Origami & Folding
what one sheet can do
Concept

Crimp — where it appears

A fold that takes the smallest sector at a vertex and folds it away between its two neighbours. It is a physical move, a machine primitive, and the step in a reduction that decides whether a vertex folds.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

4 creases, assignment MVMVthe strip0.200.200.200.200.20MVMV3 availableafter crimp 10.200.200.20MV1 availableafter crimp 20.20nothing left2 crimps, each removing two creases4 creases is an even number, and that is not a coincidencethe merged segment measures outer minus middle plus outer

A machine that can only crimp

Change the atom and the whole picture changes. A machine whose single move folds two adjacent creases at once reaches strips no simple-fold machine reaches, is defeated by strips they handle easily, and cannot fold an odd number of creases at all — for reasons that are pure arithmetic.

complexity · Simple foldability the waterbomb tessellation's odd vertexdegree six, and this site prints nine of them on one sheetevery assignment64passes all four conditions30has a flat folded state1812 labellings satisfy every condition the subject has and have no flat folded state

Crimp it away and ask again

Four conditions decide whether a vertex folds flat, and they decide it exactly at a vertex whose sectors are all different sizes. Everywhere else they over-count: two markings of every tied four-crease vertex, twelve of the degree-six vertex this site prints nine of on one sheet. What decides the case is not a fifth condition but a procedure — fold the smallest sector away and ask the smaller vertex.

flat-folding · Crimping of the markings that fold, how many folded objects each one makesmarkings that foldexactly one objectthe most any one makesthe preliminary base4 creases · four equal sectors, the first vertex anybody folds881a halved four-crease vertex4 creases · degree four with its two smallest sectors equal — the case the lemma is silent at661the waterbomb tessellation's odd vertex6 creases · degree six, and this site prints nine of them on one sheet18126a Yoshimura vertex6 creases · degree six with every sector equal, and twenty-two of them on the printed pattern30122the preliminary base's centre8 creases · degree eight, and the vertex at the middle of the first base anybody folds112164a vertex at no particular angles6 creases · degree six, drawn from the census and rounded to a tenth of a degree881at four creases the marking names the object; above it, it need not

One marking, many objects

A crease pattern with every mountain and valley written on it is spoken of as though it named a folded model. At four creases it does. At six it need not, and at the eight-crease vertex in the middle of the first base anybody folds, a single marking can be folded into four genuinely different objects — same creases, same letters, four answers.

flat-folding · Layer multiplicity every condition holds here6 creases4 creasesevery condition holds at the vertex on the paper — and one crimp later the smallest sector has the same letter on both sidesthe four conditions all hold · a stacking does not exist

A short reason to say no

When a folding question comes back yes it brings an object anybody can check. When it comes back no it usually brings nothing but the assurance that a search looked everywhere. At one vertex that is false: a refusal comes with a witness one or two steps long, out of a search space of a hundred and twelve, and the witness is a vertex the crease pattern does not contain.

complexity · Hardness of folding the bar is how many moves survive the conditions at a vertexa move flips two creases meeting at one point, which is what pushing a vertex through doesthe square patch0216 pairs tried at each of two letterings · 0 legal · 0 leave the verdict alonethe elongated patch6270 pairs tried at each of two letterings · 6 legal · 6 leave the verdict alonethe hexagonal patch8360 pairs tried at each of two letterings · 8 legal · 8 leave the verdict alonethe triangular patch8360 pairs tried at each of two letterings · 8 legal · 8 leave the verdict alonethe rhombille patch16756 pairs tried at each of two letterings · 16 legal · 16 leave the verdict aloneevery one of them leaves the lettering on the side of the question it was already on

Every move leaves the verdict

The only change a folder can make to a lettering without breaking it is to push one vertex through, flipping two creases at once. Try every such move on five tessellation patches, from two different letterings each: nineteen of two thousand nine hundred and sixty-four survive the conditions, and not one of the nineteen turns a lettering that agrees with itself into one that does not, or the other way about.

flat-folding · Local moves

Named alongside it

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

Vertex degreeThe big-little-big lemmaDecision procedureLayer orderingMaekawa's theoremAssignmentBuried creaseCertificateConfiguration spaceConnectivityFlat-foldabilityFlat folded state

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