Posts Tagged ‘polypennies’

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.

Polyform Link Roundup

March 25th, 2012

There have been a few recent developments worth noting in the world of polyform puzzles:

Rodolfo Kurchan has posted Puzzle Fun #25. Some good new coloring problems using multiple sets of polyominoes.

David Goodger has been doing some good work on triangular and hexagonal grid polysticks.

George Sicherman is continuing to make advances in the realm of polyform compatibility problems. He also recently posted a catalog of the polypennies up to order 6.

KSO Glorieux Ronse is a school in Belgium that has, over the past decade, conducted a wonderful educational experiment by posting contests based on polyomino problems that could be engaged with by their own students just as much as the world’s puzzle solving elite. (The latter tended to win, of course.) Their 50th contest, which they state was their final one, was held late last year. They solicited the polyform puzzle community for problems to use in the contest and got quite a few, including one from me. No word yet on the results of the contest, (or their previous one for that matter) but the problems there are still pretty interesting.

I’ll be at the 10th Gathering for Gardner (G4G10) this week, and I expect that I’ll come back with quite a lot to think and post about. If you’re going to be there, my talk on Flexible Polyforms has tentatively been scheduled for the Thursday morning session. I hope to see some of you there!


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.


January 23rd, 2012

A while back, (before I started this blog) I was exploring polyforms using unit-radius circles as their base cell type, which I called “polypennies”. We can think of these as “flexible” polyforms: since connections between the circles can occur at arbitrary angles, we consider two polypennies to be equivalent if we can continuously move the circles around each other without changing which circles are adjacent. (As with other polyform types, rotations and reflections are also considered equivalent.)

The pentapennies

I called these polyforms “polypennies” rather than “polycircles” because “pennies” captured the equal size of the cells. (ETA: I forgot that I raided the word from the term “penny graph,” which has been used as an alternative to “unit coin graph” to describe the adjacency graph associated with a particular configuration of non-overlapping unit radius circles.) I also knew that eventually I would want to get to polyforms made of circles of arbitrary size, for which I was reserving the term “polycircle”. Well, it happens that I’ve been invited to Gathering for Gardner 10, where I plan to give a talk on flexible polyforms, so eventually is now.

For polycircles with cells of arbitrary size, another dimension of flexibility is required. Two polycircles are equivalent if they can be made congruent by continuously expanding or shrinking the circles without changing adjacencies, in addition to applying the transformations allowed with polypennies. This extra flexibility means that, in addition to the polycircles that are equivalent to polypennies, there are some polycircles that could only be formed by placing circles into spaces where they wouldn’t fit if all of the circles were forced to be the same size.

As with other flexible polyforms, elegant tiling puzzles for the polycircles can be produced by attempting to maximize the symmetry of the configuration to be tiled. Here’s an example, with fourfold rotational symmetry, of a tiling puzzle containing all of the polycircles of order 1 through 4:

This was not a hard puzzle to solve, once I came up with a configuration to tile that would work. Adding smaller pieces is a time-honored trick for making polyform puzzles easier; I put in the 1- through 3-circles because I was failing to make any headway with the 4-circles alone. The extra dimension of flexibility was helpful in that one can generally resize the circles to touch more neighbors than is possible in polypenny puzzles, which tend to end up with a number of cells with only two neighbors. On the other hand, the 4-circles with a circle inside the gap between three others in a triangle were trickier to deal with than any of the 4-circles that are equivalent to 4-pennies.

Can we do better than the above? I think fivefold symmetry may be possible.

Problem #26: Find and solve a tiling puzzle for the 1-, 2-, 3-, and 4-circles with fivefold rotational symmetry.