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 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…
Our son Max attends the United States Air Force Academy in Colorado Springs, CO. The Academy is divided into 40 squadrons of about 100 cadets each. Each squadron has its own patch, and many of the squadrons display a wall plaque version of the patch at the squadron’s CQ desk (Cadet in Charge of Quarters).
Max asked me if I could create a replica of his squadron’s patch in plywood, about 22″ across. I created the vector drawing in Affinity Designer, and cut it out of a combination of 1/2″ and 1/4″ birch ply to get a layered effect. Several of the cadets assembled it, and painted it:
They did a nice job! In cause you’re wondering, Squadron 2’s logo references the F-102 Delta Dagger
, a 1960s era fighter-interceptor.
Next thing you know, a couple of his buddies from his water polo team wanted plaques for their CQ desks. Squadron 28 (featuring a stylized SR-71 Blackbird) opted for two tones, alternating stain with a pleasant natural tone.
Squadron 11 went for a subdued look, and selected a dark stain for the entire plaque. I haven’t stained birch before, but this Fine WoodWorking forum
has some suggestions for getting the best results.
I must admit that I am partial to the bright colors of the original patch, so I’ve shown a Photoshopped version below.
There was interest in a Squadron 8 patch as well, based on the F-15 Eagle
. I prototyped it in Affinity Designer based on a patch from the USAFA website:
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. One challenge is to match all edges.
Below is a prototype we created in acrylic. The picture is a hexagon, but not a solution.
An arbor is a small group of trees. I think of it as smaller than a forest, bigger than a grove. Ann Arbor (A2
to locals) loves its trees, and the co-founders of Ann Arbor named their new tree town after their wives, who shared a first name.
Like many of our designs, the Arbor Letter design has mathematical inspiration that has connections to nature. If you look carefully at the trees, you’ll note they are self-similar. Whenever a branch forms, the branch is a copy of the tree, reduced in scale. So the trees are fractals.
At first, we were taken with large (12”) diameter designs that we hung on the wall. And we have a pair of 20″ diameter with our surname initials for our front doors. But reduced to 3.5″, with a loop added, it makes a beautiful, delicate ornament. We now make the design in a variety of sizes, and have written software for composing variations in the shape, positioning, and number of trees.