Pascal's Triangle Interactive
The Legend
Pingala, an ancient Indian mathematician (circa 300-200 BCE), wrote the Chandaḥśāstra, a treatise on Sanskrit prosody. In analyzing metrical patterns of syllables (laghu/light and guru/heavy), Pingala discovered combinatorial patterns that form what we now call Pascal's triangle—over 1,800 years before Blaise Pascal! The arrangement was called Meru Prastara ("Staircase of Mount Meru"). Pingala also described what we recognize as the Fibonacci sequence in the context of counting metrical patterns.
The Mathematics
Pascal's Triangle Construction: Start with 1 at the top. Each number is the sum of the two numbers above it.
Binomial Coefficients: Row n, position k contains C(n,k) = n!/(k!(n-k)!), the number of ways to choose k items from n.
Recurrence: C(n,k) = C(n-1,k-1) + C(n-1,k)
Row Sums: The sum of row n equals 2^n (total number of subsets).
Connection to Prosody: C(n,k) counts the number of n-syllable meters with exactly k heavy syllables. Sanskrit poets used this to enumerate possible metrical patterns.
Fibonacci Link: Sum the shallow diagonals of Pascal's triangle to get Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13...