Gliders on the Hat and Spectre Tilings

a computational essay · every simulation on this page is live

The tiling below is a hat tiling — a plane covered by copies of a single 13-sided shape, discovered in 2023, that can tile the plane but only aperiodically: there is no wallpaper repeat — no block you could cut out and stamp across the plane to reproduce it. Travelling across it, a tile every two generations, is a glider.

A glider is a cellular-automaton pattern with a fixed population that moves. On an ordinary grid — Conway's Game of Life — gliders are commonplace, because the grid looks the same everywhere: a pattern that works once works forever. An aperiodic tiling makes no such promise. The ground beneath this glider keeps changing: local shapes it has crossed before do come round again, but never on a fixed schedule, so surviving one step carries no guarantee for the next. In the early stages of this project, nothing survived that: every traveller we found or evolved died within a hundred tiles of home, and for the smallest rule family — which we searched exhaustively — we could show that nothing better existed.

The panel is not a video. It is the same cellular-automaton engine that ran the discovery search, compiled to WebAssembly, flying the exact committed record of this glider — seed, rule, and tiling regenerate deterministically from a few hundred bytes. Press ⤓ record for those bytes, or open reproduce this flight for the command that re-flies it natively, cross-checked against the same run on one large fixed patch. Its population is flat on patches sixteen-fold apart in radius, and it has been flown one million tiles without drifting from its lane.

The glider as a worldtube

There is an older way to look at a cellular automaton, familiar from Wolfram's one-dimensional diagrams: draw time as a spatial axis. For a two-dimensional automaton that means stacking each generation as a layer of prisms, and for most automata the result is an unreadable solid block. Not here. Our objects are a dozen cells in an empty universe, so the history of a glider is a thin worldtube climbing through space-time: its speed is the tube's slope, a collision is a vertex where one tube ends on another, and the compass law is the statement that all tubes lean along a handful of quantized directions.

Watch the launch: two tubes rise together, braided a few tiles apart, until one strikes the other's wake at generation 42 and terminates — a worldline vertex. The survivor settles into a straight climb at exactly two generations per tile, and the tube's lean never wavers from its compass direction while the terrain streaming beneath it never settles into a cycle. Drag to orbit; the camera climbs with the glider.

This is the companion essay to the paper Gliders on Aperiodic Monotilings. Thirteen sections follow — in the contents rail, starting with The game — and every panel in them is built the same way: from the committed records, live.