The Crossed Diagonals

The last afternoon of term, a paper balloon sat on the classroom windowsill, gone soft in the heat, one flat side sagging where the air had left it. Someone had folded it in the spring and forgotten it there. It still held its shape well enough that you could see what it had been.

A child picked it up on the way out, put the little hole to her mouth, and blew. The paper snapped back into a cube. That sound — a dry pop of a sheet remembering its own corners — is older than anyone in the room, older than the school, older than most of what we call tradition in paper.

The Crossed Diagonals is the base under that balloon. Nobody knows who made it first. It appears in Japanese practice, in European classrooms, in the hands of people who never thought of themselves as folders at all, and it moves the way songs move: mouth to mouth, hand to hand, losing its author on the way. The pattern is common property. What we made here are the measurements, the drawn steps, the story, the music, and the recording.

Two bases from one drawing

Fold a square along both diagonals and both midlines, and you have a decision to make rather than a shape. The same set of lines gives you the preliminary base or the water bomb base, depending only on which creases go up and which go down. Turn one inside out and it becomes the other. A crane and a balloon begin from the same drawing, separated by a choice about direction.

This is why the geometry matters less than it looks. Kawasaki's and Maekawa's conditions tell you whether a flat state exists; they do not tell you which of the two it will be. That part is a matter of pushing the sides in with your thumbs and feeling the sheet decide.

There is also a physical honesty in this family. The paper cannot close as a set of stiff plates hinged together — it has to bow slightly on the way, and it does, which is exactly why the finished thing can hold air. Because of that bowing we did not measure how tall it stands or how far it travels; a moving model would have shown a lie. The page gives you drawings, step by step, in the order a hand works.

What the square holds

Sixteen creases. Three mountains, five valleys, and eight boundary edges around the square. Nine vertices, of which exactly one falls in the middle, where the folds cross.

Every angle at that crossing measures 45°, and the ratio check runs from 45.0° to 45.0° — no variation at all, which is unusual and pleasant to see written down. The pattern reaches a flat state at its interior crossing, and it passed the five checks we run on everything; that is all we will say about the checks.

What remains is character. Few creases, no clever counting, a symmetry a child can hold in her head after one showing — and inside it, a cube of air.

Which is what she was carrying home when the bell went.