*“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.

Witness the triangle, a fond old friend. This humble two-dimensional figure is, by all accounts, one of realest and most tangible objects in all of mathematics.

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.

They’re inescapable.

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.

Triangles most helpful in vacuums where they can be used to construct spherical cows.

Your musings are like bright stars, beautiful blue and green and red. I am so reading this to my class tomorrow! Encore! Encore!

Thank you. I immediately reposted this. So succinct, so right. Yes, Mr. & Mrs. Billy’s parents, he really does need to learn the Unit Circle.

Think it might be too soon to be teaching my 5 year old daughter that triangles don’t exist? Then again, I’d probably lose the argument anyway.

“Math is deliberately useless, and that’s what makes it so useful.”

Underwater basket-weaving is deliberately useless. So is Harry Potter. Why is math different? What distinguishes math from other “fantasy” that makes it so useful?

Your question has occurred to many people, and it’s not an easy one to answer. Here’s a take on it by the physicist Eugene Wigner:

https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

“Still more astonishing is that world of rigorous fantasy we call mathematics” – Gregor Bateson 🙂

*Gregory

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Show me three.

Where I strongly object is where you use the phrase “mere abstraction”, qualifying the word “abstraction” with the word “mere”. I find it as objectionable as referring to a scientific theory as “just a theory”. Abstraction is one of the most amazing, beautiful, and powerful processes ever! There’s no “mere” about it!

Today I learned that all of Ben’s cleaning elves are ionized.

More water-soluble that way.

Ok, but if the triangles we have don’t actually exist mathematically, because they don’t meet those criteria you listed, then lines aren’t real either, right? Or any shape, for that manner?

Is this in math theory? I’m in calculus now, so I haven’t gotten there yet…

Also, I love your blog! I’m a high schooler and I totally love math in a world where not many people seem to. Really cool to find a blog all about how amazing math is and be reminded that lots of people like it too! It makes me so excited for everything in math I have yet to learn!

Emma

You’re exactly right – lines, points, and other shapes are just as “unreal” as triangles, by this argument. (For what it’s worth, I don’t think most schools offer any specific “math theory,” class, but this triangle argument goes back thousands of years. It was central to Plato’s philosophy, for example.)

Anyway, thanks for reading, and I’m glad you’re enjoying the blog! I think you’ll find there’s lots of math enthusiasts like you out there. 🙂

True, lines aren’t real. They don’t occupy any space for one. And they are infinite: have you ever _seen_ an infinite line? 🙂

Shapes that you draw aren’t identical to the shapes that you are talking about in your proofs. Your drawings are imperfect. That’s why geometry is the science of making good conclusions from bad drawings.

Beautiful!

Sorry about getting philosophical, but it’s these kind of things that make you question what existence even means, doesn’t it? Sure, triangles can not be touched, but neither can love (you may say that love exists in mind, but the same could be said of triangles then.) At another level, given that it is likely that the universe has some perfect mathematical structure (even though we do not know what it is yet), from the point of view of math, the universe is simply one giant mathy thing, and it is the math the comes before reality.

I love the poem. May I ask where you got it from or whether you wrote it?

My own feeble handiwork…

I very much like it. I told it to a student that I was tutoring. We got deep into various myths that we use in physics, points, lines, triangles, fields, etc…

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Question, what if you define an area using a triangle. I.E. not a drawing, but using points and lines. Like cutting a piece from a whole, like drawing a triangle on a piece of paper but not using any visual representation for those lines. Like defining points on a coordinate plane and using those points to define the lines that make up the triangle. Then declare those lines(that aren’t drawn) and everything in them to be the triangle. Would you consider that a triangle? Anyway, great article.

Ever head of Descartes? Because this is what his fifth meditation is all about, and you just copied his idea without even referencing him! If this had been about any other shape, I could pass it off as a coincidence, but you used a triangle in your explanation which is the same example he used and set off a siren in my head….

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