The Quest to Find Rectangles in a Square

Lisanne Taams, a pupil astatine Radboud University successful the Netherlands, is moving connected a Ph.D. about, successful her words, “computing motives of moduli stacks of vector bundles connected stacky curves.”

“It took maine 2 years to adjacent accidental that properly,” Ms. Taams said. But, she added, specified heights of abstraction lone elevated her delight arsenic she precocious spent clip connected a much factual contemplation: counting the ways that a quadrate tin beryllium divided into akin rectangles.

She recovered this geometric puzzle connected Mathstodon, a assemblage wrong the societal web Mastodon. Created successful the outpouring of 2017 by 2 mathematicians successful England, Christian Lawson-Perfect and Colin Wright, registered accounts connected Mathstodon totaled astir 3,000 successful September. Since then, with the Twitter exodus, the fig has since accrued to astir 13,000.

The puzzle was posted successful December by John Carlos Baez, a mathematical physicist astatine the University of California, Riverside.

“There are 3 ways to disagreement a quadrate into 3 rectangles with the aforesaid proportions!” Dr. Baez wrote.

He illustrated the reply with 3 images that helium borrowed from Wikipedia:

In the representation connected the left, the rectangles are 3 times arsenic agelong arsenic they are wide, helium explained successful an email. In the mediate image, the rectangles are one-and-a-half times arsenic agelong arsenic they are wide. “The 3rd solution is trickier,” Dr. Baez said. The rectangles are “about 1.75487 times arsenic agelong arsenic they are wide, though 1 rectangle is turned astir truthful it’s abbreviated and squat,” helium added.

Dr. Baez noted that the fig 1.75487 is of involvement to mathematicians. “It’s the quadrate of the ‘plastic ratio,’” helium said, “which is simply a fig that has a batch of properties akin to the much celebrated ‘golden ratio.’”

Having laid that foundation, Dr. Baez asked his Mathstodon followers: “What if you chop a quadrate into 4 akin rectangles? What proportions tin they have?”

Among the archetypal to instrumentality the bait was Rahul Narain, a machine idiosyncratic astatine the Indian Institute of Technology, Delhi. “I was connected Mastodon earlier it was cool,” helium says successful his bio (he joined successful December 2017). “And the information that each the chill radical are present present is simply a spot unsettling.”

Dr. Narain sketched retired a systematic strategy for solving the puzzle, though helium hoped idiosyncratic other would transportation it out. As helium said successful his reply to Dr. Baez, “I truly person different things I request to enactment connected close now, I can’t spend to get nerd-sniped immoderate much than this!”

There turned retired to beryllium 11 solutions — 11 ways a quadrate tin beryllium divided into 4 likewise proportioned rectangles. The solutions gradually accumulated with important input from Ian Henderson, an autarkic bundle developer successful the Bay Area, and Daniel Piker successful Bristol, England, who works arsenic a plan systems expert processing bundle for architects astatine Foster + Partners.

And a batch of different radical besides helped, Dr. Baez said. “That’s wherefore it was fun.”

Ms. Taams recovered 11 solutions by manus and soon discovered that she had made a fewer mistakes. She past decided to fto the machine bash the work. She wrote bundle and generated immoderate images. But erstwhile she checked the advancement online, “I saw different radical already had a batch much pictures,” Ms. Taams said.

Mr. Piker, who enjoys making geometric animations, had drawn each 11 options:

The simplicity of the occupation is what drew him. “I thought it was benignant of chill that determination was thing truthful elemental that seemingly hadn’t been looked astatine before,” Mr. Piker said.

However, helium added, “The maths rapidly went beyond my understanding.”

He could marque consciousness of a impervious posted by Ms. Taams, though it was not thing that helium would person easy produced. She posted an 11-part thread — with method passages composed with LaTeX, a technological typesetting connection — showing that this humble geometry puzzle is connected to much superior and ceremonial mathematics.

In different words, she came up with a impervious that the ratio of the agelong sides to the abbreviated sides are “algebraic numbers,” a large taxable successful fig theory.

“I don’t deliberation we’ve gotten anyplace adjacent the bottommost of this yet,” Dr. Baez said. “But it’s a bully step.”

Sarah Hart, a mathematician astatine Birkbeck, University of London — whose book, “Once Upon a Prime,” which explores connections betwixt mathematics and literature, comes retired successful April — called this rectangulation recreation “awesome” and “lovely.” (She had not partaken successful the Mathstodon pursuit.)

“What makes a occupation lovely?” she said. “That’s a tricky one.” For Dr. Hart, it helps if the occupation is casual to describe, and casual to play with — “you tin get your hands soiled consecutive distant with elemental examples.” And if it “becomes deliciously analyzable and challenging.”

Dr. Hart besides noted that “a batch of the astir absorbing problems travel from recreations similar this.”

Ms. Taams recovered her impervious computationally, and past pondered it further. The computations produced a acceptable of equations, she said. “And past you wonder, ‘Oh, are these each the equations? Yes oregon no?’” She convinced herself the reply was “Yes” by conscionable looking astatine 3 examples. “It’s a small spot hard to reason why. If you look astatine the pictures, you benignant of spot it.”

(More formally, Ms. Taams proved that erstwhile conducting a akin rectangulation of a quadrate — that is, dividing it into a number, N, of akin rectangles — the ratio is an algebraic fig of grade astatine astir N.)

The online treatment astatine 1 constituent turned to a akin probe into “squaring the square” by William Tutte and his collaborators successful the 1930s, which is related to electrical circuit theory.

“It turns retired you tin deliberation of the tallness and width of each quadrate arsenic related to voltage and existent successful an electrical circuit — and utilizing this, you tin find ways to ‘square the square’ utilizing electrical circuit theory,” Dr. Baez said by email. “Something similar this is besides existent for the rectangle dissection problem, but we haven’t exploited it yet.”

David Eppstein, a machine idiosyncratic astatine the University of California, Irvine, commented that the word “guillotine partition” is the modular mode to picture the process of recursively slicing off, vertically oregon horizontally, rectangular pieces from a square. Jules Hedges, a machine idiosyncratic astatine the University of Strathclyde, Glasgow, suggested that “Mondrian” — aft the Dutch abstract creator Piet Mondrian — mightiness besides beryllium a fitting sanction for this process. This prompted Stefano Gogioso, a machine idiosyncratic astatine the University of Oxford, to constituent retired that, “in instrumentality learning, Mondrian tree/forests is utilized to denote a definite benignant of classification performed by guillotine cuts.”

Returning to the substance of the 11 four-way rectangulations: This effect was confirmed with 2 batches of code, 1 by Dr. Narain, enumerating and investigating each the guillotine partitions, and 1 by Mr. Henderson utilizing a much progressive approach.

They didn’t halt there.

“The question appeared successful my mind, ‘Oh, what astir five? What astir six?’” Mr. Henderson said.

Both helium and Dr. Narain recovered 51 solutions erstwhile it came to dividing a quadrate into 5 akin rectangles:

Mr. Henderson recovered 245 imaginable rectangular proportions that disagreement a quadrate into six akin rectangles, and 1,371 options for 7 akin rectangles. Initially, helium gave up connected 8 rectangles — helium tried, but the programme conscionable kept running. Eventually, it ran retired of memory.

“I got it retired of my system,” helium said.

But past helium circled backmost and went troubleshooting and realized that determination was thing amiss with the codification for 8 rectangles. “In immoderate case, with this fixed, it does really decorativeness running,” Mr. Henderson said successful his email. “There are (according to the code, astatine least) 8,506 antithetic facet ratios for 8 rectangles.” He whitethorn effort for nine.