Crystallography & Symmetry
patterns derived from their groups
Generator

The hat: eight kites, thirteen sides

The hat: eight kites, thirteen sides
The hat: eight kites, thirteen sides. The shape a search over the eight-kite polykites returns, drawn on the kite grid it lives in — the Laves tiling [3.4.6.4], in which every hexagon is cut into six kites. The eight kites of the shape are tinted and its outline is drawn heavy. Thirteen sides result, of two lengths only: a half and root three over two, in units of the hexagon's circumradius, with one side of twice the shorter length where two kite edges lie in a line. Its interior angles are 90, 120, 240 and 270 degrees. Nothing about the shape was chosen: it is the one octakite that clears every filter in the search.

The shape a search over the eight-kite polykites returns, drawn on the kite grid it lives in — the Laves tiling [3.4.6.4], in which every hexagon is cut into six kites. The eight kites of the shape are tinted and its outline is drawn heavy. Thirteen sides result, of two lengths only: a half and root three over two, in units of the hexagon's circumradius, with one side of twice the shorter length where two kite edges lie in a line. Its interior angles are 90, 120, 240 and 270 degrees. Nothing about the shape was chosen: it is the one octakite that clears every filter in the search.

5 essays call hat-tiling. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

The hat: eight kites, thirteen sides. The shape a search over the eight-kite polykites returns, drawn on the kite grid it lives in — the Laves tiling [3.4.6.4], in which every hexagon is cut into six kites. The eight kites of the shape are tinted and its outline is drawn heavy. Thirteen sides result, of two lengths only: a half and root three over two, in units of the hexagon's circumradius, with one side of twice the shorter length where two kite edges lie in a line. Its interior angles are 90, 120, 240 and 270 degrees. Nothing about the shape was chosen: it is the one octakite that clears every filter in the search. Order without repetition

One tile, and no period

Every aperiodic pattern in this collection so far needs two shapes. A search over the eight-hundred-and-seventy-three ways of gluing eight kites together, filtered by nothing but whether a shape tiles and whether it repeats, returns exactly one — and it is the shape announced in 2023.

 Unreflected copies stop at 1 ring. Copies of the hat, all of the same handedness, covering a core of 1 ring of hexagons — 9 tiles, every cell covered once. At 2 rings the same search runs to exhaustion and returns nothing: there is no such covering, and the failure is a proof for that region rather than a search that gave up. The reflected copy is not a convenience of the drawing; the tiling cannot proceed without it. Order without repetition

The tile that needs no reflection

One shape tiles the plane and never repeats, and it does it with copies of both hands. Cut the tiles out of card and that is nothing; ask for it in a molecule, where handedness cannot be undone by turning something over, and it is the whole question.

 One hat patch laid out as hats, as equilateral tiles and as turtles. A patch of 36 hats found by exact cover on the kite grid, 4 of them reflected and drawn in the second colour, laid out three times. Every edge keeps its direction; short edges and long edges are given their own lengths. At short 1 and long √3 the tiles are hats, at equal lengths they are the equilateral member of the family, and at short √3 and long 1 they are turtles. In all three the same tiles touch the same neighbours along the same edges, and each layout was checked to be a tiling: 0 gaps and 0 overlaps, 0 gaps and 0 overlaps, 0 gaps and 0 overlaps among 1500 sample points, and every interior vertex surrounded by a full turn. Order without repetition

The hat and the turtle are one tiling

The hat has short sides and long sides; the turtle has the same turns with the two lengths exchanged, and looks nothing like it. Take a patch of hats, keep every edge pointing the way it points, stretch the short edges and shrink the long ones, and the patch becomes a patch of turtles — every tile touching the same neighbours along the same edges.

 A patch of hats scatters a pattern that repeats. The diffracted intensity of the 1217 points of a patch of 183 tiles laid out as the hat, over 2 by 2 cells of the kite grid's reciprocal lattice, whose edges are the faint lines. Every local maximum above a hundredth of the central peak is a disc with area proportional to its intensity; 12 reach the central peak's full height. 72 maxima are drawn. Adding a reciprocal lattice vector to the scattering vector changes the intensity by at most 1.1e-15 of the central peak, so each cell holds the same pattern. Order without repetition

How much of the hat is a crystal

Put a scatterer on every corner of a patch of hats and the diffraction pattern repeats exactly, because every corner sits on a lattice. Inside each repeat the strongest reflections are those of an ordinary crystal with partly filled sites, and by Parseval's identity they carry sixty-three per cent of what the pattern holds. The aperiodicity the hat is famous for lives in the remaining third, in reflections a hundred times weaker.

 Identical layers, each turned by an angle no number of turns undoes. Plan views of 4 layers of a stack. Each layer is the same square lattice with one cell shaded and one direction drawn, and each is turned from the one below through the angle whose cosine is three fifths, about 53.13 degrees. That angle is not a rational part of a full turn, so no number of layers brings the drawn direction back to where it started. A tiling of space with this structure has a symmetry that turns one layer onto the next and climbs one layer, and it has no translation. Order without repetition

Aperiodic is two words in space

A tile is aperiodic when none of its tilings is periodic, and periodic has been read two ways: a tiling with a translation, or a tiling with infinitely many symmetries. In the plane those are one condition, provably. In space they come apart, and a prism found in 1988 sits exactly in the gap.

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