These Typefaces Are Actually Puzzling
The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of mysterious origin. “That variety of suits,” stated Martin Demaine, an artist in residence at the Massachusetts Institute of Technological innovation. “It’s a puzzle in which the phrase ‘puzzle’ comes from.”
His son, Erik Demaine, an M.I.T. laptop or computer scientist, agreed. “It’s a self-describing etymology,” he explained.
The father-son duo is most famed 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 classic specimen begins as a circular piece of paper, which, when folded together concentric circles, routinely twists into a saddle curve. The Demaines’ trio of pieces, “Computational Origami,” was portion of the 2008 “Design and the Elastic Mind” show at the Museum of Contemporary Artwork in New York and now resides in its long term selection.
These times, however, the Demaines are extra targeted on “algorithmic puzzle fonts,” a suite of mathematically impressed typefaces that are also puzzles. The major software is pleasurable. One particular font, a homage to the mathematician and juggler Ron Graham, who died in 2020, draws its letters from the patterns of movement traced by balls thrown into the air throughout juggling tricks.
A further font, proposed by the computer system scientist Donald Knuth (practically all fonts contain collaborators), has as its distinguishing attribute that all letters can be “dissected” — slash into parts and rearranged — into a 6-by-6 square.
In a 2015 paper, “Exciting With Fonts: Algorithmic Typography,” the Demaines discussed their motivations: “Scientists use fonts just about every working day to convey their research as a result of the published term. But what if the font itself communicated (the spirit of) the exploration? What if the way text is composed, and not just the textual content by itself, engages the reader in the science?”
Influenced by theorems or open troubles, the fonts — and the messages they compose — can normally be study only soon after solving the linked puzzle or collection of puzzles.
Acquire, for instance, a new font in their selection that debuts today: the Sudoku Font. The inspiration came in the fall of 2019, when Erik Demaine co-taught the study course “Fundamentals of Programming” (with the personal computer scientist Srini Devadas). All through just one 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 whilst 50 %-paying awareness Mr. Demaine mused about no matter if it may well be possible to make a font dependent on Sudoku — that is, based mostly on the puzzles whose unique remedies would in some way expose letters of the alphabet.
Soon after taking part in close to with various opportunities, the Demaines made a Sudoku puzzle font that works as follows: To start with, begin with a person of their Sudoku puzzles and resolve it. Subsequent, draw a line connecting the longest path of squares with consecutive figures (ascending or descending but only edge-adjacent squares, not diagonal). That line draws the form of a letter within just the grid of the puzzle. A series of Sudokus therefore solved can expose a message, like so:
The total suite of puzzle fonts is readily available, with various degrees of interactivity, on Dr. Demaine’s web site The Demaines hand-created the letter designs, but utilised a laptop to produce the letter-embedding Sudoku puzzles.
“It was challenging to style and design letters that nonetheless enabled the puzzle to be solvable, and with no adding additional stray connections to the longest route,” Dr. Demaine explained. “This was pretty a tough font to design, both for the human and the laptop or computer.”
Math + art = fun
The Demaines started this puzzle-font experiganza all-around the switch of the century with a dissection puzzle — a puzzle whereby just one form, or polygon, is sliced up and reassembled into other geometric styles. Their inspiration was a issue posed in 1964 by Harry Lindgren, a British-Australian engineer and amateur mathematician: Can every letter of the alphabet be dissected into parts that rearrange to sort a sq.?
In 2003, setting up on former perform, the Demaines proved that, of course, without a doubt it was achievable, and they revealed the result. (Normally, a puzzle font will come with a corresponding exploration paper.) This very first foray was a puzzle only in the feeling that the Demaines had been perplexed for a though about how to style the font. And they built the obstacle extra puzzling by introducing an additional criterion: They wished not basically a dissection font, but a “hinged dissection” — a exclusive form of dissection whereby the pieces are connected (hinged) at their vertexes, forming a shut chain that rearranges, in this situation not only into the sought after square but also into every single other letter of the alphabet.
They succeeded in their quest by deploying the mathematics of “polyforms,” shapes built from a number of copies of a polygon, this sort of as a triangle. Additional precisely, they utilized a polyform with the improbable identify “polyabolo” (popularized by Martin Gardner, who was a arithmetic columnist for Scientific American). A polyabolo is made from congruent ideal isosceles triangles. A square can be cut into two right isosceles triangles and those people two triangles can in flip be minimize into four ideal isosceles triangles, and these four triangles into 8, and these 8 into 16, 16 into 32, 32 into 64, 64 into 128, and so forth.
By this process, the Demaines produced their Dissection Font. Every 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 acquiring the ideal hinged dissection — a related chain of triangles that can morph from just one letter into any other — needed that just about every letter be dissected into 128 triangular parts (creating it a “128-abolo”).
Reflecting on this work out in an electronic mail, the Demaines claimed: “The pleasurable for us was combining art and math with each other, aiming for excellent layout (recognizable as letters and on the lookout steady throughout the alphabet) in just difficult mathematical constraints (mounted space and doing the job with polyabolo designs).”
The philosophy of getting trapped
20 yrs on, these humble beginnings have sprouted into a incredible entertaining home of fonts, with artistic media as assorted as rods of glass, string artwork and coins.
Think about the Tiling Font: Every single letter “tiles the aircraft,” which means, as the Demaines explain, “that infinitely numerous copies of that one shape can fill two proportions with no leaving any gaps among the tiles.” Perfect for a toilet renovation.
With the Conveyer Belt Font, just about every letter is shaped by the closed loop of a conveyor belt that curves around strategically put wheels. (The font name is intentionally spelled “conveyer” instead than “conveyor,” as the font “conveys” letters and words and phrases.)
The Conveyer Belt Font was prompted by a still-unsolved issue posed in 2001 by the Spanish mathematician Manuel Abellanas: If there are several two-dimensional and nonoverlapping wheels, or disks, of equal measurement, can they all be wrapped (related) with a taut conveyor belt, these types of that the belt touches all the wheels but doesn’t intersect by itself?
The Demaines tried using to resolve this trouble and bought trapped. They distracted themselves by designing the font. “That’s constantly been an crucial part of our philosophy,” Mr. Demaine claimed. “If we get stuck on a problem, we like to come across an artistic way to stand for it.”
The Demaines also obtain that puzzles are a nice way to initiate newcomers into the enjoyment of formal arithmetic. The Checkers Font (in which the letters are formed from paths of leaping moves) arrived into currently being when Spencer Congero, a pc science graduate student at the University of California, San Diego, bought in touch with the plan. The Spiral Galaxies Font (dependent on the Japanese pencil-and-paper puzzle of the same name unique solutions to puzzles sort letters) was a collaboration with Walker Anderson, then a scholar at Central Bucks West Large College in Doylestown, Pa., and a member of the United states of america World Puzzle Championship team.
The puzzle font was Mr. Anderson’s gateway to mathematical exploration now he’s an undergrad learning math at M.I.T. For the Demaines, these types of collaborations are induce for celebration: 1 additional particular person productively “corrupted” into the planet of theoretical computer science.
Creative constraint
Specified their track record with origami, the Demaines have in a natural way designed a couple fonts riffing on the nuances of folding, which include the Origami Maze Font, the Easy Fold & Lower Font, the Fold & Punch Font and an Difficult Folding Font.
The Demaines also made the decision, for a adjust, to produce a minimalistic font demanding only a one fold.
Lest that simplicity make the unsolved font far too very simple to browse, they additional a restriction: The letters have to be illegible right before folding. Most of their typefaces, in fact, are dependent close to comparable constraints. The Demaines like to make the activity hard but not preposterously so they really don’t want far too a great deal flexibility or adaptability, due to the fact the allure is in the challenge, but they do want the task to be attainable.
With these parameters, they devised the One particular-Fold Silhouette Font. The silhouette element borrows from a 1900-era “Rabbit Silhouette Puzzle,” in which five cards with cutouts of numerous animals stack up to make the silhouette of a rabbit. The A person-Fold Silhouette Font works in a related way. Picture a transparent sheet, with black markings:
The central vertical crease invites you to fold the sheet in 50 % (from right to remaining, as if you had been turning the webpage of a reserve).
And surprise, the text is uncovered!
With the Strip-Folding Font, a sequence of letters is folded from a extensive strip of paper — the constraint right here was that each individual letter had to be foldable working with only horizontal, vertical and diagonal folds.
Font physics
Final tumble, the Demaines released their Tetris Font, which is a continuation of their experiments into the computational complexity of the iconic slipping-block online video recreation. (In 2002 Erik Demaine was conferred the title of “Tetris Grasp” by the Harvard Tetris Culture, in honor of his “intellectual contribution to the art of Tetris,” for a foundational paper, “Tetris Is Difficult, Even to Approximate.” )
The upshot of the new final result is this: They have proved, in playing the offline model of Tetris (whereby the participant has full information and facts in progress about the identification and get of items that will drop) that the video game is “NP-complete” — that means that no effective resolution algorithm exists, even with as several as 8 columns or four rows. And extra pretty much, as Dr. Demaine explained on his site, NP-completeness indicates “it’s computationally intractable to figure out no matter if you can endure, or obvious the board, offered an preliminary board configuration and a sequence of n pieces to come.”
In the beginning, the resourceful constraint for this font was that each and every letter be built as a stacking of a single copy of all 7 Tetris styles. Then the Demaines realized it would be neat to animate the font, with letters dropping into development like items in the game — so that just about every piece put also had to be supported by the earlier items, with no excessive overhangs, therefore obeying “Tetris physics.” This necessitated a bit of redesign, sometimes with the help of a pc device (“BurrTools”) that assembled ideal styles from fundamental unit items.
“When us humans received trapped discovering a great alternative, we’d put some of the shapes we’d been hoping into BurrTools, and it would help tutorial our lookup,” Dr. Demaine mentioned. “Q” and “M” were being among the the last letters to slide into area.
Last but not least, attempt to fathom All the things Font, also just launched these days into the wild. It was inspired by these eye charts with “Es” on just about every line. In the mathematical font context, the letter “E” is what is identified as a “canonical form” — just about every letter of the alphabet can be folded into an “E,” and “E” in switch can be folded into every single letter. Which means, in the end, that each letter can fold into each and every other letter. (A normal canonical kind for protein chains, which fold into a variety of shapes, is the helix.)
So, had this posting been penned in Everything Font — with every letter bearing a crease sample (folding directions) for an additional letter — there would be a different short article encoded herein.
The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of mysterious origin. “That variety of suits,” stated Martin Demaine, an artist in residence at the Massachusetts Institute of Technological innovation. “It’s a puzzle in which the phrase ‘puzzle’ comes from.”
His son, Erik Demaine, an M.I.T. laptop or computer scientist, agreed. “It’s a self-describing etymology,” he explained.
The father-son duo is most famed 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 classic specimen begins as a circular piece of paper, which, when folded together concentric circles, routinely twists into a saddle curve. The Demaines’ trio of pieces, “Computational Origami,” was portion of the 2008 “Design and the Elastic Mind” show at the Museum of Contemporary Artwork in New York and now resides in its long term selection.
These times, however, the Demaines are extra targeted on “algorithmic puzzle fonts,” a suite of mathematically impressed typefaces that are also puzzles. The major software is pleasurable. One particular font, a homage to the mathematician and juggler Ron Graham, who died in 2020, draws its letters from the patterns of movement traced by balls thrown into the air throughout juggling tricks.
A further font, proposed by the computer system scientist Donald Knuth (practically all fonts contain collaborators), has as its distinguishing attribute that all letters can be “dissected” — slash into parts and rearranged — into a 6-by-6 square.
In a 2015 paper, “Exciting With Fonts: Algorithmic Typography,” the Demaines discussed their motivations: “Scientists use fonts just about every working day to convey their research as a result of the published term. But what if the font itself communicated (the spirit of) the exploration? What if the way text is composed, and not just the textual content by itself, engages the reader in the science?”
Influenced by theorems or open troubles, the fonts — and the messages they compose — can normally be study only soon after solving the linked puzzle or collection of puzzles.
Acquire, for instance, a new font in their selection that debuts today: the Sudoku Font. The inspiration came in the fall of 2019, when Erik Demaine co-taught the study course “Fundamentals of Programming” (with the personal computer scientist Srini Devadas). All through just one 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 whilst 50 %-paying awareness Mr. Demaine mused about no matter if it may well be possible to make a font dependent on Sudoku — that is, based mostly on the puzzles whose unique remedies would in some way expose letters of the alphabet.
Soon after taking part in close to with various opportunities, the Demaines made a Sudoku puzzle font that works as follows: To start with, begin with a person of their Sudoku puzzles and resolve it. Subsequent, draw a line connecting the longest path of squares with consecutive figures (ascending or descending but only edge-adjacent squares, not diagonal). That line draws the form of a letter within just the grid of the puzzle. A series of Sudokus therefore solved can expose a message, like so:
The total suite of puzzle fonts is readily available, with various degrees of interactivity, on Dr. Demaine’s web site The Demaines hand-created the letter designs, but utilised a laptop to produce the letter-embedding Sudoku puzzles.
“It was challenging to style and design letters that nonetheless enabled the puzzle to be solvable, and with no adding additional stray connections to the longest route,” Dr. Demaine explained. “This was pretty a tough font to design, both for the human and the laptop or computer.”
Math + art = fun
The Demaines started this puzzle-font experiganza all-around the switch of the century with a dissection puzzle — a puzzle whereby just one form, or polygon, is sliced up and reassembled into other geometric styles. Their inspiration was a issue posed in 1964 by Harry Lindgren, a British-Australian engineer and amateur mathematician: Can every letter of the alphabet be dissected into parts that rearrange to sort a sq.?
In 2003, setting up on former perform, the Demaines proved that, of course, without a doubt it was achievable, and they revealed the result. (Normally, a puzzle font will come with a corresponding exploration paper.) This very first foray was a puzzle only in the feeling that the Demaines had been perplexed for a though about how to style the font. And they built the obstacle extra puzzling by introducing an additional criterion: They wished not basically a dissection font, but a “hinged dissection” — a exclusive form of dissection whereby the pieces are connected (hinged) at their vertexes, forming a shut chain that rearranges, in this situation not only into the sought after square but also into every single other letter of the alphabet.
They succeeded in their quest by deploying the mathematics of “polyforms,” shapes built from a number of copies of a polygon, this sort of as a triangle. Additional precisely, they utilized a polyform with the improbable identify “polyabolo” (popularized by Martin Gardner, who was a arithmetic columnist for Scientific American). A polyabolo is made from congruent ideal isosceles triangles. A square can be cut into two right isosceles triangles and those people two triangles can in flip be minimize into four ideal isosceles triangles, and these four triangles into 8, and these 8 into 16, 16 into 32, 32 into 64, 64 into 128, and so forth.
By this process, the Demaines produced their Dissection Font. Every 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 acquiring the ideal hinged dissection — a related chain of triangles that can morph from just one letter into any other — needed that just about every letter be dissected into 128 triangular parts (creating it a “128-abolo”).
Reflecting on this work out in an electronic mail, the Demaines claimed: “The pleasurable for us was combining art and math with each other, aiming for excellent layout (recognizable as letters and on the lookout steady throughout the alphabet) in just difficult mathematical constraints (mounted space and doing the job with polyabolo designs).”
The philosophy of getting trapped
20 yrs on, these humble beginnings have sprouted into a incredible entertaining home of fonts, with artistic media as assorted as rods of glass, string artwork and coins.
Think about the Tiling Font: Every single letter “tiles the aircraft,” which means, as the Demaines explain, “that infinitely numerous copies of that one shape can fill two proportions with no leaving any gaps among the tiles.” Perfect for a toilet renovation.
With the Conveyer Belt Font, just about every letter is shaped by the closed loop of a conveyor belt that curves around strategically put wheels. (The font name is intentionally spelled “conveyer” instead than “conveyor,” as the font “conveys” letters and words and phrases.)
The Conveyer Belt Font was prompted by a still-unsolved issue posed in 2001 by the Spanish mathematician Manuel Abellanas: If there are several two-dimensional and nonoverlapping wheels, or disks, of equal measurement, can they all be wrapped (related) with a taut conveyor belt, these types of that the belt touches all the wheels but doesn’t intersect by itself?
The Demaines tried using to resolve this trouble and bought trapped. They distracted themselves by designing the font. “That’s constantly been an crucial part of our philosophy,” Mr. Demaine claimed. “If we get stuck on a problem, we like to come across an artistic way to stand for it.”
The Demaines also obtain that puzzles are a nice way to initiate newcomers into the enjoyment of formal arithmetic. The Checkers Font (in which the letters are formed from paths of leaping moves) arrived into currently being when Spencer Congero, a pc science graduate student at the University of California, San Diego, bought in touch with the plan. The Spiral Galaxies Font (dependent on the Japanese pencil-and-paper puzzle of the same name unique solutions to puzzles sort letters) was a collaboration with Walker Anderson, then a scholar at Central Bucks West Large College in Doylestown, Pa., and a member of the United states of america World Puzzle Championship team.
The puzzle font was Mr. Anderson’s gateway to mathematical exploration now he’s an undergrad learning math at M.I.T. For the Demaines, these types of collaborations are induce for celebration: 1 additional particular person productively “corrupted” into the planet of theoretical computer science.
Creative constraint
Specified their track record with origami, the Demaines have in a natural way designed a couple fonts riffing on the nuances of folding, which include the Origami Maze Font, the Easy Fold & Lower Font, the Fold & Punch Font and an Difficult Folding Font.
The Demaines also made the decision, for a adjust, to produce a minimalistic font demanding only a one fold.
Lest that simplicity make the unsolved font far too very simple to browse, they additional a restriction: The letters have to be illegible right before folding. Most of their typefaces, in fact, are dependent close to comparable constraints. The Demaines like to make the activity hard but not preposterously so they really don’t want far too a great deal flexibility or adaptability, due to the fact the allure is in the challenge, but they do want the task to be attainable.
With these parameters, they devised the One particular-Fold Silhouette Font. The silhouette element borrows from a 1900-era “Rabbit Silhouette Puzzle,” in which five cards with cutouts of numerous animals stack up to make the silhouette of a rabbit. The A person-Fold Silhouette Font works in a related way. Picture a transparent sheet, with black markings:
The central vertical crease invites you to fold the sheet in 50 % (from right to remaining, as if you had been turning the webpage of a reserve).
And surprise, the text is uncovered!
With the Strip-Folding Font, a sequence of letters is folded from a extensive strip of paper — the constraint right here was that each individual letter had to be foldable working with only horizontal, vertical and diagonal folds.
Font physics
Final tumble, the Demaines released their Tetris Font, which is a continuation of their experiments into the computational complexity of the iconic slipping-block online video recreation. (In 2002 Erik Demaine was conferred the title of “Tetris Grasp” by the Harvard Tetris Culture, in honor of his “intellectual contribution to the art of Tetris,” for a foundational paper, “Tetris Is Difficult, Even to Approximate.” )
The upshot of the new final result is this: They have proved, in playing the offline model of Tetris (whereby the participant has full information and facts in progress about the identification and get of items that will drop) that the video game is “NP-complete” — that means that no effective resolution algorithm exists, even with as several as 8 columns or four rows. And extra pretty much, as Dr. Demaine explained on his site, NP-completeness indicates “it’s computationally intractable to figure out no matter if you can endure, or obvious the board, offered an preliminary board configuration and a sequence of n pieces to come.”
In the beginning, the resourceful constraint for this font was that each and every letter be built as a stacking of a single copy of all 7 Tetris styles. Then the Demaines realized it would be neat to animate the font, with letters dropping into development like items in the game — so that just about every piece put also had to be supported by the earlier items, with no excessive overhangs, therefore obeying “Tetris physics.” This necessitated a bit of redesign, sometimes with the help of a pc device (“BurrTools”) that assembled ideal styles from fundamental unit items.
“When us humans received trapped discovering a great alternative, we’d put some of the shapes we’d been hoping into BurrTools, and it would help tutorial our lookup,” Dr. Demaine mentioned. “Q” and “M” were being among the the last letters to slide into area.
Last but not least, attempt to fathom All the things Font, also just launched these days into the wild. It was inspired by these eye charts with “Es” on just about every line. In the mathematical font context, the letter “E” is what is identified as a “canonical form” — just about every letter of the alphabet can be folded into an “E,” and “E” in switch can be folded into every single letter. Which means, in the end, that each letter can fold into each and every other letter. (A normal canonical kind for protein chains, which fold into a variety of shapes, is the helix.)
So, had this posting been penned in Everything Font — with every letter bearing a crease sample (folding directions) for an additional letter — there would be a different short article encoded herein.