The Virahāṅka–Fibonacci numbers

The sequence 1, 2, 3, 5, 8, 13, 21, 34, … is one of the most celebrated in all of mathematics, appearing throughout Art, Science and Mathematics. Remarkably, it was first discovered in the context of ART — specifically, poetry.

In the poetry of many Indian languages — Prakrit, Sanskrit, Marathi, Malayalam, Tamil and Telugu among them — each syllable is classified as either LONG or SHORT. A long syllable is pronounced for exactly twice as long as a short one: sung, a short syllable lasts one beat and a long syllable two.

That raises a question the ancient poets asked seriously: how many rhythms are there with 8 beats, made of short syllables of 1 beat and long syllables of 2 beats? Put mathematically: in how many different ways can one write 8 as a sum of 1s and 2s?

Start small and count. For n = 1 there is 1 way. For n = 2 there are 2 (1 + 1, and 2). For n = 3 there are 3 (1+1+1, 1+2, 2+1). For n = 4 there are 5.

For n = 5 there is a SYSTEMATIC way that beats listing. Write '1+' in front of every rhythm having 4 beats, and '2+' in front of every rhythm having 3 beats. Together those give every rhythm of 5 beats — and there are 5 + 3 = 8 of them.

Why does that work? Every 5-beat rhythm must begin with either a 1 or a 2. If it begins with 1, the rest is a 4-beat rhythm; if it begins with 2, the rest is a 3-beat rhythm. So the number of 5-beat rhythms is the number of 4-beat rhythms plus the number of 3-beat rhythms — and each term is the sum of the two before it. For 6 beats: 8 + 5 = 13.

So 8 beats gives the 8th term: 1, 2, 3, 5, 8, 13, 21, 34 — thirty-four rhythms. This method was first given by the Prakrit scholar Virahāṅka around 700 CE, in the form of a Prakrit poem. He was inspired by the earlier work of Piṅgala, around 300 BCE, and followed by Gopala around 1135 CE and Hemachandra around 1150 CE. So these numbers were worked out in India long before Europe. In the West they are called the Fibonacci numbers, after the Italian who wrote about them in 1202 CE — some 500 years after Virahāṅka, and not the first, second or even third to describe them. The numbers turn up in nature too: the petals of a daisy generally number 13, 21 or 34.