What If π Were 2?
One day my kid looked up from a worksheet and asked: “what would circles look like if π wasn’t 3.14?” I didn’t have a good answer. So I built one.
Here’s the equation for a circle: x² + y² = r². Every point, same distance r from the center. That’s the whole thing. Now look closely — there’s no π anywhere in it.
π shows up later, once you ask how big the circle actually is — circumference C = 2πr, area A = πr². Those are things you calculate about the shape, not something baked into it. So “change π” can’t mean editing that equation — there’s nothing in there to change. It has to mean something sneakier: build a shape where area ÷ radius² is an actual dial, then crank it to 2 and see what falls out.
Turns out mathematicians already built that dial — they just call it p, the exponent in a p-norm, math’s general-purpose way to measure distance. Ordinary straight-line distance, the kind circles are made of, is p = 2. Taxicab distance — city blocks, no cutting corners — is p = 1. Pick any p and you get your own flavor of distance, and your own flavor of “circle.” Sorry, Euclid, you don’t own the word.
Drop that exponent into the circle’s equation:
|x|^p + |y|^p = r^p
Plug in p = 2 and you get the circle you already know. Dial p down toward 1 and the curve straightens, corner by corner, into a diamond. At p = 1 — tips sitting at (±r, 0) and (0, ±r) — that diamond’s area comes out to exactly 2r². Not roughly. Not “close enough for a worksheet.” π = 2, full stop. And it’s not a shape I invented for this post — it’s the circle of taxicab geometry, the one you’d get if New York cabs ran the universe.
Enough telling you. Here’s the thing itself — drag the slider, or just type a π value, and it finds the matching p and draws it live. Dip below 2 and the shape folds into thin, spiky stars. Push past it and it rounds out toward a square by the time you hit 4.
Here’s the twist: area was just one way to look at it. π is just as legitimately defined by circumference instead — and measured that way, the same diamond gives you a completely different number.
Same shapes as before, just measured by walking the edge instead of filling in the inside. The circle is the only point where area-π and circumference-π agree — nudge p either way and they go their separate ways. “Which one’s the real π” is a trick question, same energy as “is a hot dog a sandwich.” Depends what you’re measuring, and both camps will die mad about it.
The idea keeps going, too. Curve the space itself — draw circles on a sphere — and the ratio bends on its own, no new shape required. Squash a circle into an ellipse, though, and nothing happens: area’s still πab no matter how hard you pancake it. π shrugs.
My kid got an answer. I got a rabbit hole. Worth it.