“Do you know the Land O’Lakes butter box?”
This is an odd question for Daniel Alexander. He’s a professor of mathematics at Drake University, and he’s trying to explain the complex mathematical phenomenon known as fractals.
Given the context, butter cartons feel more than a bit off topic. It’s not obvious how they have anything to do with beauty. Or mathematics. Or the beauty of mathematics, for that matter, which is, after all, what prompted Alexander’s rhapsody on fractals.
But as it turns out, the Land O’Lakes butter box is a fractal itself.
“I wish we had the real box here,” Alexander said, his attention now encapsulated by a yellow and green carton on his computer screen. “She’s holding a butter box, and on the picture of the butter carton is the butter carton that she’s holding. So, there’s her image of her there, and then within the box that she’s holding there’s another one, and then within that box there’s another one.”
Fractals are self-similar objects. When you zoom in on one part of the fractal, the new orientation resembles the whole. And like a set of Russian nesting dolls with an infinite number of figurines, fractals regress infinitely.
“I find those pictures—when I understand them, first of all—I find them interesting to look at,” Alexander said. “I mean, ok, there’s a beauty in there for me. I also find the mathematics that describes them is of itself really, really intriguing. It’s a thing of beauty in and of its own self.”
Let’s be honest. Most people wouldn’t swipe right for mathematics. Beauty is subjective, but it’s associated with art, music and Tinder profiles pics, not with the quadratic formula.
Surely, the image that represents the mathematics is beautiful, in that it follows the rules of aesthetics. Perhaps the Mandlebrot set, a famous fractal pictured above, visually piques your fancy, even though it’s the product of this function,
an equation whose aesthetic value is questionable at best.
But mathematicians find beauty in equations, in proofs and in logic. What’s more, there’s an entire mathematical literature dedicated to the topic. And as it turns out, mimicking the mathematician’s perspective on beauty is as simple as finding fractals in butter boxes.
The little black dress of mathematics
Alexander took out a sheet of paper and wrote:
It’s called Euler’s formula—pronounced ‘oiler’—after the 18th century mathematician Leonhard Euler who discovered it.
Now stay with me here. Euler’s formula is often referenced for its mathematic beauty. It unifies five invariably important constants in a single, true expression.
π — Pi ( 3.14159), a famous irrational number.
e — Euler’s number ( 2.71828), another famous irrational number.
i — The imaginary unit ( = √(-1)), a number not counted among the real numbers. (I know, right.)
1 — A single entity, the multiplicative identity.
0 — The additive identity, the neutral number.
Not only is Euler’s formula useful—it describes the relation between trigonometric functions (i.e., sine and cosine) and complex functions (functions involving both real and imaginary numbers). It’s humble, unadorned, elegant, yet no less breathtaking. The little black dress of mathematics.
“That confluence of concepts, there’s nothing visual here,” said Christopher Porter, a philosopher and mathematician at Drake. “A visually impaired individual can appreciate the beauty of this in the same way that someone seeing off the board can…It’s something of the unexpectedness, surely the simplicity.”
Perhaps the best way to demonstrate the beauty of mathematical minimalism is to contrast it with something not simple at all.
No beauty in this beast. With its entanglement of 39 symbols, the Bethe-Bloch equation—pronounced ‘beh-ta block’—is a dense portrayal of truth. Inelegantly, it describes the energy loss of a charged particle as it enters a medium and no, you’re not supposed to understand it.
Mathematicians may not find this equation beautiful because it is so dense. Much like how minimalists find beauty in their art, the simplicity of equations contributes to their aesthetic appeal.
The same goes for proofs: the arguments establishing the truthfulness of a statement. Porter said proofs are sometimes as equally ineloquent as the Bethe-Bloch equation.
“They’re like brute force,” Porter said. “When you’re done with them you feel like you really got yourself muddy. Typically, they say a proof with seven cases, that’s so inelegant. But you’ve just got to do it. You put your head down, you go through all the cases…no one is going to say that’s beautiful.”
On the other hand, there are surprisingly simple proofs. Porter will sometimes call those proofs ‘cute.’
“What makes it cute? It didn’t require a lot of steps,” Porter said. “I didn’t fill up the board with lots of symbols, I just made one simple little idea that I used that was unexpected, where you do something in just the right way and then boom you get it; there that’s cute.”
Fireflies and metronomes
JoAnne Growney is a mathematician and poet in Pennsylvania. For over a decade, she’s spent six hours a week writing a blog called Intersections — Poetry with Mathematics, a site dedicated to blurring the line between the two disciplines.
Growney finds beauty in the interconnectedness of mathematics. To demonstrate, she described a poem called Figures of Thought by Howard Nemerov, which comments on the recurrence in nature, architecture and art of a mathematical phenomenon called the logarithmic spiral.
“I think the beauty comes from the fact that there are multiple meanings,” Growney said. “The logarithmic spiral, it’s the pattern of a seashell or the pattern on the leaves, it’s basically a graph on a piece of paper. And the multiple meanings are part of the beauty and the density, the compactness of the language.”
“You could make a religion out of it, if you wanted to.”
Likewise, fireflies and the human heart rate might seem utterly unaffiliated, but mathematics binds them with a concept called phase locking. Fireflies flash their luminous derrieres in synchronized unison, a system modeled by a set of functions known as the Arnol’d Family. This same family of functions models the electrical pulses of a beating heart, how humans breathe and sleep and how metronomes gradually synchronize over time.
The laws of mathematics aren’t restricted to a professor’s scribbles on a chalkboard. Mathematicians discover patterns that relate to the world in diverse ways. In response to this interconnectedness, Drake’s Alexander joked that God’s a mathematician.
“So, all of a sudden you start with this really simple thing and it connects to all these other things and it seems to have some resonance in nature,” Alexander said. “So why does this happen? You could make a religion out of it if you wanted.”
“This same family of functions models the electrical pulses of a beating heart, how humans breathe and sleep and how metronomes gradually synchronize over time.”
Math is everywhere. For that reason, one finds the logarithmic spiral in a pinecone or a galaxy, or the human heartbeat in a firefly display.
But unlike the mathematical laws that govern the universe, beauty doesn’t need to be proven. In all things and Tinder, beauty is subjective. Just as Picasso swipes right on cubism or One Direction on smiling at the ground, the mathematician swipes right on a proof, or the fractal in a butter box.
“When I’ve wrestled with a deep proof and understood it, I get the warm fuzzies the same way that I would when I listen to music that I really like,” Alexander said. “It’s just this feeling I get of body and mind excitement. To me, that’s one of the marks of beauty. Is it something that moves you and changes how you think of things, even if it’s just momentarily.”
“In all things and Tinder, beauty is subjective.”