Matpic
mathematics, in pictures
Concept

Irrational rotation

Turning a circle repeatedly by an angle that is not a rational part of a full turn, so no point is ever revisited.

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

0123456721 steps of a rotation by φ − 1 of a turnthe gaps between neighbouring points take 2 distinct values — never more than three, at any number of steps

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 0123456740 steps of a rotation by √2 − 1 of a turnthe gaps between neighbouring points take 3 distinct values — never more than three, at any number of steps

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

dynamics · pigeonhole

Named alongside it

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

IterationOrbitApproximationContinued fractionsEntropyEquidistributionGolden ratioMeasurePhyllotaxisPigeonholeRecurrenceReversibility

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