The hat and the turtle are one tiling

One tile, and no period finds the hat by a search over the shapes that eight kites can make. The tile that needs no reflection then draws what the shape belongs to: keep the hat’s thirteen turns, let its short sides and its long sides take any two lengths, and the polygon still closes, because the turns come to a full circle whatever the lengths are. The hat is the member with short sides 1 and long sides √3. The member with the lengths the other way round, short sides √3 and long sides 1, is called the turtle.

That essay passes over the most important fact about the family in a clause: the tilings of any two members correspond tile for tile. The hat and the turtle do not look like relatives. The hat is a lopsided, stubby shape with a notch in it; the turtle is broad and has a head. A claim that a tiling by one is secretly a tiling by the other is a claim about something the pictures do not show, and it is worth making exactly and then checking.

It is true, and it is a construction rather than a coincidence. A patch of hats can be laid out again as a patch of turtles, or of any member of the family, by changing the lengths of its edges and nothing else — and the result is still a tiling, with the same tiles touching the same neighbours.

Two kinds of edge, thirty degrees apart #

The hat lives on a grid, and the grid supplies the whole argument.

Cut every hexagon of the ordinary hexagonal tiling into six kites, each running from the hexagon’s centre to the middle of one side, out to a corner, and back through the middle of the next side. The result is the Laves tiling [3.4.6.4], and a kite in it has two kinds of edge. A long edge runs from a hexagon’s centre to the middle of a side, and has length √3/2 when the hexagon’s side is 1. A short edge runs along half of a hexagon’s side, and has length ½.

The two kinds point in two different families of directions. Long edges point along the six directions from a centre through the middles of the sides; short edges point along the sides themselves, which are thirty degrees away. No short edge is ever parallel to a long one, and every edge of every polykite, the hat included, is one kind or the other.

The hat’s boundary, traced along the kites, is fourteen kite edges: eight short and six long. Two of the short ones lie end to end along a straight line, which is why the polygon has thirteen sides rather than fourteen. And in a tiling by hats on the kite grid, two tiles that share an edge share a kite edge, and they agree about its kind — a short edge of one tile is a short edge of its neighbour. In the patch drawn above that was checked at every one of the edges two tiles share.

That is the fact the family stands on, and it is a stronger fact than the closing of the hat. The boundary closes because the fourteen edges add to nothing. It does not follow from that alone that the eight short edges add to nothing by themselves and the six long ones by themselves; in a general polygon built from two kinds of edge they would not. For the hat they do, and the reason is that the turns alone decide the directions: a closing that holds for every pair of lengths a and b is a statement that a times the short sum plus b times the long sum is zero for every a and b, which forces each sum to be zero.

A patch lays out again if its tiles do #

Now take a whole patch rather than one tile.

A patch of hats is a drawing in the plane: points, edges between them, and tiles bounded by loops of edges. Forget where the points are and keep only which edges join which points, what kind each edge is, and which way it points — a structure with the distances thrown away, with a direction and a kind left on each edge. Then put the points back by a rule: start from one point, and walk along the edges, moving by the edge’s direction times one length for a short edge and a different length for a long one.

That rule gives every point a single position exactly when every closed walk comes back to where it started. A closed walk moves by some sum of short edges and some sum of long edges, and it returns for every choice of the two lengths only if each sum is zero by itself. So the question is whether, round every closed loop in the patch, the short edges and the long edges close separately.

Round every tile they do; that is the figure above. And in a patch shaped like a disc, every closed walk is a sum of tile boundaries, traversed with signs, because a disc has no holes for a walk to go round. So every closed walk closes in each kind separately, and the rule gives every point one position for every pair of lengths.

In the patch of thirty-six hats, which has two hundred and sixty-eight points, that was checked edge by edge rather than inferred: walk outwards from one point to give each point a short part and a long part, then test every edge of the patch against the walk. The largest disagreement is a rounding error in the fifteenth decimal place.

Each point’s position is therefore a short part and a long part added together, and the layout at any pair of lengths is a weighted sum of the two. At short 1 and long √3 the weights are the ones the kite grid has, and the layout returns every point of the searched patch to its own place on the grid. Every other pair of lengths is the same patch, drawn differently.

Nothing that makes it a tiling moves #

A layout that gives every point a position is not yet a tiling. Its tiles could fold over each other, or open gaps, or turn inside out. None of that can happen here, and the reason is that the things a tiling depends on locally are not the things the lengths change.

Every edge keeps its direction, so every angle keeps its size. The angle a tile makes at a corner is the angle between two edges, and neither edge has turned. At a point in the interior of the patch the angles of the tiles meeting there added to a full turn before the lengths changed, and they add to a full turn after. Every edge keeps a positive length, so no tile turns inside out. A tile is a loop of edges turning by the same angles as before, and a loop with positive sides and the same turns encloses positive area.

Those two facts are checked rather than trusted, at seven pairs of lengths spread across the family. At every one of the hundred and sixty-six interior points the angles still add to a full turn. Every tile has positive area. And fifteen hundred points scattered over a disc well inside the patch were each tested against every tile: at every member, none was covered by no tile and none by two.

At short √3 and long 1 every tile of the patch has the turtle’s perimeter and the turtle’s area, which is the check that the tiles there really are turtles rather than some other member drawn at a similar size. So a patch of hats laid out with its lengths exchanged is a patch of turtles, and the thirty-six turtles touch each other exactly where the thirty-six hats did.

What moved, and what did not #

The layouts differ in every respect a picture shows. The tiles change area and perimeter; the patch as a whole changes proportions, becoming wider or narrower as one kind of edge grows against the other. The figure at the head of this essay does not look like three drawings of one thing.

What does not change is everything below the line: which tiles touch which, along which edges, which tiles are the reflected ones, how many neighbours each tile has. Those were counted once, on the graph, because no choice of lengths can alter them. The tiling is the graph with its directions and kinds; the shape of a tile is a coordinate on it.

That reframes what was proved about the hat. The proof that hats tile only aperiodically works by grouping tiles into clusters, grouping the clusters into larger clusters of the same kinds, and showing the grouping is forced and goes on forever — the same shape of argument as inflation, where a tiling reproduces itself at a larger scale. Every step of it is a statement about which tiles touch which. A period would be a translation carrying the tiling onto itself, and a translation carries the graph onto itself; laying the graph out at other lengths turns that translation into a translation of the new layout. So a periodic tiling by turtles laid out at the hat’s lengths would be a periodic tiling by hats. Aperiodicity belongs to the combinatorics, and the whole family inherits it — provided every tiling by a member really does come from a hat tiling, which is the half of the correspondence the construction above does not supply, and which is part of the published proof.

So does every count of local configurations. How many patches of each size a tiling contains is a question about which tiles surround which, and a layout at other lengths does not change a single one of those surroundings; the number of different neighbourhoods of each radius is the same for the hat and for the turtle, even though no neighbourhood of one is congruent to any neighbourhood of the other. Order that is not periodicity is order in this combinatorial sense before it is anything geometric.

The proportion of reflected tiles goes with it. The reflected tiles are a label on the graph, so a patch has the same number reflected at every member, and the limiting proportion in the plane — one reflected tile for every φ⁴, about 6.85, unreflected ones, a published value that the hat’s own essay measures roughly and quotes exactly — is a property of every member’s tilings, the turtle’s included.

The turtle is ten kites #

The layouts carry one more measurement worth stopping on, because it says the turtle is not an arbitrary member of the family.

A kite of the grid has area √3/4, about 0.4330, when the hexagon it was cut from has side 1. The hat is eight kites and its area at its own lengths is eight times that, 3.4641. Laid out at the turtle’s lengths, every tile of the patch has area 4.3301, which is ten kites to the last digit the arithmetic carries. The equilateral member’s tiles have area 4.0981, which is not a whole number of kites at all.

That is what the published account says in other words: the turtle is itself a polykite, built from ten kites, and it lies on a kite grid of its own. The two members at the ends of the interval between them — the one whose points sit on the original grid and the one whose points sit on a finer, rotated grid — are the two that are made of kites. Every member strictly between them tiles by exactly the same combinatorics and is made of nothing in particular.

The turtle’s being a polykite matters for how anyone could have found it. A search over polykites, like the one that found the hat, searches shapes made of whole kites, and the turtle is one of those shapes at ten kites; the equilateral member, and every member between, is invisible to such a search however far it runs. So the family was not found by searching. It was found by noticing that the hat’s boundary is a sequence of turns with two lengths, and asking what happens when the lengths move.

The member where the rule dissolves #

There is one member at which all of this stops being the whole story, and the figure at the top drew it without comment: the one with both lengths equal.

What kept the construction honest everywhere else is that a short edge could only ever meet a short edge. That was never a rule anybody imposed; it was a fact of the geometry, because the two kinds of edge have different lengths, so a short edge laid against a long one leaves a piece of the long one sticking out. The difference in lengths is a matching rule that the shape enforces by itself. Matching rules and what actually forces aperiodicity is the essay about the difference between a decoration somebody has to obey and an edge that simply will not fit, and this is the second kind, built into the tile. Penrose’s rhombs need their arrows drawn on because two rhombs with equal sides will fit together in ways the arrows forbid; the hat needs no arrows because its two kinds of side are different lengths, and the lengths do the forbidding.

At equal lengths it dissolves. A short edge of the equilateral tile is exactly as long as a long one, and nothing in the shape stops them meeting. So the equilateral tile has all the tilings the construction gives it — every hat tiling laid out at equal lengths, aperiodic like the rest — and it has others, in which short edges meet long ones, which no layout of a hat tiling contains. Some of those others are periodic. The equilateral member is therefore not an aperiodic monotile: it tiles the plane without a period, and it tiles the plane with one.

That is the member the chirality result turned on, and the one that matters for a crystal built from one hand. The periodic tilings of the equilateral tile all use its mirror image; replace its straight edges by curves, so that each edge fits only the edge it was cut against, and both the reflected copies and the short-to-long contacts are excluded at once. What is left tiles only aperiodically and never uses its mirror image, which is the spectre. None of that is computed here — the equilateral tile’s extra tilings are not on the kite grid at all, and a search over the grid cannot see them — but the construction above says precisely which contacts the curved edges have to forbid.

The ends of the family #

As one length shrinks towards nothing the tiles thin, and the sweep above shows them doing it. At the limit the tiles degenerate: with the long length set to zero, every pair of points joined by a long edge lands on the same point, and the patch’s two hundred and sixty-eight points collapse onto a hundred and forty. The degenerate tiles at the two ends are shapes the published work calls the chevron and the comet, and both tile periodically. The family is aperiodic in its interior and nowhere else: at one interior point and at both ends it admits a period.

What was computed, and how #

Computed here: the patch of thirty-six hats, by exact cover on the kite grid; each tile’s boundary as a loop of kite edges, each edge classified as short or long by its length; the agreement of the two tiles on every shared edge; the separate closing of short and long edges round the hat; a layout of the patch by walking outwards, giving every point a short part and a long part, and the check of every edge against that layout; and, at seven pairs of lengths, the areas of all tiles, the angle sums at every interior point, and the covering of fifteen hundred sample points. Also that the hat’s own lengths return the searched patch, and that at the turtle’s lengths every tile has the turtle’s perimeter and area.

A patch of thirty-six tiles is not the plane, and nothing above decides whether the hat tiles the plane at all; no procedure decides that for any shape. What the construction establishes is a statement about every patch at once: whatever patch of hats exists, the same patch of turtles exists, and the checks were run on one of them.

Quoted: that the hat tiles the plane only aperiodically; that every tiling by a member with unequal lengths arises from a hat tiling in this way; that the equilateral member and the two degenerate ends admit periodic tilings; and the spectre. All are from the work of Smith, Myers, Kaplan and Goodman-Strauss.

Who found the family #

David Smith found the hat in 2022 by cutting shapes out of card and seeing how far they would go, and the paper announcing it, with Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss, appeared in March 2023. It already contained the family: the authors noticed that the hat and the turtle were two members of one continuum, and gave the argument that any two members with unequal lengths have the same tilings, which is what lets a single computer-assisted proof about clusters serve the whole family at once. The spectre followed at the end of May.

The kite grid is much older than the hat. Every shape that tiles by itself has to be found somewhere, and the Laves tilings were classified in the nineteen-thirties as the duals of the uniform tilings, long before anyone thought a single shape drawn on one might never repeat.

Where this goes: which members keep their points on a grid #

At the hat’s lengths every point of the patch is a point of the kite grid — that is where the patch was found. A point’s position is a short part plus a long part, and every long kite edge is the sum of two short ones, so both parts lie on the triangular lattice the short edges generate — the lattice underneath the hexagons the kites were cut from. Scaling the two parts by different factors puts the points on one lattice exactly when the ratio of the factors is rational. At the hat that ratio is one; at the turtle, where the short edges are stretched by √3 and the long ones shrunk by √3, it is a third, and the points again lie on a lattice, a finer and rotated one. At equal lengths the ratio is √3, the two parts generate a set of translations that is dense in the plane, and no lattice holds the points. The hat and the turtle are members whose points sit on a lattice, and the equilateral member is not. What that does to the pattern of spots a tiling scatters — whether an aperiodic tiling with its points on a lattice diffracts like a crystal, like a quasicrystal, or like neither — is a question about positions rather than about contacts, and it is the one thing on which the members of this family genuinely differ.