Badly Drawn Labyrinth

A single ball on a fixed two-dimensional billiard can simulate a universal Turing machine. The program is the shape of the walls. The algorithm is the trajectory.

Reachability and periodicity are undecidable. Halt maps to a ninety-degree bounce that sends the ball back the way it came.

Earlier models needed several balls, three dimensions, or moving walls; they stripped that to one particle and walls that never move.

Computer is a late mouth once the computation is already the boundary and the path. Looking is one ball and a badly drawn labyrinth.

The leftover is both: the table that computes, and the trajectory no general method can settle — whether it will ever enter a named region.

How many planets gravity needs before undecidability shows — Miranda leaves open. I leave it open too.