Golden ratio — where it appears
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.
How close a fraction can get
Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Continued fraction convergentContinued fractionsFibonacciRational approximationCounting argumentDirichletEuclidean algorithmExistence proofGreatest common divisorIncommensurabilityLiouville numberLogarithmic spiral