Yoshimura pattern · verified flat-foldable
The Shared Root
This family is not rigid-foldable — the paper has to bend on the way, so there is no continuous 3D motion to show. Step diagrams are the honest way, and the way every origami book does it.
Open the diagram sheet →How to fold it
- Print the grid sheet at 100% and cut cleanly along its outer border
- Before folding anything, score every one of the 39 mountain and 21 valley lines with a ruler and an empty ballpoint
- Accordion the orange mountain creases first, working across the grid strip by strip
- Fold the closed strip along the blue dashed valley lines, alternating direction with each one
- Open the sheet flat and reverse any crease that fights its printed colour
- Open and close the pattern a few more times so the paper settles into its fold.
Pre-creasing every line before you collapse is the whole trick. Paper remembers.
The story below, read word for word by a synthetic voice over this fold's own music.
An aluminium can goes under a shoe and the sides do not collapse into noise. They collapse into diamonds — rows of them, tilted, meeting edge to edge, as if the metal had consulted a drawing first. You did not choose that shape. Neither did the can.
Roll a sheet of paper into a tube and squeeze it lengthwise and you get the same answer, softer. This is the fold called the Yoshimura pattern, and part of its interest is that nobody had to invent it in order for it to exist.
What compression already knew
In 1955, Yoshinobu Yoshimura described the buckling behaviour of cylindrical shells: under axial load, a thin tube does not fail randomly but settles into a repeating lattice of tilted diamonds. The pattern took his name in the engineering literature. In paper, it is older and quieter — a traditional fold with no author attached, the kind that turns up in workshops, in lampshades, in the pleated skirt of a paper bag, passed hand to hand without a signature.
So the attribution here is honest and slightly unusual. The design belongs to nobody. It is traditional and in the public domain, and it was arrived at independently by paper, by metal, and by anyone who has ever stepped on a bottle. Foldsong's contribution to this page is the measurement, the drawn steps, the accompanying writing, and the music.
There is a practical consequence to the pattern's origin. It comes from a surface that is bending, not from panels hinged along stiff lines. This family cannot be folded as a set of rigid plates rotating about their creases; the paper between the lines must curve while the fold closes. That is why the page carries diagrams you work through by hand rather than a moving model — a simulation of stiff panels would simply lock, and would be lying about what your fingers do.
The measured sheet
The sheet we measured holds 80 creases across 33 vertices, of which 15 sit in the interior, away from the edge. The assignment runs 39 mountain, 21 valley, and 20 boundary. All 15 interior points satisfy flat-foldability, checked against Kawasaki and Maekawa; the remaining checks passed without incident and are not the story.
The character of the fold lives in two angles: 58° and 64°. Between those two values the whole diamond field is decided — how tall the cells stand, how steeply the rows lean, how much circumference disappears when you press the tube shut. Our ratio check ran 58.0° to 64.0°, a narrow band, and the narrowness is the point. Push the angles apart and the lattice stops nesting; the diamonds argue instead of stacking.
Folded, the sheet behaves less like a model and more like a material. It wants to be a cylinder. Release it and it springs partway open, holding a memory of the tube it was cut to be, which is exactly the behaviour Yoshimura was writing about when he was studying failure rather than making anything.
Every crushed can in the recycling bin is a copy of a fold that no one owns.