There’s Something about Pentagons

Last month, researchers Casey Mann, Jennifer McLoud, and David Von Derau at the University of Washington Bothell found a new pentagon that tiles the plane, and the crowd went wild. It’s tough for a piece of research mathematics to get news coverage, but this plucky little pentagon was the perfect media-friendly story. Everyone likes shapes, and they can understand what it means to tile the plane. The problem also has an interesting history, and it’s a good way to show people how mathematicians think: once we answer a question in one context, say for regular polygons, we see what changes when we change the context.

Representatives of the 15 types of pentagons that tile the plane. The newest one is in the bottom right corner. Image: Ed Pegg Jr, via Wikimedia Commons.

Several good math writers covered the story of the little pentagon that could. Alex Bellos wrote about it for the Guardian, NPR had a nice article by Eyder Peralta, Kevin Knudson wrote a post about it in his recently-launched Forbes column, Katie Steckles covered it for the Aperiodical, and Robbie Gonzalez wrote a short post for io9.

Lost in some of the news coverage was the fact that most of the pentagonal tilings are not individual pentagons that work but infinite families of pentagons, so there are infinitely many pentagon tilings, not just 15. It’s a subtle point but one that has caused a bit of confusion. Luckily, Wolfram has an interactive demonstration of the different types of tilings that helps clear things up. Reshan Richards also has a pentagonal tiling module on his Explain Everything app. (I do not have Explain Everything, so my mention of it here is not an endorsement or review.)

The question of pentagonal tilings is two questions in one: what pentagons can tile the plane, and how can they tile the plane? In other words, this is the distinction between tiles and tilings. The current discovery is of a new tile, a different shape that tiles the plane. Frank Morgan has a post about the other question: how can tiles be arranged into tilings? Specifically, he has mentored undergraduate students who are studying tilings involving two of the different pentagonal tiles. The post is full of pictures of these two tiles in all sorts of different tiling patterns with names like Christmas Tree, Toothy Smile, and Space Pills. His group’s AMS article about their work (pdf) has even more illustrations.

I first learned about the history of pentagonal tilings when Math Munch wrote about Marjorie Rice in 2013. Rice, one of the big names in pentagonal tiles, was not a mathematician but learned about the problem from a Martin Gardner article in Scientific American. It is rare that true amateurs make breakthroughs in research mathematics, but she discovered four new classes of pentagons that tile the plane, and her story is heartwarming and inspirational. Ivars Peterson, the Mathematical Tourist, has a post about her and the tiling of the MAA’s entryway.

While I was researching this post, I ran across another article from Peterson about pentagons and tilings. This one is about the seemingly paradoxical Biosphère dome in Montreal. It is a dome, hence spherical rather than flat, but it appears to be tiled with regular hexagons. “Where does the curvature come from?” Peterson asks. “I know the pentagons are there, and I have tried to find them, but I have had very little success in locating even one.”

I wonder if we’ve now found all the pentagonal tiles, or if someone else will have success locating another one.