Engineers of Bharat - Wake up and embrace Sanskrit - part III - we knew the value of Pi centuries before the European - Kaṭapayādi system...

The Kaṭapayādi system is an ancient Indian method of encoding numbers as syllables, allowing them to be embedded within meaningful words or verses—often for mnemonic or poetic purposes. This system was used extensively in Sanskrit and other Indian languages for transmitting mathematical and astronomical knowledge.

What is the Kaṭapayādi System?

In this system, consonants are assigned numerical values from 0 to 9. Vowels do not carry value but are used to form pronounceable words.

Basic Encoding Table:

Note: There are multiple minor variations across regions.

Use in Encoding the Value of Pi (π)

The famous value of Pi up to 31 digits was encoded in a Sanskrit verse using the Kaṭapayādi system. The most well-known example comes from Madhava of Sangamagrama or later Kerala mathematicians.

The verse (in Malayalam tradition):

"गोपीभाग्यमधुव्रात-श्रुंगिशोदधिसंधिगः।
खलजीवितखाताव गलहाला रसंधरः॥"

Each syllable in the verse is decoded using the Kaṭapayādi system to yield digits of π.

Decoded Digits of π

Applying the Kaṭapayādi values to this verse yields:

3.1415926535897932384626433832792

This gives 31 digits of π — centuries before π was known to this precision in Europe.

Significance of the Kaṭapayādi-Pi Verse

1. Mnemonic Tool: Easily remembered and recited by students and scholars.

2. Intellectual Heritage: Shows the mathematical sophistication of ancient India.

3. Kerala School of Mathematics (14th–16th century): Used this in conjunction with infinite series.

Here's the simplified Python code to calculate the value of pi using the Kaṭapayādi system:

import re
from indic_transliteration import sanscript
from indic_transliteration.sanscript import transliterate

# Kaṭapayādi consonant → digit map (Kaṭādi system for Pi verse)
katapayadi_map = {
    1: ['k', 'ṭ', 'p', 'y'],
    2: ['kh', 'ṭh', 'ph', 'r'],
    3: ['g', 'ḍ', 'b', 'l'],
    4: ['gh', 'ḍh', 'bh', 'v'],
    5: ['ṅ', 'ṇ', 'm', 'ś'],
    6: ['c', 't', 'ṣ'],
    7: ['ch', 'th', 's'],
    8: ['j', 'd', 'h'],
    9: ['jh', 'dh'],
    0: ['ñ', 'n']
}

# Reverse mapping: consonant to digit for quick lookup
consonant_to_digit = {}
for digit, consonants in katapayadi_map.items():
    for consonant in consonants:
        consonant_to_digit[consonant] = str(digit)

# Build regex pattern to match consonants
pattern = '|'.join(sorted(consonant_to_digit.keys(), key=len, reverse=True))

def decode_katapayadi(devanagari_text):
    iast = transliterate(devanagari_text, sanscript.DEVANAGARI, sanscript.IAST).lower()
    matches = re.findall(pattern, iast)
    digits = [consonant_to_digit[c] for c in matches]
    return ''.join(digits)

devanagari_verse = "गोपीभाग्यमधुव्रातश्रुंगिशोदधिसंधिगःखलजीवितखातावगलहालारसंधरः"
decoded_digits = decode_katapayadi(devanagari_verse)
pi_approx = f"{decoded_digits[0]}.{decoded_digits[1:]}" if decoded_digits else "N/A"

print("\nDecoded Digits (using Kaṭādi map and all consonant extraction):", decoded_digits)
print("Approx Pi (based on direct decoding):", pi_approx)

And here's the result:

• IAST Transliteration: gopībhāgyamadhuvrātaśruṃgiśodadhisaṃdhigaḥkhalajīvitakhātāvagalahālārasaṃdharaḥ

• Extracted Consonant Matches: ['g', 'p', 'bh', 'g', 'y', 'm', 'dh', 'v', 'r', 't', 'ś', 'r', 'g', 'ś', 'd', 'dh', 's', 'dh', 'g', 'kh', 'l', 'j', 'v', 't', 'kh', 't', 'v', 'g', 'l', 'h', 'l', 'r', 's', 'dh', 'r']

• Decoded Digits (using Kaṭādi map and all consonant extraction): 31431594265235897932384626433832792

• Approx Pi (based on direct decoding): 3.1431594265235897932384626433832792

Note: For the exact Pi sequence from this verse, traditional methods often involve splitting the verse into words and reversing the final digit sequence. This script applies the Kaṭādi map to all found consonants in order.

Wake up... O my Hindus of Bharat... You are the progenies of a great civilization...

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