Study Math With Kusama

A small, nerdy way to look at her art: learning a little mathematics inside the dots, nets and mirrors.11 min read4 days ago--Press enter or click to view image in full sizeA whole building wrapped in Yayoi Kusama’s polka dots: the City Gallery Wellington during her exhibition. Photo: Wainuiomartian, via Wikimedia Commons, CC BY-SA 4.0.For nearly a century Yayoi Kusama covered the world in dots, nets and mirrors. She passed away recently, and I will not try to squeeze her life into a few lines here, since other people will do that far better. I want to remember her in a different way, one that maybe suits her best. I am going to take a few of her works and read a little mathematics inside them. The kind you can explain to anyone, even to people who never got along with numbers.Kusama was not a mathematician and had no interest in being one. But her obsession with repetition, with filling every surface, with the infinite, brushes up against some of the simplest and most beautiful ideas in mathematics, whether she meant it to or not. Let us use her work as a blackboard.One note, so the game stays honest. Lesson 3 leans on a real study that measured her canvases with tools borrowed from physics. Lesson 2 is a small experiment I ran myself, and you can rerun it. The rest is a way of reading the art with mathematics, not a set of published measurements. Wherever I am reasoning by eye instead of quoting a number, I say so.Lesson 1. The mirror rooms, and the infinity that endsPress enter or click to view image in full sizeYayoi Kusama, “You Who Are Getting Obliterated in the Dancing Swarm of Fireflies” (2005), an infinity mirror room. Photo: WendyAvilesR, via Wikimedia Commons, CC BY-SA 4.0.You step into one of her rooms. Mirrors in front of you and behind you, tiny lights hanging in the dark. You turn, and you see your reflection repeated to infinity, smaller and smaller, sinking into the black.But is it really infinite? Here mathematics says something subtler, and more beautiful.Two mirrors facing each other bounce the image back and forth. First reflection, then the reflection of the reflection, then the reflection of that, and on and on. In theory the images are infinite. In practice they are not, because no mirror is perfect. Each one gives back, say, 95 percent of the light it receives and swallows the rest. So every jump between the two mirrors is a little dimmer than the one before.If the first image has brightness 1, the second is worth 0.95, the third is 0.95 × 0.95, which is about 0.90, the fourth about 0.86, and so on. The total light reaching your eyes is the sum of all these pieces:1 + 0.95 + 0.95² + 0.95³ + …This is called a geometric series, and it hides the surprise that catches everyone off guard the first time. Adding up infinitely many numbers can give a finite answer. The rule is this:1 + r + r² + r³ + … = 1 ÷ (1 − r), as long as r is a number between 0 and 1.With r equal to 0.95, the sum is 1 divided by 0.05, which is 20. Infinitely many images, but a finite amount of light. And that is exactly what you see. The far images become so faint that they vanish into the dark. Not a true infinity, but an infinity that quietly burns out.It is the same idea as Zeno’s old paradox. To cross a room you first have to cover half of it, then half of what is left, then half again, forever. It looks impossible to arrive. And yet 1/2 + 1/4 + 1/8 and so on adds up to 1, and you do reach the other side. In Kusama’s room your reflection never actually reaches infinity, for the very same reason that you always manage to cross the room.Try it yourself. Stand between the two mirrors of an elevator or a fitting room. Count how many copies of yourself you can make out before they dissolve into the dark. That number tells you, roughly, how good those two mirrors are.Lesson 2.

She said she had seen the world covered in dots since she was a child, and she put them everywhere: canvases, bodies, trees, pumpkins, whole rooms.Let us ask a question that sounds trivial. Are those dots placed at random?Mathematics has a precise answer to “what does random even mean”, and it exposes an illusion. When you scatter dots in a truly random way, our brain sees clumps: crowded patches and empty patches. They look grouped, not random. It is a famous trick of perception. True randomness looks messy, while order looks like chance.To tell the situations apart, there is a simple trick. For each dot, look at how far away its nearest neighbor is, then average all those distances and compare the result with what you would expect if the dots were simply thrown down at random. The ratio between the two is called the Clark and Evans index. I will call it R:R near 1: a random arrangement, a true roll of the dice.R greater than 1: spaced out, orderly dots.R less than 1: dots bunched into clusters.I actually measured itI went looking for a photo of one of Kusama’s dot panels to measure, but the freely licensed ones out there are either three dimensional scenes or walls seen at an angle, where perspective distorts the distances. So I did what the authors of the study in the next lesson also do when they test their method. I generated fields of dots on the computer and measured them.