Area
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.
Adding up rectangles until they stop being rectangles
The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.
A matrix is a picture of what happens to the grid
Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.
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.
Bayes' theorem is a picture of a square
A test that is 99% accurate returns a positive result. The chance it is right can easily be under one in five, and the reason is visible the moment the population is drawn as a square rather than described as a formula.
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.
Area is the undoing of slope
Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.
Completing the square, by completing a square
The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.
The door that was not opened
Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.
Named alongside it
The objects these essays reach for when they reach for this one.
DissectionLimitBayes' theoremConditional probabilityFundamental theoremHypotenusePiRiemann sumRight triangleAccumulationAlgebra tilesAltitude