The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of mysterious origin. “That sort of matches,” mentioned Martin Demaine, an artist in residence at the Massachusetts Institute of Technological know-how. “It’s a puzzle the place the word ‘puzzle’ will come from.”
His son, Erik Demaine, an M.I.T. computer scientist, agreed. “It’s a self-describing etymology,” he reported.
The father-son duo is most well known for mathematical investigations into paper folding, with “curved-crease sculptures” — swirling loops of pleated paper that resemble intergalactic interchanges. Curved origami dates to late 1920s Bauhaus a basic specimen begins as a round piece of paper, which, when folded alongside concentric circles, immediately twists into a saddle curve. The Demaines’ trio of parts, “Computational Origami,” was component of the 2008 “Design and the Elastic Mind” exhibit at the Museum of Contemporary Art in New York and now resides in its everlasting collection.
These times, on the other hand, the Demaines are extra concentrated on “algorithmic puzzle fonts,” a suite of mathematically impressed typefaces that are also puzzles. The principal software is enjoyment. One particular font, a homage to the mathematician and juggler Ron Graham, who died in 2020, attracts its letters from the patterns of motion traced by balls thrown into the air in the course of juggling tricks.
Another font, proposed by the laptop scientist Donald Knuth (practically all fonts entail collaborators), has as its distinguishing attribute that all letters can be “dissected” — slash into parts and rearranged — into a 6-by-6 square.
Inspired by theorems or open up complications, the fonts — and the messages they compose — can ordinarily be read only immediately after fixing the connected puzzle or collection of puzzles.
Just take, for instance, a new font in their collection that debuts currently: the Sudoku Font. The inspiration came in the tumble of 2019, when Erik Demaine co-taught the program “Fundamentals of Programming” (with the pc scientist Srini Devadas). All through a person course, Dr. Demaine and his 400 freshmen and sophomores programmed a Sudoku solver — they wrote code that solved a Sudoku puzzle. Dr. Demaine’s father sat in on the lecture that working day, and while fifty percent-having to pay awareness Mr. Demaine mused about irrespective of whether it may be feasible to make a font centered on Sudoku — that is, based on the puzzles whose distinctive solutions would somehow reveal letters of the alphabet.
Immediately after enjoying around with different alternatives, the Demaines developed a Sudoku puzzle font that works as follows: Initial, start out with one particular of their Sudoku puzzles and clear up it. Future, draw a line connecting the longest route of squares with consecutive figures (ascending or descending but only edge-adjacent squares, not diagonal). That line draws the condition of a letter in just the grid of the puzzle. A sequence of Sudokus so solved can expose a message, like so:
The full suite of puzzle fonts is out there, with different levels of interactivity, on Dr. Demaine’s website. The Demaines hand-built the letter styles, but used a pc to create the letter-embedding Sudoku puzzles.
“It was tough to structure letters that nonetheless enabled the puzzle to be solvable, and without incorporating added stray connections to the longest path,” Dr. Demaine mentioned. “This was very a challenging font to structure, both of those for the human and the computer system.”
Math + art = enjoyment
The Demaines began this puzzle-font experiganza close to the switch of the century with a dissection puzzle — a puzzle whereby a single condition, or polygon, is sliced up and reassembled into other geometric styles. Their inspiration was a trouble posed in 1964 by Harry Lindgren, a British-Australian engineer and novice mathematician: Can just about every letter of the alphabet be dissected into pieces that rearrange to sort a sq.?
In 2003, creating on past function, the Demaines proved that, sure, without a doubt it was feasible, and they revealed the final result. (Generally, a puzzle font comes with a corresponding investigation paper.) This initially foray was a puzzle only in the sense that the Demaines ended up perplexed for a although about how to design and style the font. And they designed the obstacle far more puzzling by introducing an more criterion: They wished not merely a dissection font, but a “hinged dissection” — a unique variety of dissection whereby the items are related (hinged) at their vertexes, forming a shut chain that rearranges, in this circumstance not only into the desired sq. but also into every single other letter of the alphabet.
They succeeded in their quest by deploying the arithmetic of “polyforms,” styles built from multiple copies of a polygon, such as a triangle. More precisely, they made use of a polyform with the improbable identify “polyabolo” (popularized by Martin Gardner, who was a mathematics columnist for Scientific American). A polyabolo is built from congruent correct isosceles triangles. A sq. can be reduce into two suitable isosceles triangles and those people two triangles can in transform be reduce into four right isosceles triangles, and all those 4 triangles into 8, and all those eight into 16, 16 into 32, 32 into 64, 64 into 128, and so forth.
By this strategy, the Demaines created their Hinged-Dissection Font. Each letter of the alphabet is dissected into 32 triangles (rendering it a “32-abolo”) that can be rearranged into a 4-by-4 square, or any other letter. But attaining the sought after hinged dissection — a linked chain of triangles that can morph from one letter into any other — expected that every letter be dissected into 128 triangular items (building it a “128-abolo”).
Reflecting on this exercising in an e mail, the Demaines stated: “The fun for us was combining art and math alongside one another, aiming for good design and style (recognizable as letters and hunting dependable throughout the alphabet) inside of tough mathematical constraints (set spot and performing with polyabolo designs).”
The philosophy of finding stuck
20 a long time on, those humble beginnings have sprouted into a wonderful fun residence of fonts, with artistic media as various as rods of glass, string artwork and cash.
Think about the Tiling Font: Each and every letter “tiles the plane,” meaning, as the Demaines reveal, “that infinitely lots of copies of that 1 form can fill two proportions without having leaving any gaps in between the tiles.” Great for a toilet renovation.
The Conveyer Belt Font was prompted by a nevertheless-unsolved trouble posed in 2001 by the Spanish mathematician Manuel Abellanas: If there are numerous two-dimensional and nonoverlapping wheels, or disks, of equal dimension, can they all be wrapped (linked) with a taut conveyor belt, this sort of that the belt touches all the wheels but doesn’t intersect itself?
The Demaines tried out to fix this challenge and got caught. They distracted them selves by developing the font. “That’s generally been an crucial portion of our philosophy,” Mr. Demaine said. “If we get stuck on a challenge, we like to locate an artistic way to characterize it.”
The Demaines also discover that puzzles are a wonderful way to initiate newcomers into the enjoyment of official mathematics. The Checkers Font (in which the letters are fashioned from paths of jumping moves) arrived into currently being when Spencer Congero, a computer science graduate pupil at the University of California, San Diego, got in touch with the concept. The Spiral Galaxies Font (based mostly on the Japanese pencil-and-paper puzzle of the exact identify distinctive methods to puzzles kind letters) was a collaboration with Walker Anderson, then a college student at Central Bucks West Large University in Doylestown, Pa., and a member of the United states of america Earth Puzzle Championship crew.
The puzzle font was Mr. Anderson’s gateway to mathematical exploration now he’s an undergrad studying math at M.I.T. For the Demaines, these kinds of collaborations are result in for celebration: Just one extra particular person successfully “corrupted” into the world of theoretical laptop or computer science.
Creative constraint
Supplied their name with origami, the Demaines have normally created a handful of fonts riffing on the nuances of folding, together with the Origami Maze Font, the Simple Fold & Slice Font, the Fold & Punch Font and an Difficult Folding Font.
The Demaines also made a decision, for a improve, to make a minimalistic font requiring only a one fold.
Lest that simplicity make the unsolved font far too very simple to read through, they extra a restriction: The letters should be illegible just before folding. Most of their typefaces, in fact, are based mostly all around comparable constraints. The Demaines like to make the undertaking tricky but not preposterously so they don’t want as well a great deal freedom or overall flexibility, considering that the attract is in the problem, but they do want the job to be attainable.
With these parameters, they devised the A single-Fold Silhouette Font. The silhouette ingredient borrows from a 1900-era “Rabbit Silhouette Puzzle,” in which five playing cards with cutouts of various animals stack up to make the silhouette of a rabbit. The 1-Fold Silhouette Font works in a comparable way. Picture a transparent sheet, with black markings:
The central vertical crease invitations you to fold the sheet in fifty percent (from proper to still left, as if you were being turning the page of a reserve).
And shock, the text is unveiled!
Font physics
The upshot of the new result is this: They have proved, in participating in the offline version of Tetris (whereby the player has complete information and facts in progress about the id and order of items that will fall) that the sport is “NP-complete” — meaning that no efficient option algorithm exists, even with as number of as eight columns or four rows. And extra nearly, as Dr. Demaine described on his web-site, NP-completeness usually means “it’s computationally intractable to figure out no matter whether you can endure, or apparent the board, supplied an preliminary board configuration and a sequence of n items to come.”
In the beginning, the creative constraint for this font was that every single letter be manufactured as a stacking of one particular duplicate of all 7 Tetris shapes. Then the Demaines understood it would be neat to animate the font, with letters dropping into development like items in the activity — so that each and every piece put also experienced to be supported by the past parts, with no extreme overhangs, so obeying “Tetris physics.” This necessitated a bit of redesign, in some cases with the assist of a computer system tool (“BurrTools”) that assembled preferred designs from simple unit parts.
“When us human beings bought caught finding a fantastic alternative, we’d place some of the shapes we’d been striving into BurrTools, and it would support guideline our search,” Dr. Demaine stated. “Q” and “M” were being amid the past letters to drop into spot.
So, had this report been penned in Everything Font — with every letter bearing a crease pattern (folding guidelines) for a different letter — there would be a different report encoded herein.
