We were recently introduced to an interesting set of tiles by Barry Cipra. Barry writes about mathematics and science, designs games, and has a keen interest in recreational mathematics.
Barry is inspired by the art of Sol Lewitt. One of Lewitt's methods was to begin with a particular geometric idea that could be varied by a change in orientation, color, or value, and explore all the possible arrangements. If you're not familiar with Lewitt, we recommend taking a look at the MOMA retrospective of his work.
You can see a binary number in the corner of each, and each tile has paths that enter one edge, and exit another. There are two paths from each edge, and each path must go to an adjacent edge. The tiles are numbered, and are meant to be oriented, with the number in the lower left corner.
This already gives rise to some interesting questions:
• Why are there sixteen tiles?
• What do the numbers in the corner of each tile mean?
Cipra focuses on loops that appear in the tiles. Here is a simple example of a loop:
As you travel all the way around the loop, it enters (and exits) a tile 4 times. We say this loop has length 4.
Here are tiles with two length 8 loops. One is highlighted, the other is not.
Notice that we always have paths that seemingly end at the edges. Let's imagine that if we have a rectangular arrangement of tiles, then the left edge of the rectangle connects to the right edge, like a piece of paper that bent until it becomes a cylinder. Note: the image below takes time to load.
This allows our paths to continue. But the path could still exit at the top or the bottom of the cylinder. So let's stretch and bend the cylinder into a circle and connect the top to the bottom, making a donut, or torus. Note: the image below takes time to load.
Now, the paths have all becomes loops! This inspires more questions about the 16 tiles arranged in a square:
• Is there a tiling with just one loop, that uses every path segment?
• What would be the length of a loop that contained every path segment?
• Is there a tiling where all the loops have length 4?
• Is there a tiling where all the loops have length 8?
• Is it possible to create a loop of length 6?
• How many ways are there to tile the torus?
What if we consider only using a subset of the tiles to make a smaller square or rectangle? How would you modify the questions above? For example, if you use only one of the tiles, is there a tile with a loop having length less than 4? What if you use 2x3 tiles? 3x5?
There is much more to be explored here. At the 2019 MOVES conference at MOMATH, Cipra collaborated with Donna Dietz and Peter Winkler to explore some of the features of Cipra tiles. If you want to learn more, check out their article, Exploration of another Sol Lewitt Puzzle from Barry Cipra.
Of course, the best way to explore is with a set of our Cipra Loops tiles!