Origami & Folding
what one sheet can do
Concept

Repeating rule — where it appears

A crease assignment written as a rule about parities rather than crease by crease: the letter on a crease depends only on which row and column it sits in. A grid corrugation has sixty-four of them and a waterbomb tessellation five hundred and twelve, so a family small enough to sweep in full.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

the 16 repeating rules that fold, written outrows first, then the two column classes — and every one of them alternates down the columnrows · columns above|below20VV · MV|MV21MV · MV|MV22VM · MV|MV23MM · MV|MV24VV · VM|MV25MV · VM|MV26VM · VM|MV27MM · VM|MV36VV · MV|VM37MV · MV|VM38VM · MV|VM39MM · MV|VM40VV · VM|VM41MV · VM|VM42VM · VM|VM43MM · VM|VMfour ways of writing the rows times four ways of alternating the columns is sixteen, and there is nothing else

Sixty-four rules, sixteen fold

The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.

tessellation · Miura 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 16 repeating rules that fold, written outrows first, then the two column classes — and every one of them alternates down the columnrows · columns above|below20VV · MV|MV21MV · MV|MV22VM · MV|MV23MM · MV|MV24VV · VM|MV25MV · VM|MV26VM · VM|MV27MM · VM|MV36VV · MV|VM37MV · MV|VM38VM · MV|VM39MM · MV|VM40VV · VM|VM41MV · VM|VM42VM · VM|VM43MM · VM|VMfour ways of writing the rows times four ways of alternating the columns is sixteen, and there is nothing else

Half the recipe is decoration

Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.

history · Pedagogy the bar is how many repeating rules fold, and it is the same bar six timesthe same corrugation redrawn at six geometries, every one of them swept in fullas printed1648 refused, all by the count · 38 of them close a circle of foureven columns1648 refused, all by the count · 38 of them close a circle of foura one-sided ramp1648 refused, all by the count · 38 of them close a circle of foura violent taper1648 refused, all by the count · 38 of them close a circle of foursix taller rows1648 refused, all by the count · 38 of them close a circle of foura steeper zigzag1648 refused, all by the count · 38 of them close a circle of fourno vertex condition reads a column width, a row height or a row count, so none of them can move the table

The leaf's rules are the Miura's

A plicate leaf packs into a corrugation that is broad in the middle and narrow at both ends. Enumerate every repeating mountain-valley rule it admits and the table is the Miura fold's table, rule for rule, number for number — and it stays that table when the leaf is redrawn with even columns, a violent taper, taller rows or a steeper zigzag. The plant's geometry cannot reach its own letters.

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 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

Named alongside it

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

Maekawa's theoremAssignmentCorrugationMiuraDegree-fourNotationParityPedagogyLayer orderThe Yoshimura patternCertificateConvergence

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