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}.