Visual proofs of
beautiful ideas.

Start with a triangle. Replace the middle third of every edge with a smaller triangular bump. Repeat. The shape stays inside the same circle forever, so the area can't escape - but the edge gets longer by a third with every step.

Three corners. A wandering point. Pick a random corner, move halfway toward it, drop a dot. Repeat forever. The dots don't fill the triangle. They draw a Sierpinski - a perfect fractal, holes inside holes, out of pure randomness.

Drop a thousand balls through a triangular grid of pegs. Each ball flips a coin at every row, left or right, half each. The heap at the bottom isn't random. It is row N of Pascal's triangle, drawn one ball at a time.

A cube's eight corners split by parity into two regular tetrahedra. Put both back together and three Platonic solids hide in one figure: the two tetrahedra, the cube their outer points trace, and the octahedron at their crossings.

A dodecahedron has twenty corners. The right eight of them form a cube. There are exactly five ways to pick them, and the five cubes share every vertex twice over. Drag the figure and watch the five-fold dance.

Take three identical golden rectangles. Stab them through each other at the centre, each perpendicular to the other two. Their twelve corners are exactly the twelve vertices of a regular icosahedron. The d20 hiding in any dice bag was secretly the golden ratio all along.

Roll one coin around another coin of the same size. A single dot on the rim traces a perfect heart - a cardioid. And the rolling coin spins twice for every one trip around. Two surprises, one diagram.

Trisect each angle of any triangle. Take the trisecting ray nearest each side from both of its endpoints. Those pairs meet at three points. Those three points form a perfect equilateral triangle. The Greeks missed it for two thousand years; Morley spotted it in 1899.

Take any triangle. Build an equilateral triangle outward on each of its three sides. Connect the three centres. What you get is itself a perfect equilateral triangle. Always. Drag any vertex and watch the lopsided outer shape morph while the inner triangle refuses to be anything but equilateral.

Drag the three vertices anywhere. The corners change shape but their sum doesn't. Press fold and watch the three angles come down to the baseline and tile a perfect half-disc - the same straight line, every time.

Pin two points on a circle. From anywhere on the far arc, the angle they make is the same number - always. Drag P around and watch the rays swing wildly while the angle between them refuses to budge. The central angle, naturally, is twice it.

Build the right triangle, its three squares, and the little gap-triangle between the leg-squares. The whole upper half rotates 180° around the centre of the hypotenuse-square and lands on its mirror. The figure cannot be anything but the theorem.

Cut the larger leg-square with two lines through its centre, parallel and perpendicular to the hypotenuse. Four pinwheel pieces. Add the smaller leg-square whole. Five pieces slide - no rotation, no flipping - into the hypotenuse-square. They fit. That is the proof.

Drop the altitude from the right-angle vertex. The triangle splits into two smaller copies of itself - same shape, three sizes. Their areas scale as the square of any side, so adding them up writes a² + b² = c² in one line. Einstein's favourite proof.

Draw squares on every side of a right triangle. The two leg-squares morph downward into rectangles that tile the hypotenuse-square exactly. This is Proposition I.47 - the proof the western world has been teaching for 2,300 years.

Two right triangles meet at a corner. Their hypotenuses lock at 90°. The outline is a trapezoid. Compute its area two ways - once as a trapezoid, once as three triangles - and the 2ab cross-terms cancel. What's left is a² + b² = c². The only proof of Pythagoras authored by a US president.

Pascal's triangle was built with no Fibonacci rule in sight. But cut a band of cells at a shallow tilt and sum each band, and the Fibonacci numbers come walking out: 1, 1, 2, 3, 5, 8, 13, 21, ...

Add 1+1, then 1+2, then 2+3 - and divide each Fibonacci number by the one before it. The ratios bounce: above the line, below the line, above, below. Twelve steps in, all four leading digits agree on φ.

Cut an 8×8 chessboard into four pieces. Rearrange them into a 5×13 rectangle. The square had 64 cells. The rectangle has 65. The pieces did not change. The Fibonacci numbers 3, 5, 8, 13 stole a cell out of nowhere.

Cut a line in two. There is exactly one place to cut where the larger piece is itself a perfect smaller copy of the whole. That cut is the golden ratio - and the line that comes out of it contains a smaller golden line, which contains a smaller one, forever.

Draw a pentagon, draw its diagonals, and a smaller pentagon falls out. Do it again and again. The ratio of any diagonal to any side is the same number every time - the golden ratio.

Negative numbers don't live anywhere new. Every negative is a positive viewed through a mirror at zero - and multiplying by minus one is literally a half-turn around it.

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.