Fibonacci — where it appears
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.
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 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Continued fractionsGreatest common divisorGolden ratioIncommensurabilityTerminationTilingContinued fraction convergentEuclidean algorithmLamé's theoremLogarithmic spiralPeriodicityPhyllotaxis