Posts Tagged ‘pentapennies’

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.

Polypennywise

February 26th, 2012

Here are the pentapennies and tetrapennies tiling a figure with 6-fold rotational symmetry:

For a while I’ve been trying to find a tiling of a figure with 5-fold symmetry using just the pentapennies. It feels like it should be doable, but I haven’t had any luck so far. Maybe you will? Call that problem #27. As with the polycircles in my last post, I decided to stack the deck in my favor by adding smaller pieces to the tiling set. This tiling contains 85 pennies: 65 from the 13 pentapennies and 20 from the five tetrapennies. With polypenny tilings you can either use a pattern with a penny in the center, or you can leave the center open. With a penny in the center, the remaining number of pennies is divisible by six. This is nice not only because we get a little more symmetry, but also because the configuration of six pennies around the central penny is strongly connected, which means that we have more flexibility in where the polypennies can go in that region.

Unfortunately, although seven is also a divisor of 84, seven pennies don’t fit around a central penny, so this is probably as good as we can do for symmetry. Although if we went to hyperbolic geometry, seven pennies could fit perfectly around a central penny after all. But, for now at least, I’ll save my pennies and not spend them irresponsibly at non-Euclidean exchange rates.