Everything Font

by Erik Demaine and Martin Demaine, 2021

Everything to everything. This typeface illustrates how to fold any letter into any other letter, or more precisely, how to fold a piece of paper in the shape of any letter into the shape of any other letter. This lets you write one message inside another in a couple of ways. On the one hand, you could present the 6×6 crease patterns whose silhouettes look like one message (first text), and folding them reveals another message (second text). On the other hand, you could present the folded forms (as physical objects) whose silhouettes look like one message (second text), and unfolding them reveals another message (first text).

Central hub. At first glance, this typeface might require the design of 262 = 676 different foldings. Luckily, we can use a mathematical technique for proving that you can transform any shape into any shape: show how to transform any (letter) shape into a single “central” shape, and show how to transform that central shape into any (letter) shape. (This technique is sometimes called the “FedEx technique”, because one of the initial principles of FedEx was to ship all packages to a central hub and then ship them out to their destinations.) For the font, we chose the 5×5 letter E as the central shape, inspired by an early eye chart called the tumbling E chart. Thus we only had to design foldings from every letter (in a 6×6 size) into the 5×5 E, and from the 5×5 E to every letter (in a smaller size), for a total of 26×2 = 52 different designs. This software then composes pairs of crease patterns (using our FOLD software) to form the 676 possible different foldings.

Tumbling E charts. Another interesting font in the typeface is to just input a second text, and fold every letter from the 5×5 letter E. If we just display the E crease pattern (and optionally add some rotation), we obtain what might look like a tumbling E chart. Similarly, if we just input a first text, we could fold what looks like a tumbling E chart that, when unfolded, reveals a secret message. If you turn on Tumbling E Chart mode and are displaying the 5×5 Intermediate Es, they will be rotated randomly. You can reroll the randomness by clicking the Roll button.

Usage. You can turn on or off the various stages of folding with the three main check boxes. • Assuming you've input some first text, 6×6 Crease Pattern shows the original 6×6 letter with the full crease pattern drawn dashed on top. If you've input some second text, the crease pattern will show how to fold that text, with darker lines representing the first step to make an E and lighter lines showing the second step; otherwise, the crease pattern will just show how to fold an E in darker lines. • 5×5 Intermediate E shows the intermediate E shape. If you've input some first text, this will be shown as a partially folded form of the 6×6, with more intense yellow for multiple layers and dark-colored bold edges indicating folded creases. If you've input some second text, the E will have the (remaining) creases to fold into that text. • Folded Form shows the result of folding the 5×5 E into second text (if specified). • You can turn off all but one of these stages by clicking one of the Alone buttons. • You can also Swap (⟷) the first and second texts.

Efficiency. In principle, any polygon can fold into any other polygon at some scale factor, but it remains a challenging open problem to find an efficient folding, where the target polygon can be scaled as large as possible. In our designs, the final shape has height approximately 5, as tall as the intermediate E shape. To achieve this property and the original letter size of 6×6, the starting shape for a letter and the target shape for the same letter are not the same shape — the shapes are designed carefully to enable efficient folding. As a result, even folding a letter into itself is interesting.

Check out other mathematical and puzzle fonts. • Feedback or not working? Email Erik.