Descent
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Two squares, and a lattice
Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.
Every triple, on one circle
Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.
The square that cannot shrink
The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysBijectionConicContinued fractionsGaussian integersHypotenuseIncommensurabilityInfinite descentIrrationalityLatticeModular arithmeticNorm