A small rant about a big circle

π is wrong.

Not incorrect — the number 3.14159… is fine. Wrong as in a concept we've politely declined to fix for a few thousand years. If trigonometry ever felt needlessly confusing, that wasn't your fault. It was π's.

The fix has a name: τ = 2π ≈ 6.283 — the circle constant measured from the radius, the way circles are actually drawn.

θ sweeps 0 → τ

θ = 0.00

radiusthe honest center

one turnequals τ

cleanerequations & identities

Radians should make sense. π won't let them.

Quick: how much of a turn is π? You'd guess a whole circle. It's half. A full circle is 2π. Now ask for π/8 of a turn — feels like an eighth, but it's actually a sixteenth.

Every single angle makes you secretly multiply or divide by two and pray you went the right way. "Just pay attention to the direction," people say. No — that's making excuses for π. Math this simple shouldn't need a conversion step.

τ deletes the whole tax. One full turn is just τ. Half a turn is τ/2, a quarter is τ/4, a twelfth is τ/12. The bottom number literally is the fraction of the circle. No translation, no apologies.

Watch the wedge sweep around — the readout is the fraction of a full turn, straight up.

fraction of a turn

fraction of a turnwith πwith τ

full turn\(2\pi\)\(\tau\)

half turn\(\pi\)\(\tau/2\)

quarter turn\(\pi/2\)\(\tau/4\)

an eighth\(\pi/4\)\(\tau/8\)

a twelfth\(\pi/6\)\(\tau/12\)

And the sine wave stops being a squiggle you memorize. Start at τ = 0, height zero. A quarter turn (τ/4): you're at the top, height one. Half a turn (τ/2): back to zero. A full τ brings you home. It's just how high you are as you go around the circle — which is exactly what the figure up top is drawing.

Why τ and not π

We build circles from a radius.

Nobody draws a circle by picking a diameter. You pin one end, hold out a radius, and swing it around. The radius is the thing that makes the circle — so the circle's constant should be built from the radius too.

Lay that radius along the rim and count how many fit. The answer is always the same: about 6.28 of them, big circle or small. That's τ. Literally "how many radii go around."

Watch one roll out below: a single turn unspools into exactly τ radii.

one turn unrolled = τ · r

The original mistake

π measures the wrong thing.

π is the ratio of the circumference to the diameter — which is just two radii stuck together. It's a perfectly real number; it's simply answering a question nobody asked.

Define the constant from the radius instead and you get τ — and a surprising amount of math quietly gets easier. Same circle, honest measurement, one fewer stray factor of 2 haunting your formulas forever.

circle — definition

a circle = all points at distance r
           from a center

π = circumference / diameter   # the question nobody asked
τ = circumference / radius     # the one we actually mean

so τ = 2π

The famous equations were hiding a τ

Here's the kicker: nearly everywhere a \(2\pi\) shows up, it was τ in a trench coat. Pull the disguise off and the greatest hits get better, not worse.

Euler's identity

"But you'll ruin e^iπ!"

Relax — nobody's killing Euler's identity. τ makes it clearer. Start from Euler's formula, \(e^{i\theta} = \cos\theta + i\sin\theta\), and spin a full turn by setting \(\theta = \tau\).

After one full turn the height (sine) is 0 and the width (cosine) is 1. So \(e^{i\tau} = 1\): go all the way around, and you're back to 1. No mysterious −1 to decode — it just says what it does.

with π\(e^{i\pi} + 1 = 0\)

with τ\(e^{i\tau} = 1\)

a full turn lands on 1

Area of a circle

with π\(A = \pi r^2\)

with τ\(A = \tfrac{1}{2}\,\tau r^2\)

"But the π one is shorter!" Sure — and it hides the family resemblance. The τ form shows a circle's area is the same shape as half the formulas in physics class: one-half, a constant, something squared.

\(\tfrac{1}{2} g t^2\)distance fallen

\(\tfrac{1}{2} k x^2\)spring energy

\(\tfrac{1}{2} m v^2\)kinetic energy

\(\tfrac{1}{2} \tau r^2\)circle area

The bell curve — the shape behind almost all of statistics

with π\(\dfrac{1}{\sqrt{2\pi}\,\sigma}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)

with τ\(\dfrac{1}{\sqrt{\tau}\,\sigma}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)

That awkward \(\sqrt{2\pi}\) bolted to the front collapses into one tidy \(\sqrt{\tau}\).

The Fourier transform — the math inside audio, images, JPEG, and your voice on a call

with π\(F(k)=\int f(x)\,e^{-2\pi i k x}\,dx\)

with τ\(F(k)=\int f(x)\,e^{-i\tau k x}\,dx\)

The \(2\pi\) tagging along in nearly every wave equation? Always just one thing: τ.

So why name the project τ?

Find the right radius.

A lot of "how AI agents work" material is like π — technically correct, but centered on the wrong thing, so simple ideas end up looking complicated and people bounce off.

This project tries to do what τ does: find the small, honest core, then build outward so the rest becomes obvious. Pick the definition that makes everything else make sense.

tau — the idea

τ › explain agents simply

  find    the small honest core
  build   outward from there
  result  the rest looks obvious

So go ahead, have your π and eat it. But the better dessert is served every June 28th (6.28, naturally) — Tau Day. Michael Hartl's Tau Manifesto makes the full case; this project just borrows the instinct.