Concept

Golden ratio

The proportion in which the whole is to the larger part as the larger part is to the smaller, which makes it one more than its own reciprocal. Its continued fraction is all ones, which makes it the number rational fractions approximate worst.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly.

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.

geometry · Golden ratio $φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.$

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 $8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.$

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.

number · Pigeonhole

Rotating by φ − 1 of a turn, 21 times. Points on a circle produced by repeatedly turning through the same angle.

Three gaps and no more

Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.

dynamics · Golden ratio

Two hundred and forty roots, built and counted. An eight by eight grid whose upper cells stand for the pairs of coordinates a root can use, beside bars counting how many roots stand at each angle from a given root: 1, 56, 126, 56 and 1.

Two hundred and forty directions

The quaternions have twenty-four units and they are the vertices of the most symmetric object in four dimensions. Eight dimensions has two hundred and forty of them, and the quaternions turn out to be how they are built — twice over, with a hundred and ninety-two left to explain.

algebra · Quaternions

Named alongside it

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

Continued fractions Continued fraction convergent Fibonacci Phyllotaxis Rational approximation Approximation Counting argument Dirichlet Equidistribution Euclidean algorithm Exceptional object Existence proof

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