Six tilings built from two tiles, each drawn here from its mathematical definition
An aperiodic set of tiles covers the plane, but never in a repeating pattern. Fewer tiles is more interesting, and for almost fifty years the Penrose tilings at two tiles held the record. This was until the Hat aperiodic monotile was discovered in 2023, showing that one unmarked shape is enough. Here are interactive versions of some interesting two-tile systems:
Above each picture are the two tiles it uses. Drag a picture to move it. Pinch, Ctrl+scroll or the buttons zoom it; on a phone, use two fingers.
Roger Penrose, 1974
A kite has angles of 72°, 72°, 72° and 144°. A dart has angles of 36°, 72°, 36° and 216°. Both have edges of length φ and 1, where φ = (1 + √5)/2 is the golden ratio. The bare shapes also tile periodically, so the pair needs matching rules: marked corners, arcs or bumps. The patch starts from a sun of five kites. Seven rounds of subdivision split every half-kite and half-dart into smaller ones, cutting at the golden ratio.
| Kites : darts | φ : 1 in the limit |
|---|---|
| Inflation | φ ≈ 1.618 |
Roger Penrose, 1974
A thick rhomb with angles of 72° and 108°, and a thin rhomb with angles of 36° and 144°, all edges equal. Like the kites and darts, they need matching rules. The patch starts from a wheel of ten half-rhombs and goes through seven rounds of subdivision. Half-rhombs on the rim show as triangles.
| Thick : thin | φ : 1 in the limit |
|---|---|
| Inflation | φ ≈ 1.618 |
Robert Ammann, 1977; published by Grünbaum and Shephard, 1987
Two L-shaped hexagons with only right angles. The small tile is the big one scaled by Ψ = 1/√φ ≈ 0.786. Going clockwise from the longest side, the big tile's sides are 1, Ψ, Ψ², Ψ⁵, Ψ⁴ and Ψ³. Each round of subdivision turns a big tile into a big and a small tile, and a small tile into a big one, all scaled by Ψ. So the tile counts follow the Fibonacci numbers. Marks on the edges enforce the hierarchy; they are not drawn here.
| Tiles in the patch | 4,181 after 17 rounds |
|---|---|
| Inflation | √φ ≈ 1.272 |
Chaim Goodman-Strauss, 1999 and 2016
The real trilobite and crab are curved shapes with stripes. Goodman-Strauss's 2016 proof draws them on a square grid, and so does this page. Every cell holds one tile. A crab has a notch at each corner. A trilobite has a round tip at three corners and a notch at the fourth. At every corner of the grid, exactly one tile puts its tip into the notches of the other three.
The tiling comes from the chair tiling of L-shaped triominoes, as in his 1999 proof for the trilobite and cross. Centre the grid's cells on the corners of the chairs' unit squares. Then put a trilobite at the inner corner of every chair, with its three tips in the chair's three squares, and a crab everywhere else. Each square belongs to exactly one chair, so each corner of the grid takes exactly one tip. The patch has 1,024 chairs from five rounds of the chair substitution, so it has 1,024 trilobites. Show the chairs to see them: thin lines for the chairs, thick lines for the chairs twice the size. Four chairs make one chair twice the size, and their four trilobites make one larger trilobite. That hierarchy is what the matching rules force. The stripes on the real tiles carry further rules and are not drawn here.
| Trilobites : crabs | 1 : 2 |
|---|---|
| Hierarchy | chair, inflation 2 |
| Orientations used | 8 in all |
Joan Taylor and Joshua Socolar, 2010
A hexagon and its mirror image, marked with black stripes and with red and blue diameters. The colours stand for left- and right-handed flags in Socolar and Taylor's first drawing, so reflecting the tile also swaps red and blue. Rule R1: black stripes continue across every edge. Rule R2: at the two ends of any edge, the diameters that line up with it have different colours. Those diameters belong to the tiles just past each end, so R2 links tiles that do not touch. With reflections allowed, the pair is a single tile; without them, it is an aperiodic pair. A SAT solver found this patch of 469 hexagons from the two rules alone. The same encoding admits no repeating pattern on any torus up to 16 by 32 hexagons, as the theorem requires.
| Hierarchy | half-hex, inflation 2 |
|---|---|
| Order | limit-periodic |
David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss, 2023
The Spectre is one tile with fourteen equal edges, pointing in the twelve directions at multiples of 30°. Stretch a Spectre tiling: give the six directions at even multiples of 30° length a, and the other six length b. With a = b, this is the Spectre tiling. Otherwise, about one tile in 8.9 turns by an odd multiple of 30°, which swaps its edge lengths and makes it a second shape. The tiling keeps the Spectre's hierarchy. The slider stretches a level-4 supertile of 4,401 tiles.
| Even : odd turns | 4 + √15 : 1 in the limit |
|---|---|
| Overlap check | none at the ratios tested, from 0.1 to 10 |