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 220.127.116.11 and 18.104.22.168 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…
This is inspired by the Izzi puzzle, which is composed of squares. I learned about it from Professor Mark Saul of the The Center for Mathematical Talent at NYU, who developed beautiful mathematical content for i2camp.org. The Izzi puzzle consists of squares that have bisected edges that are combinations of black and white.
On my teaching blog, I explored the idea of using equilateral triangles. You need only 24 pieces to have one of each possible triangle, and they can be assembled into a hexagon. The challenge is to match all edges.
Below is the one I created in acrylic. The picture is a hexagon, but not a solution.
I created them initially using etching to create a lighter color. But I’m not a fan of etching unless it is really necessary. For one thing, it is slow. But I’ve also found that there is usually a more aesthetically pleasing alternative using low power cut lines. The new version will be available soon in our store.