Binary mathematics and combinatorial problems in Chandah-Shastra

Introduction

This document explores the contributions of Pingala, a 2nd Century BCE Indian scholar, to binary mathematics and combinatorial problems through his work on Chandah-shastra, a treatise on prosody.

Laghu and Guru: The Binary Building Blocks

• Pingala classified syllables into two types: Laghu (short) and Guru (long).
• Replacing Laghu with "1" and Guru with "0" transforms metrical patterns into binary sequences.
• This binary representation allows for the analysis of metrical structures using binary mathematics.

Binary Operations in Chandah-shastra

• Prastara: A procedure for generating all possible metrical patterns or binary sequences of a given length.
• Sankhya: The process of finding the total number of binary sequences in a Prastara.
• Nasta: Identifying the corresponding binary sequence for a given row number in an array.
• Uddsista: The reverse of Nasta, finding the row number of a given binary sequence.
• Lagakriya: Finding the number of binary sequences in the array with a given number of "1"s and "0"s.
• Adhvayoga: Determining the space occupied by the array or Prastara.

Generating Binary Arrays (Prastaras)

• Sutras 20 to 23 in Chapter 8 of Chandah-shastra provide details for generating binary arrays.
• The process involves replicating the existing array, adding a column, and filling the first set with 0s and the second set with 1s.
• This process can be repeated to generate larger binary arrays.

Nasta Algorithm

• This algorithm finds the binary sequence associated with a particular row number.
• The process involves dividing the row number by 2, placing "1" if divisible and "0" if not, and repeating until the sequence is obtained.

Uddista Algorithm

• This algorithm is the reverse of Nasta, finding the row number for a given binary sequence.
• The process involves starting with 1, scanning the sequence from right to left, doubling the current number for each "1", and doubling and subtracting 1 for each "0".

Lagakriya and Pascal's Triangle

• Pingala's Lagakriya is equivalent to generating Pascal's Triangle, which was rediscovered by Pascal in 1655 CE.
• Varahamihira's Brihatsamhita (550 CE) mentions combinations that can be obtained by choosing from a set, demonstrating an understanding of combinatorics.

Conclusion

Pingala's Chandah-shastra contains fundamental principles of binary mathematics and combinatorial problems, showcasing the advanced mathematical thinking in ancient India centuries before the development of modern computing.