Dissection Font

by Erik Demaine, Martin Demaine, Donald E. Knuth, Yushi Uno, 2018


In these fonts, each letter or digit or ampersand can be dissected (cut into pieces such that those pieces re-arrange) into a 6 × 6 square. The dissections all happen to be polyomino dissections, and they allow translation, rotation, and reflection in the piece re-arrangement. There are four different fonts, each using up to 2, 3, or 4 pieces in each dissection. Of course, with more pieces, it is easier to get nicer-looking letters. The 2- and 4-piece disconnected fonts use some disconnected pieces (but still each piece moves as a single unit), while the 2- and 3-piece connected fonts use connected pieces. In the disconnected fonts, we achieve uniform letter heights.

Origin. The 3- and 4-piece font come from a draft of The Art of Computer Programming, volume 4, pre-fascicle 9B “A Potpourri of Puzzles”, where Knuth poses (and solves) two exercises in mathematical/puzzle font design. See Knuth's December 2018 profile in the New York Times. These fonts were originally presented at Knuth's 80th Birthday Party in January 2018. The 2-piece disconnected font was released in May 2019.

Font forms. Each font can be presented in a fully solved form (“Dissection in both forms”) or in a variety of puzzle forms. “Letter without dissection” is the hardest form: the puzzle for each letter is to find a dissection with the specified number of pieces into a 6 × 6 square. In “Dissection in form of letter”, the puzzle is to find the re-arrangement of the letter into the square (a relatively easy puzzle). In “Dissection in form of square”, the puzzle is to figure out which letter the pieces can re-arrange into; this form is a puzzle font in the sense that reading the message requires solving the puzzle.

Related mathematics. It has been known since 1814 that every two polygons of the same area have a dissection (and even a hinged dissection), but finding a dissection with the fewest possible pieces (or a specified number k of pieces) is strongly NP-complete.

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