Origami & Folding
what one sheet can do
Concept

Layer multiplicity — where it appears

How many distinct folded states one marked crease pattern admits. It is not the number of assignments that fold, and the difference between the two counts is a sum rather than a coincidence.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

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 flat-foldable vertex whose sectors are multiples of 45°6 of them, to degree 8passfoldone objectthe most45·45·135·135866145·90·135·90444190·90·90·90888145·45·45·45·90·90302012245·45·90·45·45·90301812645·45·45·45·45·45·45·451121121643 of the 6 carry markings the conditions accept and the paper refuses

The whole alphabet of a grid

Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.

design · Box pleating 10 mountains, 14 valleys3.95 wide, 2.10 deep13 mountains, 11 valleys1.98 wide, 1.79 deep

The Miura folds two ways

One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.

tessellation · Miura the bar is the pairs of panels that lie over one anotherThe preliminary base288 panels · 12 rules · an ordering existsThe Miura fold22824 panels · 228 rules · not decidedThe square twist369 panels · 48 rules · an ordering existsThe hexagon twist6613 panels · 96 rules · an ordering existsThe Yoshimura pattern205565 panels · 1187 rules · not decidedFold and cut — the triangle217 panels · 15 rules · an ordering existsThe tapered corrugation28228 panels · 351 rules · not decidedThe waterbomb tessellation92652 panels · 654 rules · not decideda pattern with no bar has no two panels over one another, and its order is not a question

No height to swap

A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.

flat-folding · Layer multiplicity the bar is the average number of distinct folded statesevery printed pattern on this site has exactly one, and none of its swaps is legal2 folds1.1724 of 24 measured · 2 of 71 swaps legal3 folds1.3824 of 24 measured · 2 of 164 swaps legal4 folds2.1619 of 24 measured · 6 of 335 swaps legal5 folds2.336 of 24 measured · 0 of 121 swaps legala refused row is a sheet with too many panels to search, and refusals are counted rather than dropped

The crumple keeps its options

Every crease pattern this site prints has exactly one folded state and not one of its thirty-nine available rearrangements is legal. A sheet creased by folding it at random four times has an average of 2.16 folded states, one of them has nine, and six of three hundred and thirty-five rearrangements are legal. The sheet nobody designed is the one with room left in it.

material · Crumpling ruling out the square cell's loops, one direction at a timewhat is left splits248 arcs go, 24 remaindirection (1, 0)102 arcs go, 10 remainwhat is left splits010 arcs go, 0 remaindirection (-1, 0)102 arcs go, 10 remainwhat is left splits010 arcs go, 0 remainthe bar is how many arcs are still in play after the step

The arc that arrived twice

Which of two panels a crease calls its near one is decided by the order a face walk happened to number them, and the mirrored record is the same relation. Except on one sheet, where it is not — and that sheet turned out to be the one whose folded state comes back the other way up, which is how a duplicate in a graph became a diagnosis.

flat-folding · Layer multiplicity

Named alongside it

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

Layer orderFolded stateLayer orderingNon-crossing conditionPanelVertex degreeAssignmentBox pleatingBranchCertificateCrease assignmentCrease pattern

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