Leave it to mathematicians to look at a pizza and see a problem. While most of us consider the cheesy treat to be a solution to basically all of life’s struggles, the venerable dish has long posed a puzzle for mathematicians: how many ways are there to slice a pie?
It’s a question Joel Haddley, a math professor at the University of Liverpool, has spent years mulling over. He recently co-wrote a paper published on ArXiv detailing a new solution to this problem, and chatted with me about why mathematicians would spend time on such matters.
“I was [first] posed this puzzle at a conference in Paris as a PhD student, while eating pizza in an Italian restaurant on the Champs-Élysées,” Haddley told me via email. “I just kept thinking about it.”
The traditional triangular slices are an easy solution to dividing up a disc-shaped za into equal pieces (because nobody wants to get ripped off in the pizza divvying department). Mathematicians have pinned down another technique, too, which consists of three-sided slices where each side matches the curve of the pizza’s perimeter, like this:
All Images: Haddley and Worsley.
That technique can be extended (infinitely, in theory, though it’d be tricky to continue cutting tinier and tinier slices of za) by cutting those slices in half using the same curve:
But rather than being satisfied with this solution, mathematicians refined the problem to make it trickier. Is it possible, they wondered, to divide a disc (a flat circle, like a pizza) into monohedral tilings (slices that are all the exact same shape and size) so that some of the slices don't touch the center AND not all sides of each slice follow the curve of the perimeter?
There was one such solution: cutting the curved slices and then dividing each slice in half with a straight line, rather than a curved one.
But for years, no one could come up with another answer to the problem. The Mathematics Advanced Studies Semesters program at Penn State University even had a running challenge for mathematicians to cook up additional solutions. Now, along with co-author Stephen Worsley, Haddley has presented a new family of solutions. They look complicated at first blush but are pretty simple when you break them down. First, take a 7-sided polygon in a sort of “croissant” shape, like this:
Then stack many of them together in a spiral, like this:
You’ll get something that starts to look like a circle, except with many flat edges. If you replace the outside edge of these polygons with a curved one and then cut those slices in half like before, you can create the new tiling solutions:
The authors also showed that you can solve the problem by nicking a few triangles out these shapes, as long as they stay symmetrical:
“I've no idea whether there are or ever will be any applications,” Haddley said. “They're just nice and interesting pictures and you can have great fun with them!”
So if there are no real-world applications—aside from blowing the minds of your stoned friends when serving them a frozen pizza—why bother with puzzles like these? Haddley said he simply enjoys recreational math and, after years of chewing on this problem, the solution just dawned on him.
“A flash of inspiration came almost out of nowhere,” he said. “I shared the initial result with Stephen who had his own ideas for solutions and we started working together at it and between us came up with what you see in our paper.”
But Haddley also said he teaches a course in math education at the university. I personally can think of no better way to get students interested in monohedral disk tilings than relating it to pizza, so there could be more real world benefits than he realized.
Since penning this paper, Haddley and Worsley have come up with even more solutions, which they plan to add to their findings in a formal paper to be submitted for publication this year. But even with this latest discovery, the pizza slicing problem continues to test mathematicians, Haddley said.
“A very difficult challenge would be to prove that we have found all such solutions,” Haddley said. “We also still haven't found a tiling with the property that the centre is contained entirely within a tile, or such that the centre is on an edge of a tile. In all our examples, it is on a vertex (corner). To my knowledge, no one has, and I've no idea whether such a tiling exists, and if it doesn't exist how to prove it doesn't.”
So if you’re also a fan of pizza and math, there’s still plenty of problems left to solve. But if you’re just looking for solutions, I suggest a large with mushrooms and pineapple.