Origami & Folding
what one sheet can do
Concept

One sheet no cuts — where it appears

The rule this collection works under: a single uncut square, folded and nothing else. It is a comparatively recent convention rather than an inherited one, the oldest surviving book of paper folding does not keep it, and its value is in what it generates rather than in how old it is.

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

4×4 — 16 cranes, 9 corner joinsone sheet, and cut2×2 4 cranes 4 sides of slit3×3 9 cranes 12 sides of slit4×4 16 cranes 24 sides of slit5×5 25 cranes 40 sides of slit6×6 36 cranes 60 sides of slitcranes − joins = 2n − 1the rule the subject is usually stated under is one sheet and no cuts; theoldest surviving origami book does not keep it

The oldest book cuts the paper

The Hiden Senbazuru Orikata of 1797 is the earliest surviving book of recreational paper folding, and its famous connected cranes are made by slitting one sheet into a grid. The founding rule of the modern subject is younger than the tradition it claims to describe.

history · Folklore of folding how much choice there is in the cuttingevery subset of the joins, tested for connectivitygridcranesjoin pointssubsetsconnectedfewest joins2 × 250% of them work41211 of 1one way only3 × 36.3% of them work941614 of 4one way only4 × 44.1% of them work169512215 of 9one way only5 × 51.2% of them work251665,53678510 of 1650 wayseach interior lattice point holds four cranes at once · every subset of them tried, and the piece has to come out in one piece

Which cranes can stay joined

The 1797 book slits a square into a grid and leaves the cranes attached at the interior lattice points, each of which holds four of them at once. Of the sixteen ways to choose which of a three-by-three's four points to leave joined, exactly one leaves the piece in a single object — and it is the one that uses all four. The cutting is very nearly forced rather than chosen.

history · Folklore of folding

Named alongside it

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

Hiden senbazuru orikataKirigamiLattice identitySenbazuruThe counting problem

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