kingdom Egypt, the EarlyDynastic period of Sumeria andthe ancient Akkadian civilization.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

Excavations have led to thediscovery of a wide variety ofrelics.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

The Indus Valley Civilizationwas characterized by its extensiveurbanization.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

The well laid out street plansof Mohen-jo-daro, Harappa andother cities, as well as theorientation of the streets alongthe cardinal directions show aworking understanding ofgeometry.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

There is extensive evidenceof a standardized system ofweights and measures that wasused in different urban centers.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

There is extensive evidenceof geometrical ideas in the sealsand pottery recovered from thisera.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

There is extensive evidenceof geometrical ideas in the sealsand pottery recovered from thisera.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

Concentric circles are acommon motif.

Ananda Dasgupta Geometry in India

The Indus Valley Civilization, circa 2500-1900 BCE

So are squares, rectangles,triangle pairs joined at the apex.

Ananda Dasgupta Geometry in India

Symmetry in Harappan seals

Ananda Dasgupta Geometry in India

Symmetry in Harappan seals

Ananda Dasgupta Geometry in India

Symmetry in Harappan seals

Ananda Dasgupta Geometry in India

Symmetry in Harappan seals

Ananda Dasgupta Geometry in India

Symmetry in Harappan seals

Ananda Dasgupta Geometry in India

Symmetry in Harappan seals

Ananda Dasgupta Geometry in India

A space filling motif

Seal number M-1261A shows an intricate space filling pattern.

Ananda Dasgupta Geometry in India

A space filling motif

The basic motif is made out of four fan shaped structures, wherethe fans are formed out of two intersecting lattice of circles.

Ananda Dasgupta Geometry in India

A space filling motif

The motif tessellates space.

Ananda Dasgupta Geometry in India

A modern day surprise

Ananda Dasgupta Geometry in India

Precursors of things to come ???

This common shape is reminiscent of the Mahavedi altar describedin the Sulba-sutra!

Ananda Dasgupta Geometry in India

Part II

Numbers in the Vedas

Ananda Dasgupta Geometry in India

Numbers in the VedasKrishna-Yajur-Veda, Taittareya Aranyakam

O fire, salutations unto you once.Salutations twice.Salutations thrice . . .Salutations ten times.Salutations hundred times.Salutations a thousand times.Salutations unto you unlimitedtimes.My venerations to you, neverever hurt me.

Ananda Dasgupta Geometry in India

Numbers in the VedasKrishna-Yajur-Veda, Taittareya Samhita

The powers of ten:Hail to hundred, . . . hail to hundred thousand . . .hail to hundred million . . . hail to trillion.

Odd numbers!

Multiples of four!

Ananda Dasgupta Geometry in India

Numbers in the VedasKrishna-Yajur-Veda, Taittareya Samhita

The powers of ten:Hail to hundred, . . . hail to hundred thousand . . .hail to hundred million . . . hail to trillion.

Odd numbers!

Multiples of four!

Ananda Dasgupta Geometry in India

Numbers in the VedasKrishna-Yajur-Veda, Taittareya Samhita

The powers of ten:Hail to hundred, . . . hail to hundred thousand . . .hail to hundred million . . . hail to trillion.

Odd numbers!

Multiples of four!

Ananda Dasgupta Geometry in India

Numbers in the VedasKrishna-Yajur-Veda, Taittareya Samhita

The powers of ten:Hail to hundred, . . . hail to hundred thousand . . .hail to hundred million . . . hail to trillion.

Odd numbers!

Multiples of four!

Ananda Dasgupta Geometry in India

The powers of tenYajurveda Samhita

Ekha 1 100

Dasa 10 101

Sata 100 102

Sahasra 1000 103

Ayuta 10000 104

Niyuta 100000 105

Prayuta 1000000 106

Arbuda 10000000 107

Nyarbuda 100000000 108

Samudra 1000000000 109

Madhya 10000000000 1010

Anta 100000000000 1011

Parardha 1000000000000 1012

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

A spys tale from the Ramayana

1 Koti = 100 Lakhs = 107

1 Sankha = 105 Koti = 1012

1 Mahasankha = 105 Sankha = 1017

1 Vrinda = 105 Mahasankha = 1022

1 Mahavrinda = 105 Vrinda = 1027

1 Padma = 105 Mahavrinda = 1032

1 Mahapadma = 105 Padma = 1037

1 Kharva = 105 Mahapadma = 1042

1 Mahakharva = 105 Kharva = 1047

1 Samudra = 103 Mahakharva = 1050

1 Mahaugha = 105 Samudra = 1055

Ananda Dasgupta Geometry in India

Concept of the infiniteIsha Upanishada

Ananda Dasgupta Geometry in India

Indus Valley CivilizationNumbers in the Vedas