Concept

Equidistribution

The property of a sequence that the share of its terms landing in an interval tends to that interval's length. It is the strongest sense in which a deterministic sequence imitates a random one, and repeated rotation by an irrational angle has it.

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

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

A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear.

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

dynamics · billiards

Named alongside it

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

Irrational rotation Approximation Billiards Caustic Continued fractions Golden ratio Integrability Iteration Orbit Periodic orbit Phyllotaxis Reflection

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