## Mathematics in art

A few months ago, Manjul Bhargava came to Stanford to give a public lecture on mathematics in the arts. I was skeptical of the topic — is there anything of content to be said? — but I’d heard him give some lectures on higher composition laws and on Selmer ranks of elliptic curves before, and they had all been very impressive talks. So, I went without much of an idea of what to expect.

Bhargava’s claim in the talk was that, while many people think that mathematics is inspired by science (or, perhaps, only by other mathematics; the latter is presumably more likely to be the opinion of mathematicians), there’s actually plenty of mathematics inspired by the arts. Since Bhargava has some background in Sanskrit poetry, he told three stories of mathematics being developed in order to solve concrete problems that arise from poetry.

In Sanskrit poetry, there are two types of syllables: long syllables and short syllables. The long syllables are twice as long as the short syllables. So, suppose we wish to write a line of poetry that takes up eight beats. How many different rhythms are there that allow us to realize a line of eight beats? If we write L for a long syllable and S for a short one, here are a few of the possible rhythms:

• LLLSS
• SLSSSL
• LLLL
• SSSSSSL

And there are quite a few more.

How do we do this? Well, suppose $H(n)$ is the number of rhythms possible for a line of poetry consisting of $n$ beats. Suppose that $n$ is at least 2. Then we can remove the last syllable; either it’s an L or an S. If it’s an L, then when we remove it, we’re left with a line consisting of $n-2$ beats. If it’s an S, then when we remove it, we’re left with a line consisting of $n-1$ beats.

Hence, the number of rhythms with $n$ beats is equal to the number of rhythms with $n-1$ beats plus the number of rhythms with $n-2$ beats. In symbols: $H(n)=H(n-1)+H(n-2)$. Since we can easily check that $H(0)=H(1)=1$, we then have:

• $H(2)=1+1=2$
• $H(3)=2+1=3$
• $H(4)=3+2=5$
• $H(5)=5+3=8$
• $H(6)=8+5=13$
• $H(7)=13+8=21$
• $H(8)=21+13=34$

So, there’s our answer: there are 34 different rhythms for a line of eight beats.

You probably recognize these numbers: they’re the Hemachandra numbers, named after the 12th-century Indian scholar Acharya Hemachandra.

Oh? You object to that name? Fibonacci wrote about this sequence of numbers in the context of the growth of rabbit populations roughly 50 years after Hemachandra considered the sequence in the context of poetry patterns. (Incidentally, rabbit mating has inspired a completely different branch of mathematics in the past few years. I’d like to write about this at some point, when I understand it better.)

He also discussed how poetry led to the development of what we usually refer to as Pascal’s triangle and de Bruijn sequences.

After the talk, I noted that it makes sense that combinatorial problems arise naturally arise from the arts, but I wondered if there were such problems in other branches of mathematics. One can imagine various topics in mathematics that look as though they might have something to do with art, but they were actually developed for entirely different reasons.

For example, identification spaces in topology sort of look like art, but I doubt that they were first studied for that reason. Similarly, one can imagine that there’s plenty of mathematics that one can get out of Escher’s paintings and wood carvings. But, the causality is the opposite in this case: Escher was inspired by the geometry of the Poincaré disc to study hyperbolic tessellations, such as this one.

And then, of course, there’s the $\pi_1(SO_3)$ dance.

But there’s one other obvious instance of mathematics being developed to answer questions of artists, although I wasn’t able to come up with it on my own. (I remembered only when it came up, in a completely different context, in Edward Glaeser’s Triumph of the city, which I’m currently reading.) And that’s projective geometry.

Early paintings tend not to look very realistic when portraying depth, since they lack perspective. As a result, we end up seeing a very flat rendition of what ought to be a 3-dimensional scene.

While various attempts were made to correct this problem, it wasn’t until 1413 that Filippo Brunelleschi solved this problem conclusively by working out the right way to draw perspective paintings, and this was done by using some projective geometry. One key observation is that lines that ought to be parallel should not necessarily be drawn parallel to each other; rather, they should intersect at a point which we understand to be very far away. Here’s a sample of his paintings:

Since this was done well before mathematicians such as Desargues studied projective geometry, I think it’s fair to say that projective geometry got off the ground because of its study by artists rather than mathematicians.

Finally, knitters who make socks need to understand something of geodesics on Riemannian manifolds.

Are there any other examples of topics in mathematics that were studied in order to solve concrete problems in the arts? If there are, I’d be delighted to hear about them!