Visual proofs of
beautiful ideas.

Three cells in. One cell out. Repeat. With the right eight bits of rule, the pattern that grows is genuinely indistinguishable from random - even though every step is fully determined.

A trunk. Two branches. Each branch is a smaller copy of the tree below it. That single rule, applied to itself, builds every fractal tree you have ever seen.

Three regular shapes tile the plane: triangles, squares, hexagons. Pentagons don't. Three of them meet at a vertex, stop short, and leave a tell-tale 36° gap. The reason hides inside one number: 360.

Primes feel like a separate species. They aren't. Lay out the first 100 numbers, cross out every multiple of 2, then 3, then 5, then 7. The 25 numbers still standing are the primes - all of them.

Sin and cos aren't weird ratios from a textbook. They're the height and the width of a point going round a circle. Drag the angle and watch the two waves write themselves out, one a quarter-turn behind the other.

Cut the disc into wedges. Lay them out tip-up, tip-down, side by side. You have a rectangle of π·r by r - whose area is the area of the disc, just rearranged to where you can see it.

A4 has a peculiar aspect ratio - 1.414 to 1. Fold it in half and you get A5: same shape, half the area. Only one number on Earth makes that work, and it isn't a coincidence the printers chose it.

Every standard way to compute π begins with a circle being measured. This one doesn't. Throw darts at a board. Don't aim. Count what fell under the curve and multiply by four.

Every irrational has a fingerprint - the count of squares you cut from a rectangle of that aspect ratio, level after level. φ's fingerprint is all 1s. The smallest possible. The hardest to approximate.

Drop seeds onto a disc, each a small turn from the last. At one angle - and only one - the seeds pack into a perfect sunflower. At every other angle, the magic dies. The angle is 137.508°.

A cubed number is n stacked floors of n² cells each. Pull the floors apart and the third dimension stops being a trick of the projection - it's the stack itself.

A number is just a count of things. The digits are an artefact of which base you happened to grow up counting in. Slide the base, watch the bricks regroup.

The golden ratio is the unique aspect ratio that survives a square cut. Watch the spiral come together at exactly one number.

Multiplication isn't really repeated addition. It's stretching the number line. Drag the slider and watch.

Roll a circle along a number line. It always lands on π, no matter the size. The ratio is the point.

The sum of the first n odd numbers is always a perfect square. Watch the L-shaped layers add up.

A visual proof of Pythagoras' theorem you can scrub through.