Study Math With Kusama

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 ends

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. The dots, and the hidden shape of chance

The dot is Kusama’s signature. 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 it

I 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. Same amount (400 dots) and same area for all of them, only the arrangement changes. Here is the result, with the code attached to this article.

Look closely at the first panel. That is pure chance, R equal to 1.04, and yet it is full of clumps and gaps. It looks like anything but random, and yet it is. The middle panel is what I call the “careful hand”: a regular grid with a wobble added, the way a person fills a surface with care but without a ruler. Its R is 1.26, and that is precisely the one our eye reads as the most even and orderly. The last one, R equal to 0.47, shows what “clustered” really means.

And here is the point about Kusama. Her dots come from an obsessive vision, almost an invasion. And yet a hand that fills a surface with care tends to space the dots out and avoid overlaps, exactly like the middle panel. A decoration meant as chaos turns out, when you measure it, to be more orderly than a genuine rain of random dots. Obsession, in the end, produces order. A small aside: in “The Obliteration Room” above, the dots are placed by visitors, many different hands instead of one, so you would expect an R closer to chance than to her own canvases.

Try it yourself. Put a photo of a dotted artwork next to a handful of rice tossed on a table. Which one has the clumps? Almost always the rice, that is, true randomness. Painted dots are too well behaved to be random.

Lesson 3. The Infinity Nets, and the word “complex”

The Infinity Nets are huge canvases covered by a single gesture repeated millions of times. Small brush arcs that, overlapping, leave a net of holes emerge. No center, no edge, the same weave everywhere, until it fills everything. Kusama painted them for hours, as if in a trance.

They look complex. But what does “complex” mean, in numbers? Two researchers, Elsa de la Calleja and Roberto Zenit, asked exactly this in a 2020 study (arXiv:2012.06108). They measured Kusama’s “Net obsession” works with two different tools, and got two answers that argue with each other. This is the most instructive part of the whole story.

Note on the image: the Infinity Nets are paintings still under copyright, and I could not find a freely licensed photo. The two figures below are original illustrations I made to explain the two tools. If you want to show an actual Net, add an image from an official museum or gallery page (Tate, MoMA, David Zwirner) with the credits they require.

i. First tool: the fractal dimension

A fractal is a shape that, when you zoom in on a little piece of it, still looks like the whole. Coastlines, ferns, broccoli, our own veins. The mathematician Benoit Mandelbrot realized that measuring them needs more than the geometry you learn in school. A line has dimension 1, a surface has dimension 2. But how “dimensional” is a scribble so dense that it starts to fill the page without ever becoming a solid surface?

The answer is startling. The dimension can be a number with a decimal. A sparse net has a dimension near 1, almost a line. A very dense, even net has a dimension near 2, almost a plane. The Infinity Nets sit somewhere in between: a line turning into a plane.

How do you measure it? With a simple and clever method, box counting. You cover the image with a grid of squares, count how many squares contain a piece of the shape, then shrink the squares and count again. The way that number grows, as the squares get small, gives you the dimension.

This same method has a famous history in art. In 1999, in the journal Nature, the physicist Richard Taylor measured the fractal dimension of Jackson Pollock’s dripped paintings. It was a huge deal, so much so that some proposed using it to spot forgeries. And it was strongly challenged in 2006, again in Nature, by Katherine Jones-Smith and Harsh Mathur, who showed that even trivial doodles passed the same test. The point for us is this: applied to Kusama’s nets, box counting returns a fractal dimension similar to Pollock’s. First verdict: high complexity, like Pollock.

ii. Second tool: topology

Here de la Calleja and Zenit pull out another instrument, one that comes from a beautiful branch of mathematics: topology. It is the geometry that ignores distances and angles and looks only at what stays the same when you deform a shape like modeling clay, without tearing it and without gluing it. To topology a coffee cup and a doughnut are the same object, because both have exactly one hole.

How do you count a shape? With the Betti numbers, and for a flat image two of them are enough:

• the first counts how many separate pieces there are. If the net is all connected, it is 1. If it is made of a thousand disconnected islands, it is 1000.
• the second counts how many holes, that is, how many closed loops the shape encloses.

It is very intuitive. Topology looks at a drawing and asks: is it all one piece, and how many holes does it have? It does not ask how jagged it is.

And here comes the twist. The Betti numbers of Kusama’s nets tell a story of disconnection, not of high complexity. Seen through topology, her net is made of many repeating parts that stay fairly separate. Orderly, not intricate. Second verdict: low complexity. And, the authors note, this is the reading that agrees with the eye. When you look at them, the Nets are hypnotic but regular, not chaotic.

The same painting is “as complex as Pollock” by one ruler, and “not very complex” by the other. Who is right? Both, because they are measuring different things. The fractal dimension says how jagged a stroke is and how much space it fills. The Betti numbers say how the parts are connected to each other. A drawing can be rough and repetitive, and at the same time well separated and orderly. Kusama is exactly like that.

This is the most important lesson, and it reaches far beyond art. “Complexity” is not a quality an object owns by itself. It is a question you ask with a tool. Change the tool, and the answer changes. That is why the two researchers used two of them, because a single number, in art as in life, can lie beautifully.

Bonus. The pumpkins, little polka-dot globes

The polka-dot pumpkins hide a little geometry problem that every mapmaker knows: how do you place regular dots on a curved surface?

A pumpkin is, more or less, a surface you get by spinning a profile around an axis. And the dots on that curved belly cannot all stay the same size and spacing as they would on a flat canvas. Where the pumpkin curves and slips away from view, the dots crowd together and squash, exactly the way the parallels on a globe get shorter as they near the poles.

Behind this sits a famous result by Gauss, with a name that sounds like a magic spell, the Theorema Egregium. In plain words it says you cannot flatten a curved surface without distorting it. It is the reason every world map lies a little. Kusama’s pumpkins are, in the end, small yellow and black globes.

Kusama said she painted so as not to be swallowed by the infinite, to put dots and nets between herself and the void. It is a nice coincidence that her most personal obsession brushes against the same ideas mathematics uses to handle infinity: the infinite sums that stay finite, the chance that has a shape, the dimensions with a decimal.

You do not need to read a formula to notice it. Stand in one of her rooms and watch the little lights fade into the dark. That is a geometric series, and it converges.

Sources and notes

• Direct study on Kusama: E. de la Calleja, R. Zenit, “Fractal dimension and topological invariants as methods to quantify complexity in Yayoi Kusama’s paintings”, arXiv:2012.06108 (2020). The Pollock comparison and the Betti number result come from here. It is a preprint, so check whether it later appeared in a journal.
• Fractal analysis of Pollock: R. P. Taylor, A. P. Micolich, D. Jonas, “Fractal analysis of Pollock’s drip paintings”, Nature 399, 422 (1999). Critique: K. Jones-Smith, H. Mathur, Nature 444, E9-E10 (2006).
• Fractals: B. Mandelbrot, “The Fractal Geometry of Nature” (1982).
• Topology and Betti numbers: any introduction to topology; for the use on images, the literature on persistent homology.
• Spatial point patterns: the Clark and Evans aggregation index (1954); to go deeper, P. J. Diggle, “Statistical Analysis of Spatial Point Patterns”.
• Geometric series and Zeno’s paradox: any introductory analysis textbook.
• Surface geometry: Gauss’s Theorema Egregium.
• Photographs from Wikimedia Commons, with credits and licenses in the captions. The dot figure and the two net figures are generated by the scripts.