Concept

Billiards

The motion of a point bouncing inside a region, leaving each wall at the angle it arrived. The shape of the region decides the long-run behaviour completely, from paths that repeat forever to paths that come arbitrarily close to every point.

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

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

Two loops of equal area, and the two points where they cross. An annulus with the loop of points whose angle is unchanged by the map and the image of that loop, drawn both on the annulus and unrolled into a rectangle. The loops cross at two points, which are the fixed points.

A twist that cannot avoid two points

Turn the two edges of a ring in opposite directions without changing any area, and something in between must stay exactly where it is — not one point, but at least two, and the reason is that two loops enclosing the same area have to cross.

dynamics · fixed points

Named alongside it

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

Periodic orbit Annulus Area preservation Caustic Equidistribution Existence proof Fixed point Integrability Irrational rotation Reflection Twist map Unfolding

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