Lisanne Taams, a student at Radboud University in the Netherlands, is working on a PhD about, in her words, “computing motives of moduli stacks of vector bundles on stacky curves”. “It took me two years to even say that properly,” Taams said. But, she added, such heights of abstraction only elevated her delight as she recently spent time on a more concrete contemplation: counting the ways that a square can be divided into similarly proportioned rectangles.
She found this geometric puzzle on Mathstodon, a community within the social network Mastodon. Created in the spring of 2017 by two mathematicians in England, Christian Lawson-Perfect and Colin Wright, registered accounts on Mathstodon totalled about 3,000 in September. Since then, with the Twitter exodus, the number has increased to around 13,000.
The puzzle was posted in December by John Carlos Baez, a mathematical physicist at the University of California, Riverside, US.
“There are three ways to divide a square into three rectangles with the same proportions!” Baez wrote. He illustrated the answer with three images that he borrowed from Wikipedia.
In one, the rectangles are three times as long as they are wide, he explained in an email. In the second, the rectangles are 1.5 times as long. “The third solution is trickier,” Baez said. The rectangles are “about 1.75487 times as long as they are wide, though one rectangle is turned around so it’s short and squat,” he added.
Baez noted that the number 1.75487 is of interest to mathematicians. “It’s the square of the ‘plastic ratio’,” he said, “which is a number that has a lot of properties similar to the more famous ‘golden ratio’.”
Having laid that foundation, Baez then asked his Mathstodon followers: “What if you chop a square into four similar rectangles? What proportions can they have?”
Among the first to take the bait was Rahul Narain, a computer scientist at the Indian Institute of Technology in New Delhi. Narain sketched out a systematic strategy for solving the puzzle, though he hoped someone else would carry it out.
There turned out to be 11 solutions — 11 ways a square can be divided into four similarly proportioned rectangles. The solutions gradually accumulated with crucial input from Ian Henderson, a software developer in the Bay Area, US, and Daniel Piker in Bristol, England, who works as a design systems analyst developing software for architects at Foster + Partners.
And a lot of other people also helped, Baez said. “That’s why it was fun.”
Taams found 11 solutions by hand and soon discovered that she had made a few mistakes. She then decided to let the computer do the work. She wrote software and generated some images. But when she checked the progress online, “I saw other people already had a lot more pictures,” Taams said.
Piker, who enjoys making geometric animations, had drawn all 11 options. However, he added, “the maths quickly went beyond my understanding”.
He could make sense of a proof posted by Taams, though it was not something that he would have easily produced. She posted an 11-part thread — with technical passages composed with LaTeX, a scientific typesetting language — showing that this humble geometry puzzle is connected to more serious and formal mathematics.
In other words, she came up with a proof that the ratio of the long sides to the short sides are “algebraic numbers”. Incidentally, algebraic numbers is a major topic in number theory.
Taams found her proof computationally and then pondered it further. The computations produced a set of equations, she said. “And then you wonder, ‘Oh, are these all the equations?’” She convinced herself the answer was yes by just looking at three examples. “It’s a little bit hard to argue why,” she said. “If you stare at the pictures, you sort of see it.”
The online discussion at one point turned to a similar investigation into “squaring the square” by William Tutte and his collaborators in the 1930s, which is related to electrical circuit theory.
“It turns out you can think of the height and width of each square as related to voltage and current in an electrical circuit — and using this, you can actually find ways to ‘square the square’ using electrical circuit theory,” Baez explained by email. “Something like this is also true for the rectangle dissection problem, but we haven’t exploited it yet.”
Returning to the 11 four-way rectangulations: this result was confirmed with two batches of code — one by Narain and one by Henderson. They didn’t stop there.
“The question appeared in my mind, ‘Oh, what about five? What about six’?” Henderson said. Both he and Narain found 51 solutions to dividing a square into five similar rectangles.
Henderson found 245 possible rectangular proportions that divide a square into six similar rectangles and 1,371 options for seven. Initially, he gave up on eight rectangles; he tried, but the program just kept running. Eventually, it ran out of memory.
But then he circled back and realised there was something amiss with the code. “There are (according to the code, at least) 8,506 different aspect ratios for eight rectangles.” Henderson may now try for nine.
New York Times News Service