Hyperbolic polyforms

[Marek14]

Recently, I have accidentally found a technique how to search for isohedral polyform tilings (i.e. tilings that look identical from the point of view of each tile). The technique is robust enough to work not only in Euclidean space but also in hyperbolic tilings.

But I found something weird when exploring the polyforms in {3,7}. Shapes made out of two, four or six triangles in that tiling don't seem to be capable of tiling the whole hyperbolic plane. Shapes made from ODD number of triangles, on the other hand, work. I can't figure out why that would be.

To make it even weirder, I have found ONE exception to this: if you join two triangles together, and then add one more triangle to each exposed face, the resulting shape CAN tile the plane... but it's impossible to orient them consistently.

By this I mean, that for these tilings in general, even if the tile itself is symmetrical, you can pick a particular orientation for each one and the tiling will stay isohedral even after these choices are made -- the tiles can be treated as asymmetrical, even if they are actually not. Not here.

The tiling in question is attached -- as you can see, it's isomorphic to {4,7}.

[wendy]

In tilings of the form {3,3p}, it is possible to construct polycells of any given size, because {3,3p} divides into a tiling of triangle-stripes, where there is an unbroken Petrie zigzag in every stripe. Thus a tiling containing exactly 3 or 5 or 7 etc triangles is possible. Such cells are convex.

A similar result would befall something like {4,6}, where strips of opposite squares would form a convex region, with three of these cells at a corner.

It should be noted that the tiling of pentimos is not exactly a match for what we might call a regular tiling.

[ubersketch]

That’s actually kinda weird. Perhaps there is a generalization to all regular hyperbolic tilings? Also I see you’re having fun with HyperRogue.

[Marek14]

That’s actually kinda weird. Perhaps there is a generalization to all regular hyperbolic tilings? Also I see you’re having fun with HyperRogue.

Well, yeah. It has some neat research tools.

[ubersketch]

That’s actually kinda weird. Perhaps there is a generalization to all regular hyperbolic tilings? Also I see you’re having fun with HyperRogue.

Well, yeah. It has some neat research tools.

Haven't got around to playing with it yet. Of course I have played the main game which is one of my favorite roguelikes. (even if it wasn't roguelike it'd still be really unique)

[Marek14]

I'm experimenting with a notation for "abstract polyforms" -- a way to specify a polyform made out of regular polygons in arbitrary {p,q} tiling. It looks like this:

A polyform is a sequence of numbers. A single triangle is 1,1,1. A single square is 1,1,1,1, a single polygon in general is n 1's.

Adding another n-gon is equivalent to "increase two adjacent numbers in the sequence by 1, then add (n-2) 1's between them".

A polyform is a cyclic sequence of numbers, so by moving the numbers around and/or reflecting the sequence, we can reach the lexicographically minimal sequence; this makes it easy to check whether two polyforms are identical.

Finally, there's the operation of "closing" a vertex. This occurs if number q appears in a sequence that is supposed to be in a {p,q} tiling. In this situation, the number is erased and both adjacent numbers are replaced by a single number that is their sum (which can trigger another closing operation). Polyforms with closed vertices are no longer abstract; they can now exist only in a single tiling.

One thing that this notation can't deal with are polyforms with holes, and I'd like to extend the notation there, but since I want to use it primarily to tile the Euclidean or hyperbolic plane with polyforms and shapes with holes cannot really work as tiles, it doesn't seem too bad.

[Marek14]

So, this is interesting.

In {3,7}, there are many possible uniform coverings that use polyforms made out of 3, 5, or 7 triangles. There's only one with 6, and now, as it turns out, there is only one with 8-triangle polyform. Just look at it -- there seems to be no logical reason why this shape works and none of the others does.

[Marek14]

I made some more complex checks and I now have results for Catalan tilings of all possible triangular polyforms up to 8 triangles and squares up to 5 in tilings {3,q} and {4,q} for q<=20.

[Marek14]

Alright, so the research got a big boost thanks to MagmaMcFry's solver that uses the Dancing Links algorithm for fast and efficient search of solution space.

Which leads to fun things like these:

4_5_hexomino_18.png

(832.93 KiB) Not downloaded yet

4_5_heptomino_u6.png

(807.84 KiB) Not downloaded yet

It can even search all polyforms of a given kind!

[Marek14]

New version of HyperRogue comes with some new options that allow proper displaying of almost any tiling. This, for example, is a tetriamond made out of {3,11} triangles.

3-11-4-2-1.png

(378.24 KiB) Not downloaded yet

[Marek14]

Another example of tilings that can be rendered now. This is a domino tiling of {4,5}. Since there is no isohedral solution here, you have to use at least two nonequivalent kinds of dominos.

4-5-2x2-sf-4.png

(960.55 KiB) Not downloaded yet