Combinatorial coordinates for the aperiodic Spectre tiling

[Simon Tatham, 2023-06-16]

[Part of a series: Penrose and hats | Spectres | finite-state transducers]


In March 2023, four mathematicians discovered a polygon they called a ‘hat’, whose isometric images can tile the plane, but cannot do so periodically.

Hat monotiling: the darkest red hats are reflected

This answered a long-open question in mathematics, of whether forced aperiodicity could be achieved with only one shape of tile. (Roger Penrose had previously managed it with two tile shapes.)

The wording of the problem – isometric images – permits the tiles to be reflections of each other, as well as rotations and translations. The hat tiling made use of this freedom: the hat shape is asymmetric, and the tiling requires it to be used in both handednesses (though with one handedness nearly 7 times as common as the other).

To put it mildly, a lot of people on the Internet weren’t very happy with that. Some people felt that the problem conditions were wrong on principle – that it’s simply more natural to consider the reflected forms of an asymmetric shape to be two different shapes. Other people had more practical objections: “I want to tile my bathroom with these, but bathroom tiles need different surfaces on the front and back, so I definitely do need to order two different types of tile!”

The discoverers of the hat could quite reasonably have answered “shut up all of you, we’ve solved the problem that was actually posed”. But they did better. Two months later, the same authors announced that they’d found a variant form of their tile which solved the revised problem. The modified tile – called the ‘Spectre’ – is still asymmetric, but it can tile with only a single handedness.

Spectre chiral monotiling: nothing is reflected

Shortly after the hat tiling was announced, I had implemented support for it in my puzzle collection, adding it to the collection of grid types on which you can play the puzzle “Loopy”. In the course of this, I came up with an algorithm for generating random patches of that tiling, which also generalises to the aperiodic Penrose tiles, which I felt was so good it deserved a writeup article.

So, now the Spectre tiling has been discovered, the natural question is: does the ‘combinatorial coordinates’ algorithm described in my previous article work for Spectres as well?

The answer is yes, it does. But the structure of the tiling is different enough that the details had to be worked out all over again. In this article I present the method.

The Spectre tile

The shape of the Spectre tile is very similar to the hat. In fact, the only thing you have to do to turn a hat into a Spectre is to make all the edges the same length.

The hat tile has two different lengths of edge, differing by a factor of √3. If you stretch the short edges so that they become the same length as the long ones, nothing goes wrong (the loop of edges still comes back to its starting point without crossing), and the result has all its edges the same length.

(At least, from a certain point of view. This is only true if you count the longest edge of each shape as being divided at its midpoint, so that it’s really two consecutive edges that happen to point in the same direction. This is an unusual way to think about polygons, but fits well with the ways the tiles connect together in both tilings, as well as the grid of kites underlying the hat tiling.)

The Hat and Spectre tile shapes

All of the angles between consecutive edges of the Spectre tile are either 90° or 120° (or 180° at the straight vertex in the middle of the double-length edge):

The Spectre, with right angles marked with a square and 120° angles with an arc

There’s a small quibble with the definitions (as there seems to be with every interesting aperiodic tiling). If you take this polygon at face value, it does permit periodic tilings in a way that the hat tile doesn’t, because making all the edges the same length allows chains of tiles oriented the same way to interlock:

The Spectre polygon tiling periodically. Spectres in adjacent columns are reflections of each other.

But didn’t we just say that the Spectre only permits aperiodic tilings?

Well, that’s down to definitions. The polygon I show here is what the discoverers call a ‘weakly chiral’ aperiodic monotile: the only way to tile the plane with only one handedness of tiles of this shape is non-periodic. But allowing both handednesses introduces additional periodic tilings like the above (as well as transformations of the mixed-handedness hats tiling) …

… unless you also constrain which edges of the polygon are allowed to join to which other edges. If you number the 14 edges of the Spectre consecutively around the polygon, and require that any two tiles that meet must join an odd edge to an even edge, then the tile becomes ‘strongly chiral’: even if you’re allowed to use both handednesses, the only tilings of the whole plane (respecting the connection rules) are both non-periodic and involve only one of the handednesses.

This is exactly the same rule tweak we’ve already accepted for Penrose tiles (which also have the property that they’d admit periodic tilings too the way they’re usually drawn, since they’re just quadrilaterals). And just like Penrose tiles, you can avoid having to impose artificial restrictions if you make the tile more complicated instead. Distorting the shapes of the tile edges a tiny bit is enough to prevent the unwanted extra tilings, and produce a tile which rules out all but single-handedness non-periodic tilings of the plane by its shape alone.

The distortion can be anything you like: the precise shapes don’t matter, as long as they match. The paper suggests curving the edges alternately inward and outward, but if you prefer a straight-edged polygon, you can just add a jigsaw-piece style tab or hole on each edge, and that will work just as well:

Strongly chiral variants of the Spectre: rule that the arrows must match, or change the shape

Here’s the example tiling image from the previous section, redrawn with the curved version of the Spectre shown here:

A tiling of curved Spectres

As I say above, Penrose tiles have the same quibble – they too need to be either distorted or given extra constraints in order to rule out periodic tilings. The usual practice is to ignore this detail when displaying the tilings: just show the tiles as simple quadrilaterals without any distortion, and let the matching rules be implicit. That keeps the diagrams simpler, and also prettier.

For the rest of this article, I’ll follow the same convention, and display Spectres as plain 14-sided polygons.

Hexagonal metatiles

In the previous article I discussed the general idea of ‘substitution systems’ for generating the other types of aperiodic tiling. Generally there will be some system of multiple types of ‘metatile’, and a set of rules for expanding each metatile into multiple metatiles, to produce a larger patch of meta-tiling. You can do that as many times as you like, and then there will be a final step that converts the metatiles into the final output tiles.

For Penrose tilings I found it convenient to take the metatiles to be triangles obtained by cutting each actual Penrose tile in half, so that the final output step consisted of gluing pairs of metatiles back together. For the hats tiling, the paper presented a system of four metatile types, all differently shaped polygons.

The Spectre tiling can also be generated by the same general method. But even though the tile itself is so similar to the hat, the structure of the tiling is totally different (or else you'd have reflected Spectres just like the hats). So the metatile substitution system is completely different too!

The substitution system we’ll use for the Spectre tiling is the most complicated so far, in one way: it consists of nine different types of metatile, where the hats had only four, and the Penrose tiles had four (but in two mirror-image pairs). However, it's the simplest in another way: all nine metatiles are the same shape as each other!

Specifically, the metatiles are all regular hexagons, and they fit together into a hexagonal tiling.

How do you tell nine tile types apart when they're all the same shape? You assign them letters, of course. The paper assigns them the capital Greek letters Γ, Δ, Θ, Λ, Ξ, Π, Σ, Φ, Ψ.

With apologies to the authors of the paper, I’m going to change those names, because Greek letters are awkward on an Anglophone keyboard, and not well supported in all software contexts either. Latin letters that fit within ASCII have the best chance of being usable in identifiers in a wide set of programming languages. So for this presentation I’ll rename the tiles to G, D, J, L, X, P, S, F, Y respectively.

(Why those letters? The LaTeX ‘babel’ package defines a mapping of Latin to Greek letters, for typesetting documents containing Greek from ASCII source. These are the letters that babel maps to the paper’s original names.)

As presented in the paper, the nine hexagon types come with markings on their edges to constrain which ones can connect to which other ones. If you turned those markings into jigsaw-piece tabs and holes of different shapes, then the nine resulting jigsaw pieces would constitute an aperiodic tiling system in their own right.

Hexagons with example jigsaw edges
An example patch of aperiodic tiling with those hexagons

(Of course the jigsaw shapes, if you add any, are completely arbitrary, just like the ones shown on the Spectre tile itself in the previous section. You can make up edge shapes however you like, as long as they match. The shapes above don’t come from the original paper – I invented them myself to be easy to draw.)

The expansion system turns each of these hexagons into a pattern of either 7 or 8 hexagons, all in more or less the same layout. I won’t show the full set of expansions here – they’ll be in the next section with all the edges numbered – but here are the first two types, as examples:

[jigsawhex_G.svg] [arrow.svg] [jigsawexpand_G.svg]
[jigsawhex_D.svg] [arrow.svg] [jigsawexpand_D.svg]
Examples of hexagon-to-hexagon substitution rules

Looking at these diagrams, the first thing you probably notice is that the right-hand side of each one isn’t very hexagon-shaped. How do the expansions of these hexagonal tiles fit together?

(In particular: the right-hand sides of both those diagrams show a G and D hexagon adjacent to each other. So the expansions of G and D must also fit together somehow. But looking at those diagrams by themselves, it’s not at all obvious which part of the border of those two assemblies of hexagons fit to each other! The answer will be shown in a later section.)

The answer is: it depends on the tile. Each of the different types of edge (shown here by a different jigsaw shape) turns into a different path of edges along the outline of one of these expansion diagrams. And they don’t all expand to the same length paths of edges: they range from length 2 to 6. So the way the expansions fit together into a full tiling is very complicated!

Fortunately, we don’t need to worry about all that complexity. The mathematicians who discovered the tiling have proved that it all works, and that if you just follow the expansion rules, everything will fit together. To implement the generation algorithm, we don’t need to go through the proof ourselves: we just have to trust it.

In fact, we don't even need to worry about the different types of hexagon edge fitting together correctly, because the expansion system will automatically arrange that the hexagons only connect in approved ways. So I’ll leave the edge markings out of all the following diagrams. They’ll be complicated enough already!

Just like the metatile systems for other aperiodic tilings, this hexagon-to-hexagon expansion step can be iterated as many times as you like. But eventually you end up wanting to convert your final set of hexagons to the output Spectre shapes, to generate the one-tile aperiodic tiling that was the purpose of the whole exercise.

In this setup, eight of the nine hexagon types expand to just one Spectre. The exception is the G tile, which expands to two. I’ll show the full expansion rules in the next section.

Numbering everything

As we did for the Penrose and hat tilings, we’re going to develop a system for generating Spectre tilings using combinatorial coordinates. That is, for each Spectre in the tiling, we’ll track what hexagon type it was expanded from (and, in the case of the G hex, which of the two Spectres belonging to that hex it is); which child of what hexagon type that was expanded from, and so on for as many layers as we need. And then we’ll show an algorithm for transforming the coordinates of one tile into the coordinates of a specified neighbour, so that by repeating the transformation you can walk around the whole tiling one step at a time.

For this coordinate system, the smallest unit we’re going to deal with will have to be a single output Spectre tile. In the hats tiling we subdivided each hat into 8 cells of the underlying kite tiling, but Spectres don’t fit on any such tiling in the same way; for Penrose tiles we subdivided each output tile into two half-tile triangles because those were our metatiles, but here the metatiles are larger than a single Spectre rather than smaller. So, for the first time, we’ll use the output tiles themselves as the finest granularity in our coordinate system.

How do we specify which neighbour we want to transition to? In both previous systems, we did this by indexing the edges of the smallest-size cells: the three sides of a Penrose half-tile triangle, or the four sides of a kite. Here, we’ll do the same – but our smallest cell is the Spectre, which has fourteen edges (counting the long edge as two normal edges, which is much more convenient).

So, we start by numbering absolutely everything. We must number the edges of a hexagon; the edges of a Spectre; the hexagons appearing in the expansion of each higher-level hexagon; and the two Spectres appearing in the expansion of the G hex. The locations of these numbers are basically arbitrary – the algorithm will work just as well no matter what you choose, as long as you do it consistently. Here are the numberings I’ve used in my implementation:

Numbering all the things

(In the above expansions, the Spectre number 1 – shown with a dotted outline – appears only in the expansion of the G hex. Hexagon number 7, conversely, appears in the expansion of every hex except the G hex.)

Transitions between hexagons

With all those numbers allocated, we can now show the full expansion rules for the hexagon step.

As well as showing the pattern of hexagons (each with an orientation) that each original hexagon transforms to, we must also show how the edges match up. To do that, I’ll mark six circular blobs on the border of each figure, with a larger blob at the ‘starting’ vertex at the top of the original hexagon, and the same for the corresponding point on the output map.

[basehex_G.svg] [arrow.svg] [hexmap_G.svg]
[basehex_D.svg] [arrow.svg] [hexmap_D.svg]
[basehex_J.svg] [arrow.svg] [hexmap_J.svg]
[basehex_L.svg] [arrow.svg] [hexmap_L.svg]
[basehex_X.svg] [arrow.svg] [hexmap_X.svg]
[basehex_P.svg] [arrow.svg] [hexmap_P.svg]
[basehex_S.svg] [arrow.svg] [hexmap_S.svg]
[basehex_F.svg] [arrow.svg] [hexmap_F.svg]
[basehex_Y.svg] [arrow.svg] [hexmap_Y.svg]
Hexagon-to-hexagon substitution rules

In the above diagrams, I’ve marked all the edge numbers of the hexagons on the right-hand side. This enables you to compute internal transitions between hexagons within one of these maps.

For example, suppose your coordinates identify a G hex that is child #2 of a Y hex, and you want to find the coordinates of the hex on the other side of its edge #3:

[basehex_Y.svg] [arrow.svg] [hexmap_example1.svg]
Example transition within a single level

The expansion map for the parent Y hex shows that edge #3 of child hex #2 is adjacent to edge #1 of child #5 (type D). So if you were computing that transition, you’d rewrite the number 2 in the coordinate string to be a 5. We’d also know that we came into that D hex via its edge #1, which will be an important thing to know during recursion.

What if you want to step outside one of these maps? For example, suppose you were in that same G hex but wanted to traverse edge #0?

For this, I’ve also marked two-part edge numbers on the outside of the diagrams. Each edge of the large hexagon on the left-hand side corresponds to a sequence of edges on the right-hand side between two of the circular blobs. Each exterior edge of these maps is marked with a pair of numbers a.b, where b indicates an edge of the larger hexagon, and a is the index of a particular edge within that. So between any two circular blobs, all the indexes have the same second component, indicating that they all correspond to the same edge of the larger hex.

[basehex_P.svg] [arrow.svg] [hexmap_example2.svg] [arrow.svg] [hexmap_example3.svg]
Example transition recursing to the second level

In this example, edge #0 of the “2 (G)” hex has “0.1” written on the far side. This indicates that it’s part of the segment of boundary corresponding to edge #1 of the larger Y hex.

So, first, we recursively compute a transition for the next size of hexagon up, using the same algorithm. In this example, I’ve shown our second-order Y hex as child #1 of a P hex. Referring to the P map in turn, edge #1 of that Y is adjacent to edge #4 of the S hex at child #3. So at the second level of hexagons, we’ve just stepped out of a Y via its edge #1, and entered an S via its edge #4.

In the Y map, we can see that there are three edges of the map marked a.1 for some a: they’re marked 0.1, 1.1 and 2.1. The edge of the map we’re leaving along (corresponding to edge #0 of the G, our original example case) is marked 0.1.

The matching rules should arrange that the boundary segment on which we’re entering the S map is the same shape. And it is: there are three edges marked a.4, in the same way. So an instance of the Y and S map must appear in the overall hexagon tiling with those edges adjacent. This is shown in the diagram above, and you can see that the numbers match up in reverse order, so that 0.1, 1.1 and 2.1 on the Y map fit to 2.4, 1.4 and 0.4 respectively on the S map.

So now we know we’ve stepped off the 0.1 edge of the Y map and come in on the 2.4 edge of the S map – which lands us in its hex marked “7 (X)”, having come in via edge #4 of that hex.

Transitions between Spectres

That’s how to compute transitions between the hexagonal metatiles. But at the lowest level of the coordinate system we must deal with the actual output Spectre tiles. So we also have to consider how each lowest-order hexagon maps to Spectres.

This mapping is mostly regular. Seven of the nine hexagon types turn into a single Spectre, in a consistent orientation (relative to the hexagon’s own orientation), and the only thing that changes is how the edges of the Spectre map to the edges of the hexagon. In the following diagrams I’ve numbered the internal edges of the Spectre, and the external edges to show what segment of the Spectre boundary corresponds to each edge of the hexagon, in the same way as the diagrams in the previous section:

[basehex_D.svg] [arrow.svg] [specmap_D.svg]
[basehex_J.svg] [arrow.svg] [specmap_J.svg]
[basehex_L.svg] [arrow.svg] [specmap_L.svg]
[basehex_X.svg] [arrow.svg] [specmap_X.svg]
[basehex_P.svg] [arrow.svg] [specmap_P.svg]
[basehex_F.svg] [arrow.svg] [specmap_F.svg]
[basehex_Y.svg] [arrow.svg] [specmap_Y.svg]
Hexagon-to-Spectre substitution rules: the simple ones

Similarly to the previous section, this allows us to start by deciding which edge of one Spectre to cross, and find out which edge of another Spectre we’ve come in along.

For example, suppose we had coordinates describing a Spectre expanded from the X hexagon, and wanted to leave by – say – its edge #4. The diagram shows this to be marked 1.1, in a boundary segment containing three Spectre edges, labelled 0.1, 1.1 and 2.1. So we’d call the routine for computing a hexagon transition, which might tell us (for example) that edge #1 of this particular X hexagon was adjacent to edge #4 of a D.

Therefore, we’d expect there to be three edges on the D diagram marked as a.4 – and there are. Again, they fit together in reverse order (imagine actually rotating the diagrams to bring those boundary segments together), with 0.1, 1.1 and 2.1 on the X Spectre matching up to 2.4, 1.4 and 0.4 on the D Spectre. So if we leave the X via 1.1, we enter the D via 1.4, and the D map shows that this corresponds to its edge #11.

Those seven hexagon types are the easy ones. The final two are more complicated, so I’ve left them until last.

I’ve already mentioned that the G hex turns into two Spectres rather than one:

[basehex_G.svg] [arrow.svg] [specmap_G.svg]
Hexagon-to-Spectre substitution rule for the G hexagon

If your coordinates identify a Spectre expanded from a G hex, they must also say which of the two it is (via the numbers 0 and 1 at the centre of each Spectre). And not every transition within a G hex will go to a neighbouring hex: sometimes it will just transfer between the two Spectres within this hex. For example, if you’re in Spectre #0 of a G, and leave by edge #11, the diagram shows that that’s adjacent to edge #6 of Spectre #1 of the same G, and you can return immediately without having to recurse to compute a hexagon transition.

But the S hex does something even stranger:

[basehex_S.svg] [arrow.svg] [specmap_S.svg]
Hexagon-to-Spectre substitution rule for the S hexagon

What’s going on here? What’s that weird spur sticking out on the left of the S Spectre?

That spur should be considered as four additional boundary edges: if you imagine walking around the edge of the diagram starting on edge #0 of the Spectre itself, you walk along edges #0, #1, #2 and #3, then walk out along the spur for two steps, turn round and walk two steps back, then turn left and carry on to edge #4 of the Spectre. The two edges on the bottom side of the spur correspond to edge #1 of the S hexagon; the two edges on the top side of the spur correspond to part of edge #0.

When you’re computing an outgoing transition from an S Spectre, you can ignore this completely. If you want to leave an S spectre by edge #3, that corresponds to exterior edge 0.2, so you compute the transition out of edge #2 of the S hex in the usual way. If you want to leave by edge #4, that’s marked 3.0, so you compute a transition out of edge #0. In both cases, the spur doesn’t get involved.

But the spur gets involved when you compute an inward transition. Suppose some hexagon transition told you you were coming in along edge #1 of an S hexagon – say, the edge marked 0.1 (which must fit to 1.something in the Spectre you just left). You consult the map and discover that 0.1 is on one side of the spur, and the opposite side of the same edge is marked 5.0. Now what?

Now you simply call the hexagon transition algorithm again, to ask what’s on the other side of edge #0 of this S hexagon. In other words, if you step into an S Spectre in such a way that you land on the spur, the rules require you to step straight back out again.

(You never need to iterate this procedure more than twice.)

Generating the whole tiling

I’ve presented a system for tracking the combinatorial coordinates of a particular Spectre, within the infinite hierarchy of metatiles.

Just as in the previous article, the next step is to use this to iterate over all the Spectres in a given area of the plane, so as to construct an actual patch of tiling. There are two reasonable methods for this.

One approach is a graph-based search – breadth-first or depth-first, whichever you prefer. Choose a location for your starting Spectre, and its coordinates in the plane; then explore every edge of that Spectre, by doing the coordinate transformation to discover the Spectre on the far side of it. Each of those transformations will tell you which edge of the new Spectre adjoins that edge of the old one; that’s enough information to compute all the rest of the vertex coordinates of the new Spectre. So you check if the new Spectre is within your target area; if so, add all the remaining edges of that Spectre in turn to your list of transitions that need to be explored, and keep going until you’ve covered your whole target area.

The other approach is to traverse your area in a raster fashion, avoiding having to keep a queue of edges yet to explore (for breadth-first search), or recurse to high depth (for depth-first). You can generate all the Spectres along a particular horizontal or vertical line, by calculating which edge of the Spectre the line leaves through, and using that to calculate your next transition. One loop of this kind can be made to generate all the Spectres intersecting (say) the left-hand edge of your target rectangle, and from that, you start a series of secondary loops along horizontal lines, spaced close enough together that every Spectre must be intersected by at least one line. This allows you to store a bounded number of coordinate strings, and still (if you’re careful) generate every output tile exactly once, by automatically detecting whether any previous iteration would have encountered that tile.

The second approach is good in terms of asymptotic complexity: you can generate an arbitrarily large patch of tiling in essentially linear time and essentially constant space (discounting the size of the coordinate strings). But it involves more fiddly computational geometry than the graph search – and the graph search also has the virtue of being self-testing, in that if two paths through the tiling generate inconsistent results, you have a chance of noticing by failing an assertion, so that you can fix the bug in one of your lookup tables (or whatever).

Just as in the previous article, you don’t have to generate a lot of coordinates up front for your starting tile. Instead, you can generate just enough to get started, and every time a recursive transition computation needs a coordinate you haven’t generated yet, make it up as you go along, according to the limiting frequency distribution of the nine hex types.

Also as in the previous article, you can make a run of the algorithm reproducible by storing the full set of coordinates you ended up generating for the starting tile, once you’ve covered the whole area. Then another run with those parameters should generate the identical patch.


The advantages of the combinatorial-coordinate system are even more clear in this case than in the hats case, because the distortion imposed by the metatiling system is even more profound than it was for the hats. The four hat metatiles only distort slightly as you expand them recursively; if you wanted to use the simpler algorithm of recursively generating a large fixed patch of tiling and picking out a randomly chosen area of it, it might still have been possible to track the distorted images of the target area (with some uncertainty) and prune branches of the recursive expansion that would never land in it. The hexagons are much stranger, so an algorithm that avoids thinking about their geometry at all is extremely helpful!

In particular, there’s a detail of the metatile system I didn’t mention in the previous sections. The expansion rules for the nine hexagons reverse their handedness! If you go back and look at the expansion diagrams, you’ll see that each individual hex has its edges numbered 0, 1, …, 5 anticlockwise round the edge, but in the expansion, the edge markings go 0.0, 1.0, …, 0.1, …, n.5 clockwise. I didn’t mention this while describing the algorithm, because the algorithm doesn’t need to know or care! As long as it knows which hexes are connected to which other hexes by which edge (identified by a purely arbitrary numbering system), that’s all it needs.

The two combinatorial coordinate algorithms described in my previous article, for the Penrose and hat tiling, have a fundamental difference, because of the overlap between metatile expansions in the hat tiling. The Penrose algorithm was always stepping off the edge of one of its triangle maps and having to match that edge up to the edge of some other map to see where it had entered that one; this wasn’t too hard, because the map edges are all simple straight lines and are divided into at most two segments. The hat algorithm involves much larger maps with very complicated crinkly edges, but luckily, it never needed to step off the edge of one of those maps and figure out how that corresponds to some other equally crinkly edge – because the maps overlap, so instead we can recompute an equivalent coordinate for the same point which is already in the new map.

But the Spectre metatile expansions don’t overlap, and they do have complicated crinkly edges. So this algorithm did have to care about how the edges of one map match up to the edges of another.

Luckily, that still wasn’t very difficult, because the authors of the paper had worked it out already, and included all the necessary data in their proofs! The segments of the map edges are clearly marked in the paper’s own diagrams. So there was no trouble there.

In summary, the combinatorial-coordinates technique works just as well for the Spectre tiling as it does for the Penrose and hat tilings. I prepared an implementation of all this for my puzzle game Loopy at the same time as writing this article, and it’s working very well.

Appendix: exact plane coordinates

All the edge lengths in the Spectre tiling are the same (counting the long edge of the tile as two normal edges end-to-end), and all its angles are multiples of 30°.

This allows us to track the coordinates of all the necessary points in the plane using a system based on complex numbers, similarly to the methods I presented in the previous article.

In this case, we’ll let d be the complex number cos(π/6) + i sin(π/6), so that multiplying by d has the effect of a 30° rotation (or one twelfth of a full turn). This number is a root of the polynomial z4 − z2 + 1 (the 12th cyclotomic polynomial). So if you have two numbers expressed as polynomials in d, and multiply them to get a larger polynomial, then you can reduce the polynomial to degree at most 3, by substituting d4 for d2 − 1 everywhere it appears (and similarly d5 becomes d3 − d, etc, for higher powers). And if your original polynomials had all their coefficients integers, then the same is true of the reduced product.

In this way, you can generate an exact representation of every point that you can reach by starting from the origin and travelling a series of unit-distance steps in directions that are multiples of 30°. In particular, every vertex of a Spectre tiling can be expressed this way. So you can recognise the same vertex when you come back to it by another route, by simply comparing two tuples of 4 integers.

To convert back to separate x and y coordinates, we can use the formulae

d = cos(π/6) + i sin(π/6) = √3/2 + (1/2) i
d2 = cos(2π/6) + i sin(2π/6) = 1/2 + (√3/2) i
d3 = cos(3π/6) + i sin(3π/6) = i

so that if you have a general value a0 + a1 d + a2 d2 + a3 d3 then your output coordinates are simply

x = a0 + (√3/2) a1 + (1/2) a2
y = a3 + (√3/2) a2 + (1/2) a1

in which, if the (ai) were all integers, these coordinates are all integer multiples of 1/2.

And, just as in the previous article, this coordinate representation allows you to test a value to see if it’s within a given range, without needing floating point and its rounding errors, because the problem is equivalent to testing the sign of a general number of the form u + v √3. If u and v have the same sign, or if either or both is zero, then the answer is obvious; if u and v have opposite signs, then the answer is determined by which one has the larger magnitude, which you can decide by finding out which of u2 and 3v2 is bigger, in integer arithmetic.

Appendix: four-colourings of the Hats and Spectre tilings

In the previous article I finished off with an appendix containing an irrelevant but pretty aside. That seemed like fun, so let’s do it again!

Any tiling of the plane can be coloured with four colours, so that no two tiles of the same colour share an edge (a ‘proper’ four-colouring). This is a consequence of the famous Four-Colour Theorem (which guarantees this for finite maps in the plane), and the De Bruijn–Erdős theorem which extends it to the infinite case.

But those theorems between them don’t guarantee that a proper four-colouring is easy to construct – only that, in an abstract mathematical sense, one exists. So if you want an actual four-coloured instance of even a large finite patch of one of these tilings, it might be computationally hard to find one.

It might be – but it isn’t. In fact, there’s a simple technique for generating a natural four-colouring of each of the Hats and Spectres tilings. This particular four-colouring is easy to specify mathematically, and if you’re already generating the tiling itself based on the combinatorial coordinate system I describe in this article and the previous one, you can do it in practice as you go along, with a minimal amount of extra information to keep track of, and no increase in the computational complexity. The techniques for the two tilings are even quite similar to each other.

The basic idea is: for each tiling, you can identify a class of ‘special’ tiles, spaced widely enough apart that no two are adjacent, and with the property that if you imagine gluing each special tile to a particular one of its neighbours, then the remaining tiling of the plane is topologically equivalent to a tiling with regular hexagons. (Of course, the tiles will be differently shaped from hexagons, but each one will be adjacent to exactly six neighbours in the same pattern that hexagonal tiles connect.)

A tiling of the plane with regular hexagons can be coloured with only three colours. The colouring is essentially unique (up to permuting the three colours), and easy to compute. So we now colour the Hat or Spectre tiling with four colours, by colouring every tile according to the hexagon that it’s part of – except that our ‘special’ tiles are coloured in the fourth colour.

To put that another way: if you start by colouring the special tiles blue, and then just try to colour the rest of the tiles with the remaining three colours, you’ll find that it always turns out to work, and there’s only one choice of how to do it.

So, what are the ‘special’ tiles in question?

For the hats tiling: the hats appear in both handednesses, with one handedness being common and the other one being rare. The special tiles are the rare reflected hats: the original paper about the hats tiling observes that fusing each reflected hat with one of its neighbours leads to a tiling isomorphic to a hexagonal one.

(Using the index system from my previous article, a reflected hat is precisely a hat which has index 3 in the expansion of a red hexagonal metatile, and the hat you fuse it with is the one with index 2. It’s easy to check from the metatile expansion diagrams that no two of these fused hat pairs can ever be adjacent in the overall tiling.)

For the Spectres tiling, of course, there aren’t any reflected Spectres: the whole point is that they all have the same handedness! But there is still a class of unusual Spectres, based on orientation. All nine of the hexagon types expand to a Spectre which has the same orientation relative to the hexagon, but the G hex also generates a second Spectre rotated by 30°. And all the hexagons’ orientations, of course, differ by multiples of 60° from each other. So it follows that most of the Spectres in the tiling have an orientation that’s an even multiple of 30° (taking the reference point to be any Spectre not expanded from a G hex), but a small number – precisely the Spectres with index 1 in a G hex – have an orientation that’s an odd multiple of 30°. Those are the special ones.

So, if we’re generating one of these tilings via combinatorial coordinates, how do we also generate a consistent four-colouring according to this principle as we go?

For the hats tiling, the simplest approach I found is to pre-compute a working four-colouring for each of the four ‘kitemaps’ (showing the expansion of one second-order metatile to first-order metatiles, their contained hats, and the kites making up each hat). In this four-colouring, the reflected hats are coloured with our special fourth colour (say, blue), and the rest are coloured in any consistent way with the other three colours.

When an instance of this kitemap appears in the overall tiling, the colours won’t always match. But the blue tiles will match (because they’re the easily distinguished reflected hats), and the other three colours will be permuted in some way between the real colouring and the kitemap’s prototype colouring.

So, as we step around the coordinate system, we need to keep track of a permutation that tells us how our three real (non-special) colours relate to the prototype colouring of the kitemap we’re currently in. Given that, we can work out what colour to make each hat we come to: look up its colour in the prototype kitemap colouring, and apply the permutation.

In order to keep track of this permutation, we need to know how to update it when the coordinate transformation steps us from one kitemap to the next. It turns out that whenever this happens, you’ll be able to find a pair of adjacent non-blue hats in the overlap between the two kitemaps. Since they’re adjacent, they’ll have different colours, which means now you know how two of your real colours map to the kitemap you’ve just stepped into – and by elimination, you know the third. So this is no trouble to compute.

For the Spectres tiling, it’s even easier. Our first-order Spectre metatiles are precisely the hexagonal tiling that you get if you fuse each odd-orientation Spectre with its neighbour (specifically, the one making up the other half of its G hex). So as we step around the tiling we can directly track the three-colouring of the first-order hexagons, and use that to determine the colour of every Spectre: if it’s a G1 then it’s blue, otherwise it’s coloured the same as its hexagon.

The reason why a three-colouring of the hexagonal tiling is essentially unique is because if a given hex has one colour, say A, then the other two colours B and C must alternate around it. So all we need to keep track of is what colour our current hexagon is, and which of the other two colours goes with the odd-numbered edges and which the even.

And then, in each coordinate transition between Spectres, we need to find out whether it’s moving to a different hexagon (and not just between the two Spectres in a G hex), and if so, which edge of the old hexagon it left by, and which edge of the new hex it came in from. That’s enough to decide on the new hex’s colour (by knowing which edge we left from), and figure out how the colours are arranged around it (by knowing which edge of the new hex leads back to the old one, whose colour we remember). And those pieces of data are computed already by the transition algorithm – so no trouble!

And here are some examples of both:

Example patches of both tilings with systematic four-colouring

I don’t currently know of any comparably simple rule for generating a four-colouring of a Penrose tiling as you move around it via combinatorial coordinates. If anyone else does, I’d be interested to hear of it!