eisenstein — hat tilings

This is an aperiodic tiling by the hat (Smith, Myers, Kaplan and Goodman-Strauss, 2023). The hat is an einstein — German ein Stein, one stone: a single shape that tiles the plane, but never periodically. And every vertex of every hat sits on the Eisenstein integers ℤ[ω] (Gotthold, not Albert), so the tiling is generated with exact integer arithmetic and extends as far as you care to pan — the plane is a supertile of level 24, about 10²⁰ hats. Hence the name: an einstein, on the Eisenstein lattice.

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, hats would be smaller than three pixels, so supertiles are drawn as flat regions instead; the readout switches from hats to regions. Zoom in and the hats come back.

Colouring

The tiling is built hierarchically: hats form four kinds of metatile (H, T, P, F, with 4, 1, 2 and 2 hats), metatiles form larger copies of the same four shapes, and so on up. Each scheme colours by one feature of that structure.

reflection: light for the hat, dark for its mirror image. One hat in seven is reflected (exactly 1 : φ⁴), always inside an H, never touching another.
orientation: one of 12 colours for the hat's rotation and reflection; orientation6 ignores reflection.
meta: by the metatile the hat belongs to: H blue, T pale, P green, F gold.
meta-k: by the type of the level-k ancestor — set level. Level 1 is meta; higher levels show ever larger patches of the four types.
ancestry: a hash of the chain of ancestor types up to level: the fractal look, where every supertile of that level gets its own colour and the pattern nests.
id: a hash of the hat's position in the hierarchy: effectively random, but the same for a given seed, so panning never changes a hat's colour.
mono: one fill; with outlines on, the geometry alone.

The palette menu picks the colours. The named ones are hand-chosen; viridis, cividis, plasma, magma, inferno and turbo are the standard colour maps, sampled at as many colours as the scheme needs (four for meta, twelve for orientation). The hashing schemes shuffle their palette by seed, so the same colouring never looks the same twice.

Highlight

Draws the outlines of every supertile of the chosen level in red, on top of everything. Level 2 shows the first tier of the hierarchy; each level up is about 2.6 times larger. Off by default.

Shape

The hat belongs to a continuous family Tile(a, b) with the same angles and two edge lengths. The hat is (1, √3), the turtle is (√3, 1), and everything in between tiles the same way; pick custom and set a and b to morph the tiling live. The two endpoints degenerate and are not allowed.

Share and export

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

Credits