The tile that needs no reflection

The shape in the previous rung covers the plane and never repeats, which was the question sixty years of work had been about. It does so using copies of both hands. Turn a hat over and it is a different tile in the tiling; leave the mirror image out and the covering stops almost immediately.

Whether that matters depends entirely on what the tile is made of. A tile cut from card is turned over with a finger, and a mathematician calls a shape and its mirror image congruent without hesitation, because the plane sits inside space and a flip is a rotation there. A tile that is a molecule is in a different position: handedness cannot be undone by turning a molecule over, and a crystal grown from one enantiomer contains no copies of the other. So the question of whether the mirror image is needed is not a quibble about the definition of congruence. It decides whether the object could be built out of matter that has a hand.

How badly it needs the other hand #

The measurement is short and it is the kind this collection likes: a search that finishes.

Cover a one-ring core with copies of the hat and forbid the reflected orientations. It can be done: twelve tiles, every kite covered exactly once. Ask for a two-ring core under the same restriction and the search runs to exhaustion — every branch closed, nothing found. There is no covering of that region by one handedness, and the statement is not about the search’s patience.

Twelve tiles is not much. For comparison, the same region with both hands available is covered in twenty-one tiles with three reflected among them; four rings takes fifty-six, of which eight are reflected. The single handedness does not fail at some large scale where a subtle obstruction accumulates. It fails immediately.

How many of them there are #

The reflected copies are a minority, and how small a minority is a number with real work to do in the proof.

Two corrections have to be made before the number means anything, and they are both about what a finite search actually returns rather than about the tiling.

The first is the boundary. A tile on the edge of a patch is constrained on one side by the tiles beside it and on the other by nothing, so a patch’s outermost ring is not behaving like part of a tiling. Only tiles well inside are counted.

The second is subtler and it caught the first version of this measurement. The exact cover returns the first covering it finds, and the order it tries candidates in prefers whichever orientation was generated first — which is an unreflected one. So the first covering is not a typical covering. Reshuffling the candidate order and taking several coverings of the same region is what turns one number into a measurement with a spread.

The published value is φ⁴ ≈ 6.854 unreflected tiles for every reflected one. The measurement above straddles it at a spread that a couple of dozen interior tiles cannot narrow, and saying so is more useful than reporting the mean to three figures.

The argument that ratio is really making #

There is a reason to care about the exact value rather than the approximate one, and it is the cleanest argument in the whole subject.

In a periodic tiling, the ratio of the two handednesses is rational. A periodic tiling has a fundamental domain containing a whole number of tiles of each kind; the densities are those two whole numbers divided by the domain’s area, and their ratio is one whole number over the other. There is no room for anything else.

So a tiling whose handedness ratio is irrational cannot be periodic, and the argument needs no search, no hierarchy and no bound. If the ratio is φ⁴ — a number that is not rational, since φ is not — then no tiling of the plane by this shape has a period, full stop.

That is what the ratio is doing in the proof, and it is why an approximate measurement is the wrong instrument for it. Six point eight five four is indistinguishable, at the precision of a hundred tiles, from twenty-seven over four; every measurement here is compatible with a rational ratio and therefore with a periodic tiling with a large repeat. The exact value comes from the substitution structure — the same kind of computation that gives the golden ratio as an inflation factor, where the ratio of tile counts is the ratio of components of an eigenvector of an integer matrix, and irrational for exactly that reason.

The measurement’s job is different and it is worth being clear about: it establishes that the minority is present, is a minority, and is a minority of about the right size. The irrationality is quoted.

Turning it over is an operation, and the group knows it #

Everything above is a statement this collection has the vocabulary for already, and it is worth translating.

A reflection is one of the four motions of the plane, and it is the one that reverses orientation. This site’s own figure standard turns on the same distinction: the motif must be a comma, because a symmetric mark cannot show whether the operation that produced a copy reversed handedness, and a pattern figure that cannot show handedness cannot be checked. Every pattern plate here draws reflected copies in a second colour for that reason, and the patch figures above are drawn to the same rule.

For crystals the same distinction is a hard boundary rather than a drawing convention. Five of the seventeen plane groups contain no orientation-reversing operation, and those are the only ones a single enantiomer may crystallise in; in space the count is sixty-five of the two hundred and thirty. A protein crystal is in one of those sixty-five, and not because of a preference.

So “does the shape need its mirror image” is, for a crystallographer, the question of whether the tiling could occur in a Sohncke group — and for the hat the answer is no, at a region of two rings, decided.

The shape that does without #

Two months after the hat, the same four authors answered the question this rung is named for.

The route runs through the family. Keep the hat’s thirteen turns and let its two side lengths vary: every member of the family closes, because the turns sum to a full circle whatever the lengths are, and the tilings of any two members correspond tile for tile.

The equilateral member — every side the same length — is the interesting one. As a straight-sided polygon it tiles the plane, and it tiles it periodically as well as aperiodically, so it is not itself an aperiodic monotile. What it has is a property none of its neighbours in the family has: its periodic tilings all use reflected copies. Replace its straight edges with curves, so that an edge can only mate with the edge it was cut against, and reflected copies no longer fit at all. What is left is a shape that tiles the plane, tiles it only aperiodically, and never uses its own mirror image — the spectre, announced in May 2023.

None of that last paragraph is computed here, and the reason is worth stating rather than hiding. The machinery in these two essays lives on the kite grid, where every coordinate is an integer and every question is decided by an exact cover. The equilateral member is not a polykite: its vertices do not lie on that grid, or on any grid the same argument would work in. A search that pretended otherwise would be a search in a different problem.

What the minority is doing there #

A minority of one in seven is not a decoration on the result; it is the scaffolding. The reflected tiles are where the hierarchy that proves the whole thing is anchored.

The published argument groups the tiles into four kinds of cluster — the paper calls them H, T, P and F — and shows that any tiling by hats falls into such clusters in exactly one way, that the clusters themselves group into larger clusters of the same four kinds, and that this continues without end. A tiling with a period would have to have that period survive every level of the hierarchy, and a period cannot shrink; so there is no period. It is the same shape of argument as inflation, where a tiling reproduces itself at a larger scale and a repeat would have to be a repeat at every scale at once.

The reflected tiles are the centres of the H clusters, one to each, which is what fixes their frequency: their density is the density of H clusters, and the cluster counts at successive levels are the entries of an integer matrix’s powers. That is where the fourth power of the golden ratio comes from, and it is why the number is exact rather than fitted — the ratio of two components of an eigenvector, precisely as in the tile counts of an inflation and in the Fibonacci chain’s two spacings.

None of that hierarchy is computed in these two essays. What a covering search sees is a patch, and a patch has no levels.

A remark on how little the search knows #

It is worth noticing what the two searches in these essays actually have in common with the proofs, which is almost nothing.

The exact cover knows the shape, the grid, and one rule: every cell exactly once. It has no notion of a cluster, no notion of a scale, and no memory of what it did four tiles ago. It arrives at coverings of eighty tiles because the shape leaves it hardly any choices, not because it has understood anything.

That is the ordinary relation between a search and a theorem in this subject, and it is the relation the undecidability result makes permanent. A search produces objects and bounds; the structure has to come from an argument. The useful thing about running the search anyway is that it produces the objects an argument can be made about, and it refuses the shapes that look promising and are not — seventeen of them, in the enumeration that produced this one.

What a reflected tile is, in the group’s terms #

It is worth being exact about what “reflected” means here, because the word is doing two jobs at once.

A hat and its mirror image are congruent as subsets of the plane: there is an isometry of the plane carrying one to the other, and that isometry reverses orientation. In the language this collection uses everywhere, the two tiles lie in one orbit of the full group of motions and in two orbits of the group of orientation-preserving motions. Three reflections, and never four is the classification that makes those two groups precise: every motion of the plane is a product of at most three mirrors, and the parity of the number is exactly the handedness.

So the statement “the tiling needs both hands” is the statement that the tiling’s tiles do not lie in a single orbit of the orientation-preserving group, and it is a statement about the set of tiles rather than about any symmetry of the tiling — which has none.

That distinction matters when the tile stops being a shape and becomes an object. A tile cut from card lives in three-dimensional space and can be turned over, so the two hands are one object seen twice. A tile that is a molecule cannot: a chiral molecule and its mirror image are different compounds, separable, with different properties in a chiral environment, and no rotation of anything converts one into the other. What a molecule gives up to sit in a crystal is the essay about how much of a molecule’s own symmetry survives being put in a crystal; the question here is the reverse and sharper — whether a pattern can be built at all out of pieces that have only one hand available.

For the hat the answer is no, and the measurement above says so at a region two rings across.

Where the exactness stops #

Measured here: that one handedness covers a one-ring core and cannot cover a two-ring core; the counts of each handedness in coverings of one, two, three, four and five rings; and the majority-to-minority ratio over the interiors of four independently shuffled coverings.

Quoted: that the ratio in the plane is φ⁴; that irrationality of that ratio forbids a periodic tiling; that the equilateral member of the family tiles periodically with reflections and that its curved-edge version does not tile at all with them. All four belong to Smith, Myers, Kaplan and Goodman-Strauss.

One more thing is worth recording as absent. Nothing here has decided whether the hat tiles the plane at all — the previous rung says the same, and the reason has not changed: no procedure decides it, and a covering of five rings is a statement about five rings.

Who found it, and when #

The hat was announced on the twentieth of March 2023; the spectre followed on the twenty-eighth of May. The gap between them is short because the question was obvious the moment the first result appeared — the paper announcing the hat says so, and calls a strictly chiral aperiodic monotile the natural next object.

The word for that object had been waiting since Wang’s question of 1961, and it is a good one: an einstein, from ein Stein, one stone. The hat is an einstein if a shape and its mirror image are one shape, which is the ordinary mathematical convention. The spectre is one whether or not they are.

Where the ladder goes next #

Down, to the shape itself and how it was found: one tile, and no period is the search, the filters and the seventeen shapes that nearly survived them.

Sideways, to handedness where it is a fact about matter rather than about paper. The law that hides handedness is why a diffraction experiment cannot see which hand a crystal is, and fifteen classes may rotate light is the census of which crystals can show it at all.

And back to the question of what forces what. Matching rules and what actually forces aperiodicity sets out the distinction the spectre finally settles: a decoration on a tile is an instruction somebody has to obey, and an edge that is the wrong shape is not.

The question was answered within months #

The gap this essay is about — a shape that tiles aperiodically but needs both hands, in a setting where handedness cannot be changed — did not stay open long. The same authors closed it a few weeks after the first announcement.

The hat belongs to a continuous family. Its edges can be lengthened and shortened smoothly, and along that family sit the hat at one end, another shape called the turtle at the other, and a continuum of aperiodic tiles between them. At one particular point of the family the tile is equilateral, with all its edges the same length.

That equilateral member tiles periodically. On its own it is no use: with all edges equal, reflected and unreflected copies fit together in ways that admit a repeating arrangement, so the aperiodicity that holds everywhere else along the family fails at exactly that point.

Replacing the straight edges with curved ones fixes it. Give each edge an asymmetric shape, matched so that an edge fits only its partner, and the reflected copies no longer fit anywhere. What is left is a single tile that covers the plane, never repeats, and uses copies of one handedness only.

That shape is the spectre, and it settles the question this essay raises. A monotile requiring no reflections exists, so the obstacle to a molecular version is not a theorem about tilings. Whatever difficulty remains is chemical.

What the family says about how it was found #

The route to the answer is worth a paragraph, because it is not the route the first result suggests.

The hat was found by search. Shapes made from kite-shaped cells were enumerated and tested, and the successful one was checked by a computer-assisted argument that its tilings must be hierarchical. That is a discovery about one shape.

The family was the generalisation. Once the hat was understood, its edges could be varied, and the surprise was that aperiodicity survived the whole continuum rather than being a property of one delicate arrangement. A result about a single shape became a result about a one-parameter family, with two named members and infinitely many unnamed ones.

And the exceptional point was where the second result hid. The one member of the family that fails is the one whose failure is repairable by a device the others do not permit — because only an equilateral tile can have its edges replaced by curves that all match. The shape that was useless as a member of the family became the answer to the harder question by being modified in a way none of its neighbours allowed.

Which is a reasonable moral for a search of this kind. The interesting object was not the one the enumeration returned. It was one deformation away, at the single point where the original argument broke down.