It is April 27th, 2018, and today is commencement for over 200 undergraduate and graduate students in the Department of Mathematics. Each student received a mini fractal puzzle in Maize and Blue colors. We wish you well!

## Solving a Penrose puzzle

The Penrose tilings, P1, P2, and P3 have many different solutions. It could be different every time, although you are likely to see strong similarities in the patterns. Chet and Esther bought our P1 at the the Atlanta American Craft Council show, and wanted to know if there was a solution that used all the tiles. They sent us this image:

Because there are multiple solutions, we don’t select the number of tiles based on a particular solution. Instead, we choose the number of pieces based on the proportions of tiles that would appear if you tiled the entire, infinite plane. The idea is that on average, you can use most of the tiles. For Penrose P2 and P3, the larger tiles appear more often, and in the ratio of the golden mean, which is about 1.618 to 1. So, for example, if we put 100 smaller tiles in a box, we would put 162 larger tiles in the box.

For P1, it is a bit more complicated, since there are 6 different tile shapes: 3 are based on pentagons, one is a thin rhomb, one is based on a star shape (called a *pentacle*, and the other is part of a start (called a *half-pentacle*). Using the colors from Chet and Esther’s set, as a puzzle is expanded, the proportions will converge to 3.8% for the pentacle, 8.6% for the half-pentacle, 13.5% for the rhombus, 10.9% for the red pentagon, 27.8% for the green pentagon, and 35.4% for the blue pentagon.

So is there a solution that uses all the tiles? Maybe, but we don’t know!

## Design note #9: Dragons

Whenever we are showing at art and craft fairs, people of all ages love playing with the Koch tilings. I think the appeal is that like many fractals, they evoke nature. And there is something satisfying about getting the intricate edges to meet, and have a piece slide into place.

So we decided to try another, based on one of the dragon curves. There are several choices, some much more delicate than others. This delicate edge comes from the fact that flat fractals have infinitely long edges, and a fractional dimension between 1 and 2. The closer the dimension is to 2, the more intricate, and therefore more difficult to cut and fragile. We settled on Knuth’s terdragon because the fractal dimension is high enough to be interesting, but still makes for a sturdy tile.

It is pretty easy to learn how to attach Koch tiles to each other. The tiles cannot be the same size. There are a couple of attachment points for two Koch tiles. The Dragon tiles, on the other hand, seem to take a little longer before users readily identify how to connect them. But they are actually more flexible. Any two tiles, regardless of size, can be connected, and there are several ways to attach.

We debuted our Dragon tiles at the 2018 Joint Mathematics Meeting, and as with the Koch tiles, they were a hit with the young and not-so-young. This little guy made our day:

Recently, we’ve added frames as an option for our Dragon puzzles, which is nice if you want to be able to move your work-in-progress.

## Design note #8: Fibonacci, Sunflowers, Voronoi, and a clock

Fibonacci‘s famous sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …) has been linked to many natural phenomena. The arrangement of seeds in sunflowers is one example.

To play around with these ideas, I wrote a script in p5js that allows you to vary that angle:

So, you can create some beautiful patterns, and with the laser cutter, you could cut out the circles. But I was also interested in the idea of the dots representing the center of cells. Enter the Voronoi diagram. Given a set of points, it finds cells containing those points, with the property that any border that separates two points is equidistant from the points.

We think this evokes the idea of biological cells, and led to our abstract Cellular Clock:

Eko Hironaka was interested in a design that was more suggestive of 12 hours, and created the design for Eko’s Flower.

## Design note #7: Inversive geometry

*This is a repost of ideas that I used in the classroom.*

The design below was created using *inversive geometry*, combined with a twisting rotation.

Inversive geometry requires that we have a circle of a given radius \(r\). Consider any point in the plane, at distance \(d\), lying on a ray from the center of the circle. Then that point has an inverse point, at distance \(r^2/d\) from the center of the circle, on the same ray. Taking the inverse of a point flips points from outside the circle to inside the circle, and vice versa. Imagine the plane being made out of flexible material, and inverting is like puncturing the plane at the center of the circle, and then turning the plane inside out, with the points on the edge of the circle remaining stationary.

You can play with this in **desmos** by clicking on the graph below. The orange disk is the unit circle. The red dot can be dragged. The green dashed line segment from the red dot towards the origin leads to the purple point, which is at a distance that is the reciprocal of the red dot’s distance from origin.

What shape is traced out by the inverse point as we slide the red dot along the line? Is this always the case?

Now, consider infinite graph paper with integer coordinates. The lines consist of all vertical lines with integer \(x\) values, and all horizontal lines with integer \(y\) values. What do you get if you find the inverse of all those lines?

This mapping of the entire plane outside the disk into the unit disk (and vice versa) is a conformal mapping, which means it preserves angles. If we define distance between two points in the disk to be the same as the distance between the inverses of the points, then we are preserving distance. Angles are the same and distances are the same, so we could study the ideas in high school geometry, but staying entirely inside a disk. What project ideas does this give you?

## Design note #6: Penrose P3 tiles

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.

## Design note #5: Tiling with regular polygons

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 3.6.3.6 and 3.4.6.4 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.

## Design note #4: Koch snowflakes

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.

## Design note #3: Patches as wall plaques

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:

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.

## Design Note #2: Oriented Triangles

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.