Einstein Monotile
For over half a century mathematicians asked whether a single tile could force a pattern that never repeats — an “einstein” (German for “one stone”). Penrose got it down to two tiles in the 1970s; the one-tile case stayed open until March 2023, when hobbyist David Smith and collaborators found the “hat.” This toy builds genuine hat and spectre tilings with the exact substitution rules from the discovery papers: metatiles are recursively inflated and subdivided, so what you pan and zoom across is honest aperiodic structure, not a looping wallpaper. Colour tiles by orientation or by supertile family to expose the hidden hierarchy, and highlight the mirrored hats — the spectre, found months later, needs no mirror images at all.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- Aperiodic means more than non-repeating: the hat can tile the plane, but *only* non-periodically — no translation ever maps the tiling onto itself. The proof groups every tiling into H, T, P and F metatiles that themselves form a forced substitution hierarchy.
- The hat uses reflected copies (about one tile in seven); critics called that a loophole, so in May 2023 the same team produced the spectre, a “chiral aperiodic monotile” that tiles aperiodically using rotations alone.
- Tiles here are placed by the H/T/P/F substitution system for the hat and the nine-metatile system for the spectre, both ported from the discovery team’s reference code — each slider level inflates the patch one substitution step further, so the patch is finite but exact.
- The hat is a polykite — eight kites from the [3.4.6.4] Laves grid — and belongs to a whole continuous family Tile(a,b) of aperiodic shapes; the spectre is the special member Tile(1,1).
- Runs 100% in your browser — simulations are computed locally on your device.
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