Origami & Folding
what one sheet can do
Concept

Degree-four — where it appears

A vertex where four creases meet, which is the case almost every pattern in this subject is built from. It is the only degree at which the four local conditions decide flat-foldability completely, and as a rigid mechanism it has exactly one freedom.

Named by 13 essays across 6 fields — each of them below, with the objects they name alongside it.

degree-4 vertex, 60/90/120/90°4 of 164 creases · 25.0% surviveone degree-6 vertex8 of 646 creases · 12.5% survivethe preliminary base112 of 2568 creases · 43.8% surviveand these are only the local tests — a pattern can pass every vertexand still collide once the layers stack, which is the hard part

The same vertex, found four times

A degree-four vertex with a three-to-one assignment turns up in a buckled cylinder, in a Miura fold, in a Resch tessellation and in a crumpled sheet. It is not a coincidence and it is not influence: the flat-folding conditions are restrictive enough that a small set of vertices is nearly all there is.

history · Rediscovery parallel columnsKawasaki to 3e-14°columns fanning by 5.2°Kawasaki to 5e-14°columns fanning by 9.2°Kawasaki to 5e-14°the mountain-and-valley letters are read off the motion rather than drawn, and then put past Maekawa

The family the Miura belongs to

Move one vertex of a Miura and the sheet has no rigid folded position at all — which leaves the obvious question unanswered. What else moves? A row of paper reflected in each of a fan of lines is flat-foldable for nothing at all, and whether it also folds rigidly turns out to be a condition on a table of cosines: it has to be a column of numbers times a row of numbers.

rigid · Rigid folding the sectors are 40°, 140°, 40°, 140° — each row is one way the vertex can start to foldcrease 1crease 2crease 3crease 4mode 10.7100.7102 of the four creases movemode 200.7100.712 of the four creases movethe numbers are the four fold angles' ratios to one another as the vertex leaves the flat state

Two mechanisms at one point

Two creases drawn across each other cannot fold flat — Maekawa's count refuses them at every angle. They move perfectly well as rigid panels, and they move in two ways: bend along one line while the other stays flat, or the reverse. Every other developable vertex of degree four has two ways too, and in both of them all four creases move together at a fixed ratio. The crossing is the case where the two motions have nothing to do with each other.

rigid · Rigid folding the bar is how many rules the two tests agree aboutone reads three bits of the rule; the other folds the sheet and walks the arcsthe Miura fold64 of 6438 rules predicted to close a loop · 0 disagreementsthe tapered leaf64 of 6438 rules predicted to close a loop · 0 disagreementsthe closed form says a loop is available exactly where the columns fail to change letter and the row disagrees with them

The loop is in the rule

Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.

tessellation · Miura 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 bar is how many of a hundred random letterings agree with themselveson the orthogonal grid a box-pleated design is drawn on, at five sizes4 by 4949 interior vertices · 16 panels · found in 16 nodes6 by 67325 interior vertices · 36 panels · found in 37 nodes8 by 85849 interior vertices · 64 panels · found in 65 nodes10 by 103681 interior vertices · 100 panels · found in 100 nodes12 by 1215121 interior vertices · 144 panels · found in 145 nodes16 by 161225 interior vertices · 256 panels · found in 261 nodesevery interior vertex is a four-panel circuit, so the number of places a contradiction could sit is the number of vertices

What a grid costs in circuits

Box-pleating puts every crease on a square grid, and a square grid is the shape with the most short circuits per panel that this collection draws. On the sixteen-by-sixteen grid a designer actually works on, one mountain-valley labelling in a hundred agrees with itself. A search still finds one in two hundred and sixty-one steps.

design · Box pleating labellings a vertex keeps, against nodes a panel costs0.000.250.500.751.00481530the box-pleating gridthe tapered leafa crumple, deepeningthe waterbombthe Yoshimura, as drawnthe Yoshimura, tiltedthe twist patcheslabellings the conditions leave at a vertexthe dashed line is one node a panel, which four of these families sit on exactly

One step per panel is a table size

Four families of crease pattern search at exactly one step per panel — a grid at nine sizes, a leaf, a Miura, six crumples — and it was read as a law about patterns that fill their own sheet. It is a number: the conditions at each of their vertices admit eight labellings. Where the conditions admit four, the cost is half. Where they admit thirty, it moves again, and the same pattern at two proportions demonstrates it with everything else held still.

flat-folding · Sector angles 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 twist patches: nodes against panels050100150one a panel0 panels157every vertex of this family keeps 4 labellings

The most decided vertex here

Sixteen ways to letter four creases; Maekawa allows eight; the big-little-big lemma allows four. A twist polygon's corner is one of the few vertices in this collection where the second cut applies, so it keeps four labellings where a grid, a leaf, a Miura and a crumple all keep eight — and the family the collection long called difficult turns out to be the one whose conditions decide the most.

tessellation · Twists the box-pleating grid: nodes against panels0100200one a panel0 panels256every vertex of this family keeps 8 labellings

The designer's grid is the dearest thing here

Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.

design · Box pleating the tapered leaf: nodes against panels01020one a panel0 panels24every vertex of this family keeps 8 labellings

Nothing grown was cut out of anything

A leaf's corrugation costs twelve steps on twelve panels, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four — exactly one per panel at every geometry, which is the most any pattern here costs. A tessellation patch costs half that, and the reason is that somebody cut it out of something. A leaf's creases stop at the margin because the plant stopped there.

biology · Leaf folding the rules that fold, against the clauses they all obeya clause says an odd or even number of some chosen letters are mountainsfamilyrulesfoldparity clausesthey allowa same-or-differ recipethe Miura fold6416216yesthe tapered leaf6416216yesthe Yoshimura pattern6426132nothe waterbomb tessellation51232364noa recipe of same and differ clauses allows exactly the rules its clauses allow; that is the test

A recipe needs degree four

The Miura's letters are taught as a recipe of same-and-differ clauses, and the recipe is exact: the sixteen repeating rules that fold are precisely the rules two clauses allow. The Yoshimura's twenty-six cannot have such a recipe, because twenty-six is not a power of two. The waterbomb's thirty-two is a power of two and still has none — its clauses allow sixty-four rules and half of them fail. The difference is one vertex: at degree four the counting theorem leaves a parity, and at degree six it leaves 'not all alike', which no clause of that kind can say.

history · Pedagogy four vertices round one panel, as a chain and as a loopa combination survives the loop only if going round it brings every fold angle back to where it startedthe faceopen chain of fourclosed loopthe same at every anglea Miura face164yesa face with no two vertices alike, seed 11161yesdriven at 0.3, 0.6, 1, 1.4, 1.8 radians · a combination counts when every crease is folded and the loop closes

A loop takes choices away

A chain of four sprung degree-four vertices has sixteen combinations of branches, each a resting state, and switches between them only through the flat sheet. Close the chain into a loop round one panel and the combinations must agree when the fold angles come back round. On a face whose four vertices all differ, one combination survives; on a Miura face, four. The count is the same at every angle the face is driven to, and two surviving assignments at the same driven angle are never closer than one and a half times that angle — so they separate as the face folds and meet only when it is flat. A loop does not create the junction a region would need to switch on its own. It removes choices and leaves the switch as global as before.

biology · Insect wings

Named alongside it

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

AssignmentCorrugationSearch costConstraint propagationInterior vertexMaekawa's theoremPanelSector anglesGridLayer orderRepeating ruleVertex degree

All concepts