Eleven duals, one tile each

Duality is one instruction: put a point in every tile, join two points when their tiles share an edge, and the result is another tiling. Do it to the eleven uniform tilings and something clean happens. Each of them has one kind of vertex and several kinds of tile; each dual has one kind of tile and several kinds of vertex.

So the same eleven objects classify two things. Read one way they are the tilings by regular polygons in which every vertex is alike. Read the other they are the Laves tilings — the tilings in which every tile is alike, and all of them alike in the strong sense that a symmetry of the tiling carries any one onto any other.

How the dual is actually built #

The instruction is three words long and each of them needs a decision.

“A point in every tile.” The centroid is used — the mean of the tile’s vertices — because it is canonical, because it is exact in the same field the vertices live in, and because for a regular polygon it is the centre by every other definition as well. Any interior point would give a tiling with the same adjacencies, so the combinatorics of the dual do not depend on the choice; the shapes do, and the congruence claim below is about the centroid dual specifically.

“Join two points when their tiles share an edge.” Adjacency is looked up on exact vertex keys rather than on distance: two tiles are neighbours when two of their vertices coincide exactly and consecutively in both. There is no nearest-neighbour search and no cut-off radius, so nothing here can be tuned.

“A tile for every vertex.” The dual face around a vertex is the ring of centroids of the polygons meeting there, in cyclic order — which is available because the grower already knows that order: it is the sequence it completed the vertex with. The ring is closed only for vertices whose polygons fill a turn, so faces are taken only inside the radius within which the patch is finished, and the ragged edge of the growth contributes nothing.

The congruence is checked, not admired #

That the dual tiles are all the same shape is the claim the Laves tilings are named for, and it is exactly the kind of claim a picture cannot support. Two kites drawn at slightly different proportions look identical at figure size.

So each dual face is reduced to a number: its cyclic sequence of squared edge lengths, taken in order round the face and canonicalised up to rotation and reflection. Those lengths are exact — they live in ℚ(√2) or ℚ(√3), like the vertex coordinates they come from — so two faces have the same key exactly when their edges match, with no tolerance anywhere. Every face of every dual returns the same key as every other face of that dual. Eleven duals, one shape each, compared as integers.

Two details of that key are doing more work than they look like doing. Squared lengths rather than lengths, because a squared length in ℚ(√2) is an exact pair of rationals and its square root is not — taking the root would put a floating-point number into a comparison that has to be exact, which is the one place a tolerance would creep back in. And up to rotation and reflection, because a tile that has been turned or flipped is still the same tile: the same five edges read from a different starting corner, or read the other way round. Canonicalising means generating all the readings of the cyclic sequence and keeping the smallest, so two faces get the same key exactly when one is congruent to the other, and never merely when they are similar or nearly alike.

The orientation count is not the shape count, and the difference is the whole point of drawing the tiles on top of one another. Both of these tilings have a point group of order twelve, and in both cases every tile is congruent to every other; what differs is how much of that twelve each tile keeps for itself. The rhombille’s rhombus has two mirrors and a half turn — four operations fixing it — so the twelve orientations collapse to three, which is the three directions its long diagonal can lie in. The kite of the deltoidal trihexagonal tiling, the dual of 3.4.6.4, has one mirror and nothing else, so it appears in six orientations. In both cases the number of distinct shape keys is one. A reader settling the question by looking would be counting orientations, which is a fact about the group and the tile’s own symmetry rather than about the shape, and would report three kinds of tile here and six there.

What the dual tile is #

A dual tile surrounds one vertex of the original, so it has one side for every polygon at that vertex. That makes the shape read straight off the species: 3.12.12 duals to a triangle, 3.4.6.4 to a quadrilateral, 3.3.4.3.4 to a pentagon, and the triangular tiling — six triangles at a vertex — to a hexagon, which is the honeycomb.

Reading down that table, four of the eleven duals are tilings by a triangle, three by a quadrilateral, three by a pentagon and one by a hexagon. Only three of the eleven tiles are regular polygons — the equilateral triangle of the honeycomb’s dual, the square, and the hexagon of the triangular tiling’s dual — which is the same three that were the regular tilings in the first place, since a regular tiling duals to a regular tiling.

Each of the eleven duals has a name, and the names are mostly descriptions of the tile. The dual of the triangular tiling is the honeycomb, and of the honeycomb the triangular tiling. The dual of the trihexagonal tiling is the rhombille — the rhombus tiling that reads as a wall of cubes in projection and is the commonest floor in this subject’s ornament. The dual of 3.4.6.4 is the deltoidal trihexagonal tiling, a kite. The dual of 4.8.8 is the tetrakis square tiling, a right triangle got by quartering each square along its diagonals; the dual of 3.12.12 is the triakis triangular tiling, obtained by trisecting triangles; the dual of 4.6.12 is the kisrhombille, whose tile is a scalene triangle with three different edges and three different angles — the least symmetric tile in the list, in the tiling with the largest cell.

A tile’s own symmetry is what its edge-length and angle counts are recording. The kisrhombille’s three-of-three is a triangle with no symmetry at all; the rhombille’s one-of-two is a rhombus, which has two mirrors; the honeycomb’s one-of-one is a regular hexagon. And the tile with no symmetry belongs to the tiling with the largest group — twelve operations, p6m — which is not a contradiction but the same fact stated twice: a tile with no symmetry of its own needs twelve copies to fill a cell, and a tile with all of it needs one.

Three of them are pentagons #

This is the fact worth stopping on, because it sits directly against the rest of this collection.

Three uniform tilings have five polygons at each vertex, so three Laves tilings are tilings by a pentagon: the dual of the snub square tiling, which is the Cairo pentagonal tiling; the dual of the snub hexagonal tiling; and the dual of the elongated triangular tiling.

And this collection spends several essays proving that a periodic pattern cannot have a five-fold rotation.

Both are true, and confusing them is one of the commonest errors in this subject. The restriction forbids a symmetry: no operation of any of these tilings turns the plane by 72°. It says nothing whatever about the shape of a tile. A pentagon is a shape; five-fold symmetry is a motion; a plane can be filled with pentagons by a group that contains no five-fold rotation at all, and the Cairo tiling’s group is p4g — four-fold, with mirrors and glides, and not a trace of five anywhere.

The table above makes the difference visible in one column. None of the three pentagons has a single edge length and a single angle. They cannot: a regular pentagon’s interior angle is 108°, which does not divide 360°, so regular pentagons cannot meet edge to edge at a vertex and fill a turn — that is the angle equation refusing the pentagon at the very first step. What tiles is an irregular pentagon, and the ways it can be irregular were an open question until 2017, when the classification of convex pentagonal tilings was finally completed at fifteen types. Three of those fifteen are here.

What the notation says #

A Laves tiling is written with its tile’s vertex degrees in square brackets: the Cairo tiling is [3.3.4.3.4], the same string as the uniform tiling it duals, because the numbers that were polygons at a vertex become tiles at a corner under the swap. The bracket is doing real work — it says which corner of the pentagon touches how many other tiles, going round — and it is the reason the Laves list and the Archimedean list are usually printed with the same eleven strings.

That coincidence of notation is exactly the duality restated, and it is worth being careful about: [3.3.4.3.4] is a tiling by one pentagon whose corners are shared by three, three, four, three and four tiles; 3.3.4.3.4 is a tiling by triangles and squares whose every vertex has that sequence round it. The same eleven strings, two different lists, and one bijection between them.

The group comes through unchanged #

Duality is built out of the tiling and nothing else — a tile centre is a tile centre whatever motion is applied — so a symmetry of a tiling is a symmetry of its dual, and vice versa. That is an argument, and this collection prefers a measurement.

The measurement: the dual’s vertices are the original’s tile centres, so they are a plane point set like any other, and they were handed to the same detector. All eleven duals return the group of the tiling they came from.

Getting that answer needed one repair that says something about cells. The dual is expressed in the original’s basis, and that basis need not be the dual’s — the dual of 3.12.12 has its vertices on a lattice three times finer than the tiling it came from, and the dual of 4.8.8 acquires a centring translation. Detected in the coarse cell, the first came back with thirty-six operations and matched none of the seventeen, because a group in a supercell is a perfectly good group and is not one of the seventeen. The fix is to read the pure translations out of the detection, refine the cell until none is left inside it, and ask again. That the refined answer has no translation left in it is itself the check.

Thirty-six is worth reading rather than discarding as an error, because it is not one: it is twelve operations times a translation subgroup of order three, and the three is the ratio of the two cells. A detector handed a pattern in a cell three times too large sees every operation three times over, once for each coset of the true lattice in the drawn one, and reports the product. So the number that came back was a correct count of a group that is not a wallpaper group — wallpaper groups are quotients by the full translation lattice, and this one had been quotiented by a sublattice of it. The symptom of a cell that is too large is always this: pure translations inside the cell, and an order that is a multiple of the right one.

That is also why the refinement terminates rather than shrinking forever. Each round divides the cell by the index of the translations found in it, the index is at least two whenever any are found, and a tiling’s vertices are a discrete set — so the cell cannot fall below the shortest distance between two of them. The condition to stop on is not a size but a property: no pure translation strictly inside the cell.

The one that is its own dual #

The square tiling is self-dual and is the only member of the eleven that is. The check is direct: its dual’s vertices lie on the same lattice, one per cell, and differ from the original’s by a translation.

The other two regular tilings are each other’s dual: triangles to hexagons, hexagons to triangles. That is the plane’s version of the tetrahedron being self-dual while the cube and octahedron are a pair — the same duality, one dimension up, with the Platonic solids in place of the tilings.

Where this meets the rest of the collection #

A Laves tiling is a tiling by one tile, all copies equivalent under the group. That is precisely what a Wigner–Seitz cell is when the lattice it comes from has no basis: the region nearer to one lattice point than to any other, repeated by the lattice.

The two constructions are not the same. Dualising joins tile centres by adjacency; the Wigner–Seitz construction cuts by perpendicular bisectors and is a metric operation. They agree when the tiling’s vertices are the points of a lattice and the tiles are what a Voronoi construction would produce — which happens for the honeycomb’s dual and not in general, since a dual tile’s shape depends only on the combinatorics of the vertex it surrounds while a Voronoi cell’s depends on distances.

The overlap is the reason this collection meets the same small shapes from three directions: as parallelohedra in space, as Wigner–Seitz cells in the plane, and here as duals. The hexagon in particular is all three at once.

Who found them, and when #

Fritz Laves gave the tile-transitive list in 1931, in a paper about the geometry of crystal structures rather than about tilings — the tiles are candidate atomic environments, and a tiling with one orbit of tiles is a structure with one kind of site. That is the same motivation that produces Wyckoff positions and the same one behind the parallelohedra: a crystallographer’s interest in a shape is usually an interest in what can sit at every copy of it.

The list itself is three hundred years older than its motivation, since Kepler had the Archimedean tilings in 1619 and duality is not a hard idea. What Laves added was the other reading — and Grünbaum and Shephard’s 1987 account is what made the correspondence a theorem rather than a table printed twice.

What the round trip checked, and how #

Four things are asserted every time these figures are drawn, and each of them can fail — a figure whose assertion refuses does not appear at all.

Every dual is monohedral. All faces of each dual must present the same canonical cyclic sequence of squared edge lengths. Eleven of eleven, and a single face out of family would fail the build.

Every dual carries its tiling’s group. Detected from the tile centres, independently of the tiling’s own detection, and required to match.

A refined cell has no translation left in it. The cell-refinement above could over- or under-shoot; the check is that the finer cell’s own detection contains no pure translation, which is what “primitive” means.

And the eleven are eleven. The dual census is built from the survivors of the parity argument, so if that count moved, this one would move with it and the assertions that name eleven would fail.

Where the exactness stops #

Duality here is combinatorial and drawn with one particular choice of tile centre. The centroid is used, because it is exact and canonical; another choice of interior point gives a tiling with the same combinatorics and different edge lengths, and the congruence claim would then be false. The claim proved is about the dual as drawn from centroids.

The number of sides is a fact about the species; the shape is not. That the dual of 3.3.4.3.4 is a pentagon follows from five polygons meeting; that the pentagon has four equal sides and one different is a measurement.

The dual of the dual is the original only up to a similarity. Dualising the centroid dual does not return the tiling that was started from — it returns something combinatorially identical at a different size and, in general, with slightly different tile shapes, because the centroid of a dual tile is not the vertex it surrounds. The combinatorial statement is an involution; the geometric one is not, and nothing here claims it is.

Eleven is a count of duals of uniform tilings. The Laves tilings are usually defined the other way — tilings with one orbit of tiles — and the two definitions agree because duality is a bijection between the two conditions. That agreement is not proved here; the eleven duals are constructed, and the claim that they are all the tile-transitive tilings is not one the machinery makes.

The pentagon question, kept separate #

One last separation, since three pentagons have just been produced on a site that spends its time forbidding five.

A tiling by pentagons is not evidence of anything about the crystallographic restriction, and neither is a tiling by any other shape. The restriction is about which rotations an operation group with a lattice may contain, and the answer — orders one, two, three, four and six — is an arithmetic fact about integer matrices of finite order. A tile’s number of sides never enters it.

What the restriction does forbid is a tiling by regular pentagons, and it forbids it twice over: once because their angle does not divide a turn, so they cannot even meet at a vertex, and once because a tiling of congruent regular pentagons would carry a five-fold rotation into a periodic pattern. The two arguments are independent and both are short. Between them they close the case at the level the confusion actually lives at, which is the level of the picture rather than the level of the group.

Penrose tilings are where the distinction gets its full workout, and the reason they were surprising is precisely that they have five-fold symmetry in the diffraction pattern without having a lattice — so they evade the restriction rather than contradict it. A Cairo tiling does neither: it has a lattice and no five-fold symmetry, and it is made of pentagons.

Where the ladder goes next #

The tilings anchor has enumerated what regular polygons can do when they are laid edge to edge, and every count in it has been finite. The next question in the same shape is what happens when the polygons are dropped and only the group is kept — which is where this collection’s fundamental domains already are, and where a tiling stops being a picture and becomes a way of drawing a group.