Recently, MathPuzzle highlighted a program called MagicTile for playing with Rubik’s Cube variants on various tessellations in spherical, Euclidean, and hyperbolic geometries. One interesting hyperbolic tessellation is the {7, 3} tessellation, composed of heptagons, with three meeting at every corner. Twenty-four of these heptagons can be wrapped up into a genus-3 (that is, topologically like a torus but with three loops instead of one) “Platonic solid” called Klein’s Quartic, which John Baez has a fascinating page about.

Polyforms based on hyperbolic tessellations appear to be a relatively unexplored area. I’ve come across a couple of references on counting the n-cell polyforms for a given tessellation, but I haven’t found evidence that anyone has actually used them for puzzles. So I set myself this one. There are ten tetrahepts on the {7, 3} tessellation. Could they tile two copies of the same shape?

An implicit constraint in my solution of this problem was that the shapes could extend no farther than the second ring around a central heptagon. I searched for a solution using plastic game counters that would not have fit on any heptagons farther out on the diagrams I used as solving boards. I also knew that I was going to make images to show on my blog, and they would be clearer if there were no relatively tiny heptagons in the solution. In fact, the decision to find a puzzle in two pieces was affected by the observation that using an extra Poincaré disk would remove the need to use the tiny outer heptagons.

This was a pretty difficult puzzle to solve by hand, so I feel like I’m being a little mean using an even harder, (and perhaps impossible) variant as the numbered puzzle for this post:

#8: Find a symmetrical shape for which two copies can be tiled by the ten {7, 3}-tetrahepts. In addition to mirror symmetry, 180° rotational symmetry around the midpoint of an edge could work. (The other modes of rotational symmetry of the tessellation won’t work for a 20 cell shape.)

I would also love to see what other puzzles people can come up with on this or any other hyperbolic tessellation.