This is an aperiodic tiling by the spectre (Smith, Myers, Kaplan and Goodman-Strauss, 2023) — the chiral aperiodic monotile. The hat needs its mirror image: one hat in seven is flipped. The spectre needs no mirror at all. Its curved edges make a flipped copy impossible to fit, so a single handedness of a single shape covers the plane, and never periodically. Underneath the curves sits Tile(1, 1), the equilateral member of the hat's family, with every vertex on the Eisenstein integers ℤ[ω] — the tiling is generated with exact integer arithmetic and extends as far as you care to pan.

Moving about

drag

pan

pinch · wheel

zoom about the fingers or cursor (ctrl+wheel is finer, for trackpads)

double-tap

back to the starting view

seed

a different tiling in a different place; the shuffle button picks one at random

Far out, spectres would be smaller than three pixels, so supertiles are drawn as flat regions instead (their true boundaries — the outlines are exact, however wiggly); the readout switches from spectres to regions. Zoom in and the spectres come back.

Colouring

The tiling is built hierarchically from nine kinds of cluster, named by Greek letters Γ Δ Θ Λ Ξ Π Σ Φ Ψ after the paper. Each cluster carries one spectre — except Γ, the mystic, which carries two, the second rotated 30° against the grain of everything around it.

mystic

the mystic's companion dark, everything else light: the classic spectre picture, and this family's analogue of the hat's reflection.

orientation

one of 12 colours for the spectre's rotation — the companions are the odd twelfths.

meta

by the cluster type Γ … Ψ, nine colours.

meta-k

by the type of the level-k ancestor — set level.

ancestry

a hash of the ancestor chain up to level: the fractal look.

id

a hash of the spectre's position in the hierarchy: random-looking, seed-stable.

reflection

renders one colour here — correct, and the whole point: no spectre is ever reflected.

mono

one fill; with outlines on, the geometry alone.

Highlight

Draws the outlines of every supertile of the chosen level in red. Unlike the hat's, the spectre's supertiles have no simple polygon: what you see is the exact union boundary of their tiles, drawn along the straight Tile(1, 1) edges beneath the curves. Each level up is about 2.8 times larger.

Share and export

The button copies a link that reproduces this exact view. SVG writes the view as a vector file with the curved edges (one path per orientation, one <use> per spectre; capped at two million). PNG renders the view at 3840 × 2160.

See also

The hat page shows the original einstein and its Tile(a, b) family; the blend page switches between the two families in place, with the hat's shape presets and the spectre's curve slider.

Credits

Mathematics: A chiral aperiodic monotile. Built in Rust, compiled to WebAssembly; source on GitHub.