The Yoshimura pattern
Fold it
The sheet is on the printed page, at the size it says.
Printing this page gives the page, and adds one more: the pattern alone, at 170 mm across, on a sheet of its own. Cut a square that size, transfer or trace the lines, and fold. The size is repeated in the corner of the sheet, because a printer set to fit to page rescales silently and there has to be some way to notice.
Mountain and valley are distinguished by dash as well as by colour, so the sheet survives the monochrome laser printer it will almost certainly come out of.
What it is
| Provenance | Yoshimura, 1955 — published as buckling analysis |
|---|---|
| Creases | 26 mountain, 60 valley |
| Interior vertices | 22, every one checked |
| Panels | 65, read off the pattern |
| Folding to do | about 2380 mm of crease at this size |
| Printed sheet | 170 mm across |
What was checked
Four theorems at every interior vertex, and the faces two ways.
- Developability — the sectors around each of the 22 interior vertices sum to a full turn, so the sheet was flat before it was creased.
- Kawasaki — alternating sectors sum to a straight angle at each of them.
- Maekawa — mountains and valleys differ by exactly two.
- Big-little-big — no strictly smallest sector is flanked by two creases of the same letter.
- The faces — 65 of them, found by walking the planarised graph, and checked against Euler's formula and against the area they cover. A face walk that goes the wrong way round or merges two faces usually still satisfies Euler; it does not conserve area.
None of this decides whether the whole sheet folds flat, which is NP-hard in general. Every local condition holds. That is a different and weaker statement, and it is the one being made.
Take it away
The field's own interchange format, so the pattern is reusable outside this site.
yoshimura.fold — 45 vertices, 109 edges, 65 faces, 8240 bytes. It opens in ORIPA, Rabbit Ear and the rest of the FOLD ecosystem.
The export is short because this repository never converts anything: FOLD's
vertices_coords, edges_vertices and edges_assignment
have been the in-memory representation of a pattern here since the site's first phase. What
the file adds is the metadata that makes it openable, and the faces where they can be read.
Coordinates are in sheet widths, and the file says what one unit measures on paper.
What is argued with it
Essays that call yoshimura — read off the figure index rather than listed by hand.
Patterns nobody designed
Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.
The sheet has a thickness
Every crease pattern describes a surface with no thickness. Everything anybody builds has some, and getting it around a corner is the central problem of turning origami into hardware.
The cylinder the pattern chooses
A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.
How much line is on the paper
A crease pattern is described by its creases: how many, at what angles, in what arrangement. What a folder spends is length. The Yoshimura this site prints has eighty-six creases and 2,380 millimetres of folding on a sheet seventeen centimetres across; the Miura has thirty-eight creases and 1,049, and the fold-and-cut triangle has six and 258 — and the two counts do not rank the eight printed patterns the same way.
A corrugation agrees with itself
A Miura fold of forty-eight panels and a twist tessellation patch of forty-nine have almost exactly the same number of independent closed chains for their letters to contradict themselves round — thirty-five against thirty-six. Sixty-four per cent of the Miura's drawn letterings are consistent and thirteen per cent of the patch's. A Yoshimura at thirty-three chains manages ninety-three. The room to fail sets the scale; the construction decides where in it a pattern lands.
A twisted tube has two heights
Twist a thin cylinder and it falls into a pattern of slanted triangles that clicks between two lengths. The click is usually explained as a fold, but there is no fold. Every crease keeps its own length at exactly two heights, and in between both diagonals have to stretch or shrink. The second height exists only on a course made taller than the one that folds flat, and the Yoshimura's course, the same pattern with no lean, has its second height on the far side of the tube's own axis.
Four clicks make sixteen lengths
A Kresling tube of four courses has sixteen ways to be at rest, each course shut or open. Pressing and pulling it under one force can reach all sixteen or only five, and the difference is set by how tall the courses were made. Every course shuts at a force that falls as it is made taller, while the force that opens it rises to a peak and then falls. A stack graded below that peak opens shortest first and can only count. Graded above it, it opens tallest first and can be written like a register.
Clamped, a stack counts in pairs
A Kresling tube of four courses, its ends free to turn, can be pressed and pulled into all sixteen of its words. Clamp the ends so they cannot turn and the courses stop being independent switches, because a course that clicks shut turns by a fixed angle — eighty degrees here, whatever height it was made at — and the others must give it back. With every course of one hand the stack does not click at all, at more than twice the force that shuts any course free. With two courses of each hand it clicks in pairs, one of each hand together, so that their turns cancel: every word it reaches has as many right-handed courses shut as left-handed, and it reaches exactly four. The clamp turns a sixteen-word register into a two-switch one whose every state leaves the ends where they were.
The Yoshimura folds flat at a few heights
The Yoshimura's row height has always been treated as a free dial below the equilateral: the sectors close, the counts hold, the lemma has nothing to forbid. Asked for an actual stack of layers, its usual lettering refuses nearly everywhere. Folded, its triangles turn round one centre by twice the base angle at each step, and they close on a ring only when that angle is π over a whole number: a triangle at sixty degrees, a square at forty-five, a pentagon at thirty-six. At any other height the rosette spirals, and seven panels are enough to prove it cannot pass itself. Below π over the number of rows plus two, it does not get far enough round to meet itself, so the small patch hides this too.