Posts Tagged ‘wallpaper groups’

A Pocketful of Pentapennies

May 12th, 2017

We can think of two connected unit coin configurations (or polypennies) as being equivalent if we can transform one into the other by reflection and/or sliding coins without changing which coins are adjacent. (Coins may not overlap.)

There are 13 pentapennies. A tiling with fivefold rotational symmetry may be possible, but I haven’t been able to find one. (This is problem #27.) However, I recently found a way to tile a figure with fourfold rotational symmetry with them:

Since I’ve had trouble with five symmetries, you’d think ten would be out of the question. But I found a repeating pattern on the plane with ten symmetries that can be tiled with the pentapennies:

Notice that there are five translation symmetries. Reflecting the pattern on a vertical axis gives five more symmetries. This pattern uses the wallpaper group cm. (Conway orbifold symbol: *×) We could also try to find a tilable pattern with the same amount of symmetry using the wallpaper group p2. (Conway orbifold symbol: 2222)

Problem #45: Find a tiling of the pentapennies on a repeating pattern on the plane that has at least as many symmetries as the one above, but a different wallpaper group. I don’t think going above 10 symmetries is possible, but I’d love to be surprised.