Concept

Euclidean algorithm

Repeatedly replacing a pair of numbers by the smaller and the remainder, until nothing is left over. It finds the greatest common divisor, and a ring in which it can be run has unique factorisation as a consequence.

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

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

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

The twenty-four unit quaternions, at the corners of a four-dimensional solid. The twenty-four units of the Hurwitz quaternions drawn as the vertices of a 24-cell projected into three-space, with the ninety-six edges joining units at distance one.

The integers among the quaternions

The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.

algebra · Quaternions

Named alongside it

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

Continued fraction convergent Continued fractions Fibonacci Greatest common divisor Golden ratio Group Incommensurability Lattice Norm Periodicity Polyhedron Quaternion

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