Generator

The Yoshimura pattern — the Yoshimura pattern generator

A generator in the tessellations library, called 32 times across 9 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

yoshimura is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

`cols: 6, rows: 5`

`view: "kresling", show: "tubes", sides: 6, turn: 80, rise: 1.2`

`view: "kresling", show: "net", sides: 6, turn: 80, rise: 1.2`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

on 3 by 3, the sheet stacks at every n from 3 to 9 that is a whole number and at every n past 5, and at nothing else on this grid ×7
developability holds at every interior vertex — its sectors close to 360° — 22 checked ×2
Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 22 checked ×2
Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 22 checked ×2
the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 22 checked ×2
the interior vertex carries sectors of 60, 60, 60, 60, 60, 60°, which close to 360° and alternate to zero at any proportion at all — so the row height is fixed by the big-little-big lemma rather than by Kawasaki, and the pattern is drawn at a proportion where nothing is yet forbidden ×2
a 6-sided course turns by exactly 180° + 60° − twice its turn as made when it clicks, whatever height it was made at: 120° at 60°, 100° at 70°, 80° at 80°, 60° at 90° ×1
a course made 1.20 times the flat-folding height behaves as the two closed forms say: stress-free second state above H = 1.492, held flat between 0.983 and H, springing back below ×1
a rule with quite different letters puts every panel in exactly the same place ×1
a stack of 4 courses graded below opening's peak can be pushed and pulled into 5 of its 16 words, and the same stack graded above it into all 16, each at a length of its own ×1
along the whole path between the two states the long diagonal is stretched and the short one compressed, by 3.11% and 2.45% at the top of the barrier ×1
at a base angle of π/5 the folded triangles close on a ring and the sheet has a stacking ×1
at a row height of 1.992 half-columns the sectors still close to 360° and still alternate to zero, and every one of the 22 interior vertices fails — on the big-little-big lemma alone, with developability, Kawasaki and Maekawa satisfied at all of them ×1
at a turn of half a sector the two diagonals are one length (1.864) and the top vertex sits over the middle of its edge, which is the Yoshimura's course; at 80° the long diagonal is 2.204 and the short 1.824 ×1
at full stroke the barrier strain falls from 5.92% at 70° to 1.29% at 100°, and on every turn a shorter stroke costs less strain ×1
at this height the rosette does not close, and the search proves the sheet has no stacking ×1
clamped, the RLRL stack holds 0011 with its total turn unchanged and every diagonal at its own length, to a millionth ×1
every one of 117 sampled 6-sided designs, classified from its own pressed landscape, falls on the side of the two closed-form curves it should, and the held-flat curve found by bisecting on the landscapes at three turns agrees with the closed form to 0.0005 radii ×1
every panel not turned over has been rotated by a whole number of steps of twice the base angle ×1
four rules stack at no height drawn ×1
from two courses to eight, opening in the order of shutting reaches every one of the 2^n words and opening in the reverse order reaches n + 1 ×1
pressed from all open, the free stack shuts its courses one at a time at 28.1, 32.5, 37.8, 44.3 thousandths; clamped with alternating hands it shuts them two at a time, at 30.4 and 41.3, each between its two members' free forces; clamped and all one hand it shuts none up to 60 ×1
pressed straight down, the symmetric course holds its turn of 30° to a height of 1.23 and then turns, choosing a hand, and its diagonals reach 22.7% strain on the way; its own second state would need a turn of 210°, through the axis ×1
pressing must reach 44.2 thousandths of the diagonals' stiffness to shut the course and pulling 29.8 to open it, the first nearer the open state and the second nearer the shut one ×1
seven panels of the three-by-three sheet are already refused, and a listing of the orders of those seven finds none ×1
the 16 rules whose zigzags split two and two by band, or two and two by lean, stack at every height drawn ×1
the course made 1.79 tall and turned 80° has a second state 0.99 tall and turned 160° in which both diagonals have their own lengths again (2.204 and 1.824), and no triangle passes through another in either ×1
the force to shut falls the whole way as the course is made taller, while the force to open rises to a peak at 1.18 times the flat-folding height and then falls ×1
the letters alone force no cycle on any rule here ×1
the shallower proportion three quarters of the equilateral height, on six columns and five rows, has no stacking under the usual lettering ×1
the usual lettering passes every vertex condition at this height, whatever the height below the equilateral ×1
the word 1010 is one the stack can be pushed and pulled into, and no course in it passes through itself ×1
the word 1010 is written by 4 swings of the load, alternately pressing and pulling, and the stack ends holding exactly it ×1
the Yoshimura pattern is put past all four theorems before it is drawn ×1
under the four rules that never stack, the letters alone put more than nine panels in ten on one rising chain, 34 of 36, 51 of 55, 61 of 65 ×1
with free ends the stack reaches all sixteen words; clamped, every word it reaches has as many right-handed courses shut as left-handed, a stack of one hand reaches none but all open, and each of the 3 stacks of two of each hand drawn reaches exactly four — the same four when the force is stepped twice as finely ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

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.

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.

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.

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.

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.

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