I was sad to hear of the recent passing of Princeton professor John Horton Conway. (He’s another victim of the you-know-what that I refuse to mention in this podcast due to its over saturation of the media.) Professor Conway was a brilliant mathematician known for his interest in mathematical games and amusements. My time as an undergraduate at Princeton overlapped with Conway’s professorship there, though sadly, I was too shy at the time to actually discuss math stuff with him. Among his contributions were the “Game of Life” (that’s the mathematical game played on a two-dimensional grid, not the children’s boardgame!) and the concept of “surreal numbers”, both of which we have discussed in past Math Mutation episodes. Anyway, if you’re the type of person who listens to this podcast, you’ve probably already read a few dozen Conway obituaries, so rather than repeating the amusing biographical information you’ve read elsewhere, I figured the best way to honor him is to discuss another of his mathematical contributions. So today we’ll be talking about the “Conway Criterion” for periodic planar tilings.
You probably recall the notion of a planar tiling: basically we want to cover a plane with repeated instances of some shape. A brick wall is a simple example, though rectangular bricks can be a little boring, due to the ease of regularly fitting them in neat rows. Conway himself managed to derive some interesting discussion from simple bricks though— in the show notes at mathmutation,com, you can find a link to a video of his Princeton walking tour titled “How to Stare at a Brick Wall”. But I think hexagons are a slightly more visually pleasing pattern, as you’ve likely seen on a bathroom floor somewhere, or in a beehive’s honeycomb. Looking at such a hexagon pattern, you might ask whether there’s a way to generalize it: can you somehow use simple hexagons as a starting point to identify more complex shapes that will also fill a plane? Conway came up with the answer Yes, and identified a simple set of rules to do this.
To start with, you can imagine squishing or stretching the hexagons— for example, if you squash a honeycomb, you’ll find squished hexagons, with not all angles equal, still fill the plane. With Conway’s rules, you just need to start out with a closed topological disk. This means essentially taking a hexagon and stretching or bending the sides in any way you want, as long as you don’t tear the shape or push sides together so they intersect. You can introduce new corners or even curves if you want. You then identify six points along the perimeter. In the case of a hexagon, you can just use the six corners. But the six points you choose, let’s call them A/B/C/D/E/F, don’t have to be corners. They just have to obey the following rules:
- Boundary segments AB and DE are congruent by translation, meaning they are the same shape and can be moved on top of each other without rotating them.
- The other four segments BC, CD, EF, and FA are each centrally symmetric: they are unchanged if rotated 180 degrees around their centers.
- At least three of the six points are distinct.
It’s pretty easy to see that regular hexagons are a direct example of meeting these rules, since any two opposite sites are congruent by translation, and line segments are always centrally symmetric. But Conway generalized and abstracted the idea of a hexagon-based tiling: each of the six segments can potentially have curves and zigzags, as long as they ultimately meet the criteria. The points you use can be anywhere along the outer edge, as long as they divide it in a way that meets the criterion. If you sketch a few examples you’ll probably see pretty quickly why these rules make sense.
Thus, this is a general formula that can give you an infinite variety of interesting tile shapes to cover your bathroom floor with. At the links in the show notes you can see articles with a crooked 8-sided example, and even a curvy form that looks like a pair of fish. An article by a professor named Bruce Torrence at Randolph-Macon college also describes a Mathematica program that can be used to design and check arbitrary Conway tiles. I wouldn’t be surprised if famous artworks involving complex tilings, like the carvings of M. C. Escher or classical Islamic mosaics, ultimately used an intuitive understanding of similar criteria to derive their patterns. Though, most likely, they didn’t prove their generality with the same level of mathematical rigor as Conway.
We should also note that the Conway criterion is sufficient, but not necessary, to create a valid planar tiling. In other words, while it’s a great shortcut for coming up with an interesting design for a mosaic, it’s not a method for finding all possible tilings. There are many planar tilings that do not fit the Conway criterion. You can find lots of examples online if you search. So while it is a useful shortcut, this criterion is not a full characterization of all periodic planar tilings.
Anyway, Conway made many more contributions to the theory of planar tilings, and related abstract areas like group theory. If you look him up online, you can see numerous articles about his other results in these areas, as well as more colorful details of his unusual personal trajectory through the mathematical world. As with all the best mathematicians, his ideas will long outlive his physical body.
And this has been your math mutation for today.