Origami & Folding
what one sheet can do
Generator

The Yoshimura pattern — the Yoshimura pattern generator

A generator in the tessellations library, called 9 times across 5 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

yoshimura is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

The Yoshimura patternThe diamond pattern a thin-walled cylinder falls into when it is crushed axially. Nobody designed it — it is the buckling mode with the lowest energy, and it was named after the engineer who described it in aluminium tubes rather than after a folder. Drawn here from the construction that satisfies all three local conditions.found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter22 interior vertices · 26 mountain and 60 valley creasesmountainvalleyraw edge
Yoshimura pattern — sheet 170×122.69 mm — 26 mountain, 60 valley, 2380 mm of crease

cols: 6, rows: 5

The Yoshimura patternThe diamond pattern a thin-walled cylinder falls into when it is crushed axially. Nobody designed it — it is the buckling mode with the lowest energy, and it was named after the engineer who described it in aluminium tubes rather than after a folder. Drawn here from the construction that satisfies all three local conditions.found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter22 interior vertices · 26 mountain and 60 valley creasesmountainvalleyraw edge
Yoshimura pattern — sheet 170×122.69 mm — 26 mountain, 60 valley, 2380 mm of crease

cols: 6, rows: 5, ratio: 1.9919

The Yoshimura pattern at a height it cannot haveThe same diamond pattern with its rows made taller than the equilateral construction allows. The sectors still close to a full turn and still alternate to a straight angle, so developability, Kawasaki and Maekawa are satisfied at every interior vertex — and every one of them fails the big-little-big lemma, which is the only condition that can see a row height.found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 53°, 63°, 63°, 53°, 63°, 63°two courses and four diagonalstwo of one letter, four of the otherwhich is why it is refusedthe top sector is strictly smallestand flanked by two of one letterat all 22 interior vertices22 interior vertices · 26 mountain and 60 valley creasesmountainvalleyraw edge

cols: 8, rows: 4, mm: 170

The Yoshimura patternThe diamond pattern a thin-walled cylinder falls into when it is crushed axially. Nobody designed it — it is the buckling mode with the lowest energy, and it was named after the engineer who described it in aluminium tubes rather than after a folder. Drawn here from the construction that satisfies all three local conditions.found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter23 interior vertices · 26 mountain and 64 valley creasesmountainvalleyraw edge
Yoshimura pattern — sheet 170×73.61 mm — 26 mountain, 64 valley, 1870 mm of crease

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

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.

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.

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 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.

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.

Every generator · The tessellations field · The patterns a reader can fold