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.

To start out: the 1-, 2-, and 3-circles correspond to the 1-, 2-, and 3-pennies and have total “area” 9. The 4-circles consist of five configurations corresponding to the 4-pennies and three others with one circle inside the other three, totaling 32 circles among 4-circles and 41 circles altogether.

Yep. So a puzzle with 5-fold symmetry would have to have one circle in the center, and 8 more circles plus all of their images from rotating them 2nπ/5 around the center.