The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of unknown origin. “That kind of fits,” mentioned Martin Demaine, an artist in residence on the Massachusetts Institute of Technology. “It’s a puzzle where the word ‘puzzle’ comes from.”
His son, Erik Demaine, an M.I.T. laptop scientist, agreed. “It’s a self-describing etymology,” he mentioned.
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, robotically twists right into a saddle curve. The Demaines’ trio of items, “Computational Origami,” was a part of the 2008 “Design and the Elastic Mind” exhibit on the Museum of Modern Art in New York and now resides in its everlasting assortment.
These days, nonetheless, the Demaines are extra centered on “algorithmic puzzle fonts,” a collection of mathematically impressed typefaces that are additionally puzzles. The predominant software is enjoyable. One font, a homage to the mathematician and juggler Ron Graham, who died in 2020, attracts its letters from the patterns of movement traced by balls thrown into the air throughout juggling tips.
Another font, proposed by the pc scientist Donald Knuth (nearly all fonts contain collaborators), has as its distinguishing attribute that all letters could be “dissected” — lower into items and rearranged — right into a 6-by-6 sq..
In a 2015 paper, “Fun With Fonts: Algorithmic Typography,” the Demaines defined their motivations: “Scientists use fonts every day to express their research through the written word. But what if the font itself communicated (the spirit of) the research? What if the way text is written, and not just the text itself, engages the reader in the science?”
Inspired by theorems or open issues, the fonts — and the messages they compose — can normally be learn solely after fixing the associated puzzle or sequence of puzzles.
Take, for example, a new font of their assortment that debuts as we speak: the Sudoku Font. The inspiration got here within the fall of 2019, when Erik Demaine co-taught the course “Fundamentals of Programming” (with the pc scientist Srini Devadas). During one class, 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 day, and while half-paying consideration Mr. Demaine mused about whether or not it may be attainable to make a font primarily based on Sudoku — that is, primarily based on the puzzles whose distinctive options would one way or the other reveal letters of the alphabet.
After enjoying round with numerous potentialities, the Demaines designed a Sudoku puzzle font that works as follows: First, begin with one in all their Sudoku puzzles and remedy it. Next, draw a line connecting the longest path of squares with consecutive numbers (ascending or descending; however solely edge-adjacent squares, not diagonal). That line attracts the form of a letter throughout the grid of the puzzle. A sequence of Sudokus thus solved can reveal a message, like so:
The whole suite of puzzle fonts is obtainable, with various levels of interactivity, on Dr. Demaine’s website The Demaines hand-designed the letter shapes, however used a pc to generate the letter-embedding Sudoku puzzles.
“It was hard to design letters that still enabled the puzzle to be solvable, and without adding additional stray connections to the longest path,” Dr. Demaine mentioned. “This was quite a difficult font to design, both for the human and the computer.”
Math + artwork = enjoyable
The Demaines started this puzzle-font experiganza across the flip of the century with a dissection puzzle — a puzzle whereby one form, or polygon, is sliced up and reassembled into other geometric shapes. Their motivation was an issue posed in 1964 by Harry Lindgren, a British-Australian engineer and beginner mathematician: Can each letter of the alphabet be dissected into items that rearrange to type a sq.?
In 2003, constructing on previous work, the Demaines proved that, sure, certainly it was attainable, and so they revealed the end result. (Typically, a puzzle font comes with a corresponding analysis paper.) This first foray was a puzzle solely within the sense that the Demaines had been perplexed for a while about learn how to design the font. And they made the problem extra puzzling by including an additional criterion: They needed not merely a dissection font, however a “hinged dissection” — a particular sort of dissection whereby the items are linked (hinged) at their vertexes, forming a closed chain that rearranges, on this case not solely into the specified sq. but additionally into each other letter of the alphabet.
They succeeded of their quest by deploying the arithmetic of “polyforms,” shapes produced from a number of copies of a polygon, akin to a triangle. More exactly, they used a polyform with the unbelievable identify “polyabolo” (popularized by Martin Gardner, who was a arithmetic columnist for Scientific American). A polyabolo is produced from congruent proper isosceles triangles. A sq. could be lower into two proper isosceles triangles; and people two triangles can in flip be lower into 4 proper isosceles triangles, and people 4 triangles into eight, and people eight into 16, 16 into 32, 32 into 64, 64 into 128, and so forth.
By this technique, the Demaines created their Dissection Font. Every letter of the alphabet is dissected into 32 triangles (rendering it a “32-abolo”) that could be rearranged right into a 4-by-4 sq., or any other letter. But attaining the specified hinged dissection — a linked chain of triangles that can morph from one letter into any other — required that every letter be dissected into 128 triangular items (making it a “128-abolo”).
Reflecting on this train in an electronic mail, the Demaines mentioned: “The fun for us was combining art and math together, aiming for good design (recognizable as letters and looking consistent across the alphabet) within hard mathematical constraints (fixed area and working with polyabolo shapes).”
The philosophy of getting caught
Consider the Tiling Font: Each letter “tiles the plane,” which means, because the Demaines clarify, “that infinitely many copies of that one shape can fill two dimensions without leaving any gaps between the tiles.” Perfect for a loo renovation.
With the Conveyer Belt Font, every letter is fashioned by the closed loop of a conveyor belt that curves round strategically positioned wheels. (The font identify is deliberately spelled “conveyer” quite than “conveyor,” because the font “conveys” letters and phrases.)
The Conveyer Belt Font was prompted by a still-unsolved drawback posed in 2001 by the Spanish mathematician Manuel Abellanas: If there are a number of two-dimensional and nonoverlapping wheels, or disks, of equal measurement, can they all be wrapped (linked) with a taut conveyor belt, such that the belt touches all the wheels however doesn’t intersect itself?
The Demaines tried to unravel this drawback and acquired caught. They distracted themselves by designing the font. “That’s always been an important part of our philosophy,” Mr. Demaine mentioned. “If we get stuck on a problem, we like to find an artistic way to represent it.”
The Demaines additionally discover that puzzles are a pleasant solution to provoke newcomers into the enjoyable of formal arithmetic. The Checkers Font (in which the letters are fashioned from paths of leaping strikes) got here into being when Spencer Congero, a pc science graduate pupil on the University of California, San Diego, acquired in contact with the concept. The Spiral Galaxies Font (primarily based on the Japanese pencil-and-paper puzzle of the identical identify; distinctive options to puzzles type letters) was a collaboration with Walker Anderson, then a pupil at Central Bucks West High School in Doylestown, Pa., and a member of the USA World Puzzle Championship workforce.
The puzzle font was Mr. Anderson’s gateway to mathematical analysis; now he’s an undergrad learning math at M.I.T. For the Demaines, these kinds of collaborations are trigger for celebration: One extra individual efficiently “corrupted” into the world of theoretical laptop science.
Given their popularity with origami, the Demaines have naturally created a number of fonts riffing on the nuances of folding, together with the Origami Maze Font, the Simple Fold & Cut Font, the Fold & Punch Font and an Impossible Folding Font.
The Demaines additionally determined, for a change, to create a minimalistic font requiring solely a single fold.
Lest that simplicity make the unsolved font too easy to learn, they added a restriction: The letters should be illegible earlier than folding. Most of their typefaces, actually, are primarily based round related constraints. The Demaines wish to make the duty onerous however not preposterously so; they don’t need an excessive amount of freedom or flexibility, because the attract is within the problem, however they do need the duty to be attainable.
With these parameters, they devised the One-Fold Silhouette Font. The silhouette factor borrows from a 1900-era “Rabbit Silhouette Puzzle,” in which 5 playing cards with cutouts of assorted animals stack as much as produce the silhouette of a rabbit. The One-Fold Silhouette Font works in the same manner. Imagine a clear sheet, with black markings:
The central vertical crease invitations you to fold the sheet in half (from proper to left, as in the event you had been turning the web page of a ebook).
And shock, the textual content is revealed!
With the Strip-Folding Font, a sequence of letters is folded from a protracted strip of paper — the constraint right here was that each letter needed to be foldable utilizing solely horizontal, vertical and diagonal folds.
Last fall, the Demaines revealed their Tetris Font, which is a continuation of their research into the computational complexity of the enduring falling-block online game. (In 2002 Erik Demaine was conferred the title of “Tetris Master” by the Harvard Tetris Society, in honor of his “intellectual contribution to the art of Tetris,” for a foundational paper, “Tetris Is Hard, Even to Approximate.” )
The upshot of the new result’s this: They have proved, in enjoying the offline model of Tetris (whereby the participant has full data upfront concerning the id and order of items that will drop) that the sport is “NP-complete” — which means that no environment friendly answer algorithm exists, even with as few as eight columns or 4 rows. And extra virtually, as Dr. Demaine described on his web site, NP-completeness means “it’s computationally intractable to figure out whether you can survive, or clear the board, given an initial board configuration and a sequence of n pieces to come.”
Initially, the inventive constraint for this font was that every letter be constructed as a stacking of 1 copy of all seven Tetris shapes. Then the Demaines realized it might be neat to animate the font, with letters dropping into formation like items within the recreation — so that each bit positioned additionally needed to be supported by the earlier items, with no excessive overhangs, thus obeying “Tetris physics.” This necessitated a little bit of redesign, typically with the assistance of a pc instrument (“BurrTools”) that assembled desired shapes from primary unit items.
“When us humans got stuck finding a good solution, we’d put some of the shapes we’d been trying into BurrTools, and it would help guide our search,” Dr. Demaine mentioned. “Q” and “M” had been among the many final letters to fall into place.
Finally, attempt to fathom Everything Font, additionally simply launched as we speak into the wild. It was impressed by these eye charts with “Es” on each line. In the mathematical font context, the letter “E” is what’s known as a “canonical form” — each letter of the alphabet could be folded into an “E,” and “E” in flip could be folded into each letter. Which means, in the end, that each letter can fold into each other letter. (A pure canonical type for protein chains, which fold into numerous shapes, is the helix.)
So, had this text been written in Everything Font — with every letter bearing a crease sample (folding directions) for an additional letter — there can be one other article encoded herein.