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The same thing twice — page 5

Two constructions that look unrelated and turn out to be the same object wearing different clothes.

The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way.

Every partition, hidden in a product

Multiply out one factor for each part size and the coefficient of q to the n is the number of partitions of n. Nothing is being approximated: the product is a bookkeeping device that does the counting by multiplying.

the right triangle at an eighth of a turn: 16 directions, and a surface of genus 2. A polygonal billiard table with a long trajectory drawn on it, the finite set of directions that trajectory takes, and the arithmetic of the surface it unfolds into.

Dynamics

A table folded into a surface

Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.

A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem.

Dynamics

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

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