The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of unknown origin. “That type of matches,” mentioned Martin Demaine, an artist in residence on the Massachusetts Institute of Expertise. “It’s a puzzle the place the phrase ‘puzzle’ comes from.”
His son, Erik Demaine, an M.I.T. laptop scientist, agreed. “It’s a self-describing etymology,” he mentioned.
The daddy-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 Nineteen Twenties Bauhaus; a basic specimen begins as a round piece of paper, which, when folded alongside concentric circles, routinely twists right into a saddle curve. The Demaines’ trio of items, “Computational Origami,” was a part of the 2008 “Design and the Elastic Thoughts” exhibit on the Museum of Trendy Artwork in New York and now resides in its everlasting assortment.
Lately, nonetheless, the Demaines are extra targeted on “algorithmic puzzle fonts,” a collection of mathematically impressed typefaces which are additionally puzzles. The principle 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.
One other font, proposed by the pc scientist Donald Knuth (virtually all fonts contain collaborators), has as its distinguishing attribute that each one letters might be “dissected” — lower into items and rearranged — right into a 6-by-6 sq..
In a 2015 paper, “Enjoyable With Fonts: Algorithmic Typography,” the Demaines defined their motivations: “Scientists use fonts daily to specific their analysis by the written phrase. However what if the font itself communicated (the spirit of) the analysis? What if the way in which textual content is written, and never simply the textual content itself, engages the reader within the science?”
Impressed by theorems or open issues, the fonts — and the messages they compose — can normally be learn solely after fixing the associated puzzle or collection of puzzles.
Take, for example, a brand new font of their assortment that debuts in the present day: 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). Throughout 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 whereas half-paying consideration Mr. Demaine mused about whether or not it may be doable to make a font based mostly on Sudoku — that’s, based mostly on the puzzles whose distinctive options would in some way 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 considered one of their Sudoku puzzles and resolve it. Subsequent, 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 inside the grid of the puzzle. A collection of Sudokus thus solved can reveal a message, like so:
The complete suite of puzzle fonts is accessible, with various levels of interactivity, on Dr. Demaine’s web site. The Demaines hand-designed the letter shapes, however used a pc to generate the letter-embedding Sudoku puzzles.
“It was exhausting to design letters that also enabled the puzzle to be solvable, and with out including further stray connections to the longest path,” Dr. Demaine mentioned. “This was fairly a tough font to design, each for the human and the pc.”
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 different geometric shapes. Their motivation was an issue posed in 1964 by Harry Lindgren, a British-Australian engineer and novice mathematician: Can each letter of the alphabet be dissected into items that rearrange to kind a sq.?
In 2003, constructing on earlier work, the Demaines proved that, sure, certainly it was doable, they usually printed the outcome. (Usually, a puzzle font comes with a corresponding analysis paper.) This primary foray was a puzzle solely within the sense that the Demaines have been perplexed for some time about design the font. And so they made the problem extra puzzling by including an additional criterion: They needed not merely a dissection font, however a “hinged dissection” — a particular type of dissection whereby the items are related (hinged) at their vertexes, forming a closed chain that rearranges, on this case not solely into the specified sq. but in addition into each different letter of the alphabet.
They succeeded of their quest by deploying the arithmetic of “polyforms,” shapes made out of a number of copies of a polygon, corresponding to a triangle. Extra exactly, they used a polyform with the unbelievable title “polyabolo” (popularized by Martin Gardner, who was a arithmetic columnist for Scientific American). A polyabolo is made out of congruent proper isosceles triangles. A sq. might 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 Hinged-Dissection Font. Each letter of the alphabet is dissected into 32 triangles (rendering it a “32-abolo”) that may be rearranged right into a 4-by-4 sq., or some other letter. However attaining the specified hinged dissection — a related chain of triangles that may morph from one letter into some other — required that every letter be dissected into 128 triangular items (making it a “128-abolo”).
Reflecting on this train in an e-mail, the Demaines mentioned: “The enjoyable for us was combining artwork and math collectively, aiming for good design (recognizable as letters and looking out constant throughout the alphabet) inside exhausting mathematical constraints (mounted space and dealing with polyabolo shapes).”
The philosophy of getting caught
Twenty years on, these humble beginnings have sprouted into a superb enjoyable home of fonts, with inventive media as various as rods of glass, string artwork and cash.
Contemplate the Tiling Font: Every letter “tiles the aircraft,” which means, because the Demaines clarify, “that infinitely many copies of that one form can fill two dimensions with out leaving any gaps between the tiles.” Excellent for a loo renovation.
With the Conveyer Belt Font, every letter is shaped by the closed loop of a conveyor belt that curves round strategically positioned wheels. (The font title is deliberately spelled “conveyer” reasonably 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 all of them be wrapped (related) with a taut conveyor belt, such that the belt touches all of the wheels however doesn’t intersect itself?
The Demaines tried to resolve this drawback and acquired caught. They distracted themselves by designing the font. “That’s at all times been an necessary a part of our philosophy,” Mr. Demaine mentioned. “If we get caught on an issue, we like to search out a creative strategy to symbolize it.”
The Demaines additionally discover that puzzles are a pleasant strategy to provoke newcomers into the enjoyable of formal arithmetic. The Checkers Font (wherein the letters are shaped from paths of leaping strikes) got here into being when Spencer Congero, a pc science graduate pupil on the College of California, San Diego, acquired in contact with the concept. The Spiral Galaxies Font (based mostly on the Japanese pencil-and-paper puzzle of the identical title; distinctive options to puzzles kind letters) was a collaboration with Walker Anderson, then a pupil at Central Bucks West Excessive Faculty 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 types of collaborations are trigger for celebration: Another particular person efficiently “corrupted” into the world of theoretical laptop science.
Artistic constraint
Given their repute with origami, the Demaines have naturally created a number of fonts riffing on the nuances of folding, together with the Origami Maze Font, the Easy Fold & Reduce Font, the Fold & Punch Font and an Unattainable 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, the truth is, are based mostly round comparable constraints. The Demaines wish to make the duty exhausting 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 ingredient borrows from a 1900-era “Rabbit Silhouette Puzzle,” wherein 5 playing cards with cutouts of varied animals stack as much as produce the silhouette of a rabbit. The One-Fold Silhouette Font works in an analogous approach. Think about a clear sheet, with black markings:
The central vertical crease invitations you to fold the sheet in half (from proper to left, as for those who have been turning the web page of a e-book).
And shock, the textual content is revealed!
With the Strip-Folding Font, a sequence of letters is folded from an extended strip of paper — the constraint right here was that each letter needed to be foldable utilizing solely horizontal, vertical and diagonal folds.
Font physics
Final fall, the Demaines printed their Tetris Font, which is a continuation of their research into the computational complexity of the long-lasting falling-block online game. (In 2002 Erik Demaine was conferred the title of “Tetris Grasp” by the Harvard Tetris Society, in honor of his “mental contribution to the artwork of Tetris,” for a foundational paper, “Tetris Is Onerous, Even to Approximate.”)
The upshot of the brand new result’s this: They’ve proved, in enjoying the offline model of Tetris (whereby the participant has full data prematurely in regards to the id and order of items that can drop) that the sport is “NP-complete” — which means that no environment friendly resolution 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 determine whether or not you may survive, or clear the board, given an preliminary board configuration and a sequence of n items to return.”
Initially, the artistic 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 will be neat to animate the font, with letters dropping into formation like items within the sport — so that every piece 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 software (“BurrTools”) that assembled desired shapes from fundamental unit items.
“When us people acquired caught discovering a very good resolution, we’d put among the shapes we’d been attempting into BurrTools, and it will assist information our search,” Dr. Demaine mentioned. “Q” and “M” have been among the many final letters to fall into place.
Lastly, attempt to fathom Every part Font, additionally simply launched in the present day into the wild. It was impressed by these eye charts with “Es” on each line. Within the mathematical font context, the letter “E” is what’s referred to as a “canonical kind” — each letter of the alphabet might be folded into an “E,” and “E” in flip might be folded into each letter. Which suggests, in the end, that each letter can fold into each different letter. (A pure canonical kind for protein chains, which fold into numerous shapes, is the helix.)
So, had this text been written in Every part Font — with every letter bearing a crease sample (folding directions) for one more letter — there can be one other article encoded herein.
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