## Hyperbolic polyforms

Higher-dimensional geometry (previously "Polyshapes").

### Hyperbolic polyforms

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}.
Attachments
3-7-hexaform.png
Marek14
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### Re: Hyperbolic polyforms

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.
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wendy
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### Re: Hyperbolic polyforms

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

ubersketch
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### Re: Hyperbolic polyforms

ubersketch wrote: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.
Marek14
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### Re: Hyperbolic polyforms

Marek14 wrote:
ubersketch wrote: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)

ubersketch
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