Two players break a chocolate bar. The bar is rectangular, m by m. On each turn, a player chooses a piece and breaks it along a line — removing one or more rows or columns from one end. The top-left square is poisoned. Whoever is forced to eat it loses.
The game is deterministic. From any position, either the current player can guarantee a win (an N-position) or the opponent can (a P-position). The P-positions — the positions where you want to leave your opponent — form a pattern.
The pattern is a fractal. Specifically, the P-positions correspond to cross-sections of a three-dimensional Sierpiński octahedron (arXiv:2602.20182). Viewed at any resolution, the losing positions exhibit self-similar structure, recursively nesting copies of themselves at every scale.
The same set of positions can be generated by a second-order cellular automaton related to Rule 60 — a one-dimensional automaton that produces Sierpiński triangles. The game, the fractal, and the automaton are different descriptions of the same mathematical object. None is more fundamental than the others. The game defines a partition of positions; the fractal describes its geometry; the automaton provides its dynamics.
The conceptual point: the structure of a combinatorial game — who wins from which position — can have a geometry that is far richer than the game's simple rules suggest. Breaking chocolate is a trivial operation. The pattern of who wins is not trivial — it is fractal, recursive, and connected to structures studied for entirely different reasons in entirely different fields. The complexity is not in the game. It is in the boundary between winning and losing, which organizes itself into the same kind of self-similar structure found in dynamical systems that have nothing to do with chocolate.