The original aperiodic tile, which uses its mirror image
Explorer by David Ryan (MIT) · probe maps & essay after @SlavisTon
The Hat was the first "einstein": a single tile that covers the plane only in patterns that never repeat. David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss announced it in March 2023. The Hat needs its mirror image: about one tile in eight is flipped. The Spectre, found a few months later, needs no mirror images at all.
The Hat belongs to a family, Tile(a, b), with edges of two lengths a and b. The slider moves through the family, from the chevron, Tile(0, 1), to the comet, Tile(1, 0). The hat is Tile(1, √3) and the turtle is Tile(√3, 1). Every member with a ≠ b makes the same pattern, only reshaped, so the slider reshapes all the tiles at once. The chevron and the comet also tile in repeating patterns, and so does Tile(1, 1) once mirror images are allowed. Here they all show the same non-repeating pattern.
At the Spectre mark the tile is Tile(1, 1), the straight-edged Spectre from the Spectre page, but laid out in the hat pattern, with mirror images. The two pages show two different tilings made from one tile.
Hats turn in 60° steps and come in two hands, so the colours give each turn and hand its own colour, or split the hats by hand alone. In the key, an m marks a mirrored hat, so 0° m is a mirrored hat at 0°. Click a group in the key to give it your own colour, or roll the dice for random colours. Colour by picks what the colours show (orientation, the article's probe maps, the explorations) and Palette picks the scheme, from the original colours to Viridis, Okabe–Ito or Mondrian. Customise sets the outline and background, saves your own palettes, and exports them as a short code. Supertiles show the hierarchy that forces the pattern: hats group into the metatiles H, T, P and F, which group into larger ones, level after level. Grid lays a triangular grid through a corner. At the hat's own shape the 0° grid passes through every corner, since the hat is built from the kites of a regular grid. Move the slider and the fit breaks.
Drag to move. Scroll, pinch or double-click to zoom. Turn with the turn buttons, a two-finger twist, a right-drag, Ctrl and drag, or Shift and the arrow keys. A trackpad twist works in Safari. The address bar keeps your place, colours and shape, so you can share a view.
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