Mathematical and Puzzle Fonts/Typefaces

Erik Demaine, Martin Demaine, and others

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?

Below are mathematical typefaces we designed through inspiration which are inspired by mathematical theorems or open problems. Most include a puzzle font: reading them is itself a mathematical puzzle.

All fonts are available to play with as web applications which run entirely in your (modern) browser, including iOS and Android.

For more information, read our paper “Fun with Fonts: Algorithmic Typography” and other papers listed below, or check out media coverage in The New York Times, Science News (text available here), and Gizmodo.

• Letters, words, or text plotted by a finite sequence of integers.
• Self-plotting infinite integer sequence rejected by the OEIS.

• Each letter is made from a square of paper by a sequence of orthogonal folds followed by one straight cut.

• Each letter is an interactive Yin-Yang puzzle whose unique solution forms the image of the letter.

• Solve each Sudoku puzzle; then the longest path through consecutive adjacent numbers (differing by exactly 1 in edge-adjacent squares) draws each letter.

• Every letter-shaped piece of paper can be folded into every other letter (slightly smaller), via an intermediate "E" shape.

• Each letter tiles the infinite plane without gaps.

• Each letter is an interactive path puzzle whose unique solution forms the path of the letter.

• Each letter is an interactive Tatamibari puzzle whose unique solution forms the image of the letter.

• Each letter is built by dropping exactly one of each of the seven Tetris (tetromino) pieces in some sequence.

• Each letter is a puzzle: can you fold it into a 1 × 1 × 1 cube?

• Each letter is an impossible object called a hypercard: it can be cut and folded from a single square of paper, even though the vertical flap cannot be flattened into the horizontal plane without material overlapping.

• Each letter can be dissected into a 6 × 6 square.
• Three different fonts uses exactly 2, 3, or 4 pieces in each dissection.

• Each letter is made up of two simple symbols: a U shape and a line. Both symbols are in every letter, and every U shape has exactly the same proportions; the shapes are just rotated and/or scaled.
• Puzzle and animation fonts can be made by rotating the basic symbols.

• Each letter is an 8 × 8 cooperative always-jumping checkers puzzle. The solution sequence can be animated!
• Such puzzles are NP-complete in general, as proved in our paper “Losing at Checkers is Hard”.

• Each letter is a Spiral Galaxies puzzle whose unique solution forms the image of the letter.
• You can interactively solve the puzzles.

• Each letter is formed by folding two translucent symbols on top of each other, as in silhouette puzzles.

• Every pair of characters forms a coin-sliding puzzle, as analyzed by Demaine, Demaine, and Verrill (2000).
• You can play all 2,664 puzzles on the web or on Android and try to set a fewest-moves record!

• In one font (shown), each letter is a thread wrapped 1,500 times around 200 pins.
• In other fonts, each letter is a thread wrapped around just 22 pins, and involves puzzles related to Euler tours.

• An entire sequence of letters folds from a single strip of paper.

• Each letter is the Voronoi diagram of a set of points.

• Each letter from a square of paper by a sequence of simple folds followed by one circular hole-punch.

• Each letter is made from a square of paper by a sequence of simple folds followed by one straight cut.

• Each letter is a flat configuration of a Tangle of identical length, presented graphically or as a NSEW sequence.

• Each letter is a juggling pattern, animated by the Juggling Lab software or summarized by a single image of the ball trajectories.

• By a sequence of perfect shuffles of 26 cards labeled A through Z, the magician can present the letters of your message in order.

• Each letter is a glass cane, a cylinder of glass made by pulling and twisting an arrangement of straight lines of colored glass embedded within clear glass, as simulated by our Virtual Glass software.

• Each letter/number is a fixed-angle linkage (which model protein folding) designed so that random configurations can be uniquely decoded back to text.

• Hot glass components are arranged (in the puzzle font) so that squishing them horizontally produces a letter (in the clear font)
• Video font illustrates physical squishing to form the letters

• The font can be folded as extruded letters from a rectangle of paper, using an algorithm for folding orthogonal graphs by the same authors.

• The form without the belts (lines) is a puzzle to decode, based on a mathematical open problem by Manuel Abellanas first posed in 2001.

• One hinged chain of (128) pieces folds into any of the letters in the alphabet as well as a square.

Other Related Fonts

Here are some other fonts in the same genre as the mathematical/puzzle fonts above, but by other people.

• Each letter is a reachable pattern in the game of Peg Solitaire from an initially full 5 × 7 board.
• The puzzle is to figure out how to reach each such pattern!

• Each letter is the silhouette of a flat origami folded from a 3 × 5 box-pleated grid.
• Based on an enumeration of all such silhouettes by Matsukuwa, Yamamoto, and Mitani, which appears at ICGG 2016.

Papers

• “Orthogonal Fold & Cut”, arXiv:2202.01293, 2022.
• “Universal Hinge Patterns for Folding Strips Efficiently into Any Grid Polyhedron”, Computational Geometry: Theory and Applications, 89:101633, 2020.
• “Impossible Folding Font”, in Proceedings of 22nd Annual Conference of BRIDGES: Mathematics, Music, Art, Architecture, Culture (BRIDGES 2019), 2019.
• “2,664 Coin-Sliding Font Puzzles”, Exchange Book of the 13th Gathering for Gardner, April 2018.
• “Spiral Galaxies Font”, MOVES 2017: The Mathematics of Various Entertaining Subjects, volume 3.
• “Losing at Checkers is Hard”, MOVES 2017: Mathematics of Various Entertaining Subjects, volume 3.
• “Folding and Punching Paper”, Journal of Information Processing, 25:590–600, 2017.
• “Juggling and Card Shuffling Meet Mathematical Fonts”, Connections in Discrete Mathematics: In Honor of Ron Graham's 80th Birthday, 2018.
• “Fun with Fonts: Algorithmic Typography”, Theoretical Computer Science, 586:111–119, 2015.
• “Tangled Tangles”, MOVES 2015: Mathematics of Various Entertaining Subjects, volume 2.
• “Linkage Font”, in Exchange Book of the 11th Gathering for Gardner (G4G11), March 2014.
• “Origami Maze Puzzle Font”, in Exchange Book of the 9th Gathering for Gardner (G4G9), 2010.
• “Conveyer Belt Alphabet Font”, in Exchange Book of the 9th Gathering for Gardner (G4G9), 2010.
• “Conveyer-Belt Alphabet”, in Findings on Elasticity, 2010.
• “Hinged Dissection of the Alphabet”, Journal of Recreational Mathematics, 31(3):204–207, 2003.