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Concept

Incommensurability

The property of two lengths with no common measure — no unit, however small, dividing both a whole number of times. Its discovery is what forced the Greeks to separate number from magnitude, and it is what the Euclidean algorithm failing to terminate detects.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

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.

geometry · Pythagoras Euclid's algorithm on a 34 by 13 rectangle. The rectangle is tiled by peeling off the largest square that fits, again and again, until nothing is left.

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.

geometry · Euclidean algorithm φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.

A fraction that never closes

Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.

number · Euclidean algorithm Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

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.

number · Irrationality

Named alongside it

The objects these essays reach for when they reach for this one.

Continued fractionsFibonacciGreatest common divisorTerminationAreaContinued fraction convergentCosineCounting two waysDescentDissectionEuclidean algorithmGolden ratio

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