Roger Penrose is well known for his collaborations with Stephen Hawking studying black holes. Perhaps being a cosmologist made him interested in the work of Kepler. And perhaps that led him to thinking about tilings with pentagons. In any case, he ended up discovering a some remarkable things about tiling. In particular, he discovered Penrose tiles. He described three such tilings, called P1, P2, and P3. P3 tiles are shown below.
P3 Penrose tiles could just be two simple rhombuses (thick and thin), but you would have to follow special rules to determine whether two tile edges can be matched. But those rules can just be implemented by altering the edges to limit how you can connect the tiles. Remarkably, the resulting tiling is guaranteed to be aperiodic, which means it is not a typical repeating wallpaper-style pattern.
Amazingly, Penrose’s tilings ended up being useful in explaining physical phenomena that was discovered after Penrose discovered the tiles. You can read more about applications to crystallography, as well as the notion of inflation in the article Penrose Tiles talk across the Miles.
The Koch snowflake got me thinking about producing other sets of tiles that can be used to tesselate the plane. For the classroom, I created a prototype set of regular polygon tiles, which I used in conjunction with Jacobs’ Mathematics: A Human Endeavor. These happened to be created on a laser with lower power and airflow than mine, so some of the tiles have caramel-colored edges, which I clean up in production. But I kind of like the way it highlights the edges.
Regular tilings of the plane used only regular polygons to completely fill the plane, and this can only be done with triangles, squares, and hexagons.
However, there are many more semi-regular tilings, which allow 2 or more polygon types, always meeting in the same way at the corners. Below are the 126.96.36.199 and 188.8.131.52 tilings, named for the number of sides of the polygons that meet at each vertex in the tiling.
And note that octagons and 12-sided dodecagons can get in on the action:
These are the only regular polygons that can be used to completely tile the plane. However, as noted by Albrecht Dürer and Johannes Kepler, there are some interesting tilings you can create with pentagons. So I created a set of regular pentagons:
As we played with them, we found it a little frustrating that a slight bump tends to move everything out of alignment. So I added tabs. The construction below comes up often in graphics of Kepler’s work.
The Koch snowflake is one of the first fractal curves to be described.
Like other fractal curves, it has an infinitely long boundary, and the self-similarity is obvious as you zoom in. One of the cool things about the Koch snowflake is that it can be built from six smaller snowflakes, leaving another snowflake in the middle. That of course can also be decomposed, recursively, giving you this:
So that led to one of our first puzzles, which uses two sizes of snowflakes. I put the box of pieces in the Mathematics Commons at the University of Michigan. Both of the patterns below were created there.
Putting it together, in holiday colors…