“I’m frightened and I cannot sleep,”
the little child said.
“I fear there might be triangles
underneath my bed.”
“There might be ghosts,” the mother mused.
“I cannot speak to those.
There may be ghouls and goblins
who will nibble on your toes.
There could be long-toothed monsters
with their eyes a gleaming red.
But there’s no such thing as triangles!
They’re only in your head.”
Teaching math is a weird job. I’m paid to tell children about imaginary things. To be sure, no one mistakes me for J.K. Rowling or J.R.R. Tolkien; there are no slow-talking trees, giant spiders, or unionized cleaning elves in my line of work.
I traffic in things much stranger than that, and much less beloved.
Things like quadratic equations and non-invertible matrices. Things so abstract that—by definition— they cannot exist in the physical world.
The official story, the party line, is that mathematics is essential for everyone, as indispensable for modern life as comfy jeans and good face-soap. But honestly!
I mean, how much algebra do you use in your typical week? Unless you’re a relationship counselor for x’s and y’s, it’s probably not much.
What’s cool about math isn’t that it’s “useful.” It’s that math walks the coastline between reality and imagination, between discovery and invention.
And precisely when math is furthest from reality, that’s when it offers the best views of reality—like a mountaintop overlooking a valley.
I’m not just talking about sophisticated, obscure stuff like the “inverse hyperbolic tangent” or the “convex hull of a set.” I’m talking about all mathematics, even the most elemental, familiar stuff.
I’m talking, in fact, about triangles.
Ever since you were little, you’ve seen triangles everywhere. You find them in jack-o’-lantern eyes, corporate logos, and grilled cheese sandwiches halved diagonally. They crop up in all kinds of construction projects: the pyramids, the supports beneath the Golden Gate Bridge, the tracks of roller coasters. When architects and engineers want a shape that’s sturdy and dependable, they turn to the triangle. There’s only one problem.
Triangles don’t exist.
I don’t mean to alarm you, and I hope I’m not spoiling any fond childhood memories of geometric forms. But triangles are like Santa Claus, the tooth fairy, and Beyoncé: too strange and perfect to exist in the actual world.
By definition, a triangle is a two-dimensional figure (perfectly flat) with three sides (perfectly straight) meeting at three vertices (perfectly sharp). Under this standard, every shape we’ve mentioned, from roller coaster struts to corporate logos, is utterly and hopelessly flawed. They meet none of the criteria.
Sure, they may look compellingly perfect from a distance, like celebrities you’ve never met. But get to know them better.
Start zooming in.
See those imperfections emerge: the minor wobble in the “straight” side, the tiny round to the “sharp” corner, the slight thickness to this supposedly “flat” shape? These aren’t just coincidental features, flaws in our manufacturing process.
No physical “triangle” can ever be totally perfect. The closer you look, the more it will dissolve into jagged pixels, until—by the time you reach the level of atoms and quarks—the triangle looks nothing like its idealized geometric reputation.
There’s no such thing as triangles. There are only jumbles of matter in faintly triangle-like arrangements.
You’ve never met a real triangle, and neither have I. We’ve encountered only cheap approximations, dancing shadows, sorry knock-off versions of the true and perfect original.
Triangles, as understood by every mathematician in the world, are mere abstractions. Works of geometric fiction. Their story is not a biography; it’s a fantasy novel.
And yet… they’re so darn useful.
This is the maddening paradox at the heart of mathematics. Every mathematical object is much like the triangle: inspired by reality, but idealized beyond any physical existence.
In the words of Ian Stewart, mathematics “hovers uneasily between the real and the not-real.
Eugenia Cheng says that math studies not “real things” but rather “the ideas of things.”
And G.H. Hardy once boasted, “‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one.”
By the account of its own highest practitioners, mathematics is an absurdly theoretical and impractical discipline. And yet it is how we make buildings stand and spaceships fly.
Math is deliberately useless, and that’s what makes it so useful.