Geometry
Two squares, four triangles, and no algebra
The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.
Every square is a stack of odd numbers
Add up the odd numbers in order and the running totals are 1, 4, 9, 16, 25. This is not a coincidence, and the reason fits in a single picture.
The oldest algorithm, drawn as a tiling
Euclid's method for finding a greatest common divisor is usually presented as a loop. It is also a way of tiling a rectangle with squares, and the tiling explains why it works.
Why the list of perfect solids stops at five
There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.
One cone, four curves
The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.
The rectangle that eats itself
Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.
A circle unrolled into a triangle
Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.
Round is not the only way to be the same width
A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.
An angle that does not care where it stands
Fix two points on a circle and look at them from anywhere else on the far arc. The angle is the same from every one of those places, and it is exactly half the angle at the centre.
The plane, divided by whoever is nearest
Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.
Euclid proves it without moving anything
The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.