Origami & Folding
what one sheet can do
Concept

The Yoshimura pattern — where it appears

The diamond pattern a thin cylinder buckles into under compression. It was published in 1955 as a buckling mode and folded traditionally long before, which makes it the site's clearest case of a pattern in two records.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

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 letter20 interior vertices · 23 mountain and 56 valley creasesmountainvalleyraw edge

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.

tessellation · Found patterns what the shell produced14 interior vertices, all alike17 mountain, 40 valley57 creases carrying a letterand it folds flatchecked, not asserted11.0 sheet-widths of crease, chosen by a buckling loadmountainvalleyraw edge

Found before it was designed

Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.

history · Rediscovery the flat pattern6 columns · 5 rowsthe course, closedradius 0.16676 sides of 0.1667 close into a ring of radius 0.1667, in the sheet's own unitsthe circle through the corners is 4.72% longer than the ring, which is what a 6-sided polygonowes its circle

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.

tessellation · Found patterns the bar is how many rules send four panels round in a circleout of the rules that already fail the count at some vertexthe Miura fold3848 refused · every loop four panels · vertices of degree 4the tapered leaf3848 refused · every loop four panels · vertices of degree 4the Yoshimura pattern038 refused · not one closes a loop · vertices of degree 6the waterbomb tessellation120480 refused · every loop four panels · vertices of degree 4 and 6a loop of four needs the letters to alternate round one point, and only a degree-four vertex lets a repeating rule do that

Where a rule can close a loop

Three corrugation families have repeating rules whose letters send four panels round in a circle, and one has none at all. The one that has none is the one whose vertices are all of degree six — and the reason is that a straight line through a point carries a single letter under any repeating rule, while a strict alternation round six creases needs the two halves of that line to differ.

tessellation · Waterbomb the Yoshimura, as drawn: nodes against panels050100one a panel0 panels119every vertex of this family keeps 30 labellings

Six creases and the same straight line

The one family here whose vertices are degree six was said to break the arithmetic that every other family obeys, on the strength of a single pattern. Built as a family — six sizes from twenty-one panels to a hundred and nineteen — the Yoshimura is exactly as linear as a grid, with no decision ever withdrawn. What degree changes is the constant, and it changes it in both directions depending on one angle.

flat-folding · Vertex degree the Yoshimura at 6 by 5, at nine proportionsrow height 1.257 nodes30 labellings a vertex · 0.88 nodes a panelrow height 1.557 nodes30 labellings a vertex · 0.88 nodes a panelrow height 1.757 nodes30 labellings a vertex · 0.88 nodes a panelrow height 1.732050857 nodes30 labellings a vertex · 0.88 nodes a panelrow height 1.732050919 nodes8 labellings a vertex · 0.29 nodes a panelrow height 1.7419 nodes8 labellings a vertex · 0.29 nodes a panelrow height 1.819 nodes8 labellings a vertex · 0.29 nodes a panelrow height 219 nodes8 labellings a vertex · 0.29 nodes a panelrow height 2.519 nodes8 labellings a vertex · 0.29 nodes a panelthe equilateral Yoshimura is drawn at √3 = 1.732050808, on the dear side

A knife edge nine decimals wide

Draw the Yoshimura with its rows 1.7320508 half-columns tall and each vertex admits thirty labellings and the pattern costs fifty-seven steps. Draw it at 1.7320509 and each admits eight and it costs nineteen. The number between them is √3, which is the proportion everybody draws — and below it the sectors are unequal and the lemma is still silent, because the small ones sit next to each other.

flat-folding · Vertex degree the folded period of the Yoshimuradrawn periods across the top123456the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyesa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

The turn a column costs

The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.

tessellation · Shrinkage the clauses a recipe needs, family by familyeach recipe is checked to allow exactly the rules that fold, and nothing elsefamilyrulesfoldparitiesprohibitionsdegree-six kindsclauses in allthe Miura fold64162002the Yoshimura pattern64261223the waterbomb tessellation512323245a parity says an even or odd number of some letters are mountains; a prohibition says some letters are not all alike

Two sentences for the Yoshimura

No recipe made only of same-and-differ clauses picks out the Yoshimura pattern's twenty-six folding rules, because its vertices have six creases. A recipe allowed one other kind of sentence does, and it is short: an even number of the four zigzag classes are mountains, and no course carries the letter all four zigzags share. That is three clauses, one parity and a prohibition for each of the pattern's two kinds of vertex, and it allows exactly the twenty-six. The waterbomb tessellation, with four kinds of six-crease vertex, needs three parities and only two prohibitions, because its parities do half the prohibiting.

history · Pedagogy one strip of triangles, two tubes2 courses of a 6-sided Kresling tube, both diagonals at their own length in eachas made: 1.79 a course, turned 80°second state: 0.99 a course, turned 160°long diagonal 2.204, short 1.824long diagonal 2.204, short 1.824

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.

tessellation · Found patterns four courses holding the word 1010each course clicked shut or left open, read from the bottom upA: made 1.30 H, shut, 1.24 tallB: made 1.45 H, open, 2.16 tallC: made 1.60 H, shut, 1.86 tallD: made 1.75 H, open, 2.61 tall

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.

tessellation · Found patterns a 6 by 5 Yoshimura and the loops of three that cost ita shaded triangle is a loop of three creases whose correlations multiplyrows 1.732 half-columns tall30 letterings at each vertexevery pair correlated -0.066727 triangles, (−1⁄15)³ eachcounted ⁄ product = 0.99252

What a loop of creases carries

Multiplying each vertex's own share of letterings gives the exact share for a whole Miura, both twists and a tapered corrugation, and three quarters of a per cent too much for the Yoshimura. The difference has an exact form. The count is the product times a sum over loops in the graph of creases, each loop weighted by the correlations its vertices put on the pairs of creases it uses. A Miura vertex correlates no pair at all, so no loop can carry anything. An equilateral Yoshimura vertex correlates every pair by −1⁄15, and its 27 small triangles take the product down by 0.8 per cent before the larger loops pay a little back. Draw the rows taller than √3 and the big-little-big lemma locks three pairs and frees the rest, no loop closes, and the product is exact.

material · Kirigami the recipe each family needs, before and after the layers are counteda parity is what must add up; a prohibition is what must not all agree; closed means a same-and-differ recipe can existrules asked aboutrulesparitiesprohibitionsclosedthe Yoshimura, at the vertices2612nothe Yoshimura, with a stacking2214nothe waterbomb, at the vertices3232nothe waterbomb, with a stacking1650yes

Half the waterbomb's rules have no stack

The recipes for the waterbomb and the Yoshimura were written for the repeating rules that pass the vertex conditions, and nobody had asked those rules for a folded state. Asked, sixteen of the waterbomb's thirty-two have none, and four of the Yoshimura's twenty-six. Four panels are enough to refuse a waterbomb rule, in five lines, by forcing two creases on one folded line to interleave. And the waterbomb's surviving rules turn out to be exactly what a recipe of its kind was proved unable to describe: five same-and-differ clauses and no prohibition. Counting the layers shortens the waterbomb's recipe and lengthens the Yoshimura's, whose refusals are not local at all.

history · Pedagogy where the usual Yoshimura lettering puts its panels, foldedevery course one letter, every zigzag the other; 4 by 4; panels drawn at their folded places

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.

tessellation · Found patterns what finding a stacking costs at each height, on 4 by 4steps of the search, on a scale of powers of ten; dots mark the whole values of n, where the rosette closes2345634567n, where the base angle is π over nsteps, as a power of tenthe usual letteringzigzags split by bandzigzags split by lean

The proportion everyone draws is the hard one

The Yoshimura is drawn equilateral, and every cost and count of it has been measured there. Moved a quarter of a step along its own height dial, a search for a stacking that took 268,650 steps takes 808, and a sheet with 9,790 folded states has 92. The equilateral pattern is the most expensive member of its family on every sheet measured, because exact coincidence removes the conditions that a panel lying across a crease would impose: from 1,024 conditions to none. A drawing chosen for being easy to construct is the one instance where almost nothing constrains the layers, and a population of one chosen drawing had been reporting that freedom as the difficulty of the pattern.

complexity · Typical instances

Named alongside it

The objects these essays reach for when they reach for this one.

AssignmentBucklingCorrugationEnergy minimisationFound patternMaekawa's theoremThe big-little-big lemmaCylinderRepeating ruleSearch costVertex degreeConstraint propagation

All concepts