The Spectre is a single tile that covers the whole plane with no gaps and no overlaps. The pattern it makes never repeats. David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss announced it in 2023. It settles the "einstein" problem in its strictest form: with curved edges, not even the tile's mirror image can join in.
The tile appears in 12 orientations. Six are common and sit 60° apart. The other six are rare. Each rare tile is turned a further 30°, and it pairs with one neighbour to make a two-tile cluster called a Mystic. The colour menu groups the orientations by their angle modulo 30°, 60°, 90°, 120°, 180° or 360°. Click a group in the key to give it your own colour, or roll the dice for random colours.
With straight edges the tile is Tile(1,1). It makes this pattern only if mirror images are banned, because with mirror images it can also tile periodically. Replace every edge with the same curve and the mirror image stops fitting. The menu sets the shape of a bump on each edge. Single puts one bump on each edge and flips its direction from edge to edge. That fits because every shared edge joins an odd-numbered edge to an even-numbered one. Double puts an S-shaped pair of bumps on each edge. Height sets how far each edge swings, and zero gives straight edges. Push the height past the marked limit and the tiles collide: the red stripes show where they overlap.
Tiles group into supertiles, and those group into larger supertiles, level after level. Turn on Supertiles to see five levels of that hierarchy, drawn thicker as the level rises. Grid lays a triangular grid through a corner of the tiling and marks the other corners it passes through. It fits a few corners, then drifts off, because no grid holds the whole tiling. Click two or three corners to fit a grid through them.
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 edges, so you can share a view.
This demo needs WebGL 2, which this browser does not offer.