Origami & Folding
what one sheet can do
Concept

Strip — where it appears

A one-dimensional crease pattern: a length of paper creased across it, with nothing to fold in the other direction. It is the case where flat-foldability is decidable exactly, and a vertex on the edge of a sheet reduces to one.

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

the bar is every legal stacking; the dark part is the ones with no move out of thema swap is legal when the result is still a stacking — the two layers need not be joined by a crease3 segments2 of 6 isolated · 4 legal moves4 segments16 of 16 isolated · 0 legal moves5 segments34 of 50 isolated · 16 legal moves6 segments144 of 144 isolated · 0 legal moves7 segments366 of 462 isolated · 96 legal moves

Nothing slides past anything

A marked strip has several legal stackings and this site has counted them at length. Nobody asked whether a folder holding one can reach another by lifting a flap over its neighbour: five hundred and sixty of six hundred and seventy-two stackings have no such move at all, and whether any exists depends on the parity of the segment count.

flat-folding · Layer multiplicity the pale bar is the share that folds in a ring, the dark one in a line40 vertices from each population at each degreeangles at random, degree 425% in a ring · 70% in a lineangles at random, degree 613% in a ring · 46% in a linemultiples of 45°, degree 432% in a ring · 83% in a linemultiples of 45°, degree 630% in a ring · 87% in a linemultiples of 30°, degree 429% in a ring · 79% in a linemultiples of 30°, degree 623% in a ring · 70% in a line

A ring and a line

A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.

flat-folding · Boundary

Named alongside it

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

Flat-foldabilityThe big-little-big lemmaBoundary vertexLayer orderingLocal moveMaekawa's theoremSector angleStackingStamp folding

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