Origami & Folding
what one sheet can do
Concept

Lattice identity — where it appears

The arithmetic relating the pieces of a slit grid to the joins holding them together. For an n by n grid of cranes joined at interior lattice points, the piece count minus the join count is exactly 2n − 1 — which is the check that the joins are corner joins rather than edge joins.

Named by 4 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 two ways to come aparta crane left hanging, or every crane held and the piece still in islandsgridsubsetsnothing hangingholds togetherfailures localheld, of those passing3 × 34 joins1611100.0%100.0%4 × 49 joins512322197.8%65.6%5 × 516 joins65,5361,21578599.3%64.6%6 × 625 joins33,554,432260,625141,62199.6%54.3%a crane hangs from nothing when none of the joins at its four corners is kept — one crane, four points, no search

Nearly every cutting fails at one crane

Six by six connected cranes have twenty-five joins and thirty-three million ways to keep some of them, and an exhaustion over all of them takes a fifth of a second. Of the 33,412,811 that fail, 99.64 per cent fail at a single crane — one left holding none of the joins at its corners — which a maker can check by looking at each crane in turn. The arrangements that pass that check hold together less often as the grid grows: all of them at three by three, 54 per cent at six by six.

history · Folklore of folding the bar is the fewest joins that hold every crane in one pieceslack is how many merges those joins could make and do not: three a join, against n² − 13 × 34 joinsfloor 3 · 1 above · slack 4 · 1 way4 × 45 joinsfloor 5 · at the floor · slack 0 · 1 way5 × 510 joinsfloor 8 · 2 above · slack 6 · 50 ways6 × 612 joinsfloor 12 · at the floor · slack 1 · 1 way7 × 718 joinsfloor 16 · 2 above · slack 6 · 1,018 ways8 × 821 joinsfloor 21 · at the floor · slack 0 · 1 way9 × 928 joinsfloor 27 · 1 above · slack 4 · 308 ways10 × 1034 joinsfloor 33 · 1 above · slack 3 · 7,076 ways11 × 1142 joinsfloor 40 · 2 above · slack 6 · 2,068,604 ways12 × 1248 joinsfloor 48 · at the floor · slack 1 · 689 waysa slack of nought is a perfect tree of fours, every join merging four pieces that were separate until then

The prediction held at eight and broke at ten

A join in a slit grid of cranes merges at most four pieces, so n² cranes need at least ⌈(n² − 1)⁄3⌉ joins, and exhaustion found four and six by six meeting that floor in exactly one way. The guess was that eight by eight would too, with twenty-one joins. Counted a join at a time, it does — twenty-one, one way. Ten by ten does not: it needs thirty-four against a floor of thirty-three, and has 7,076 ways to spend them. The grids that meet the floor with no merge to spare are four and eight by eight among every size to fourteen, and sixteen by sixteen by construction, because a perfect tree of joins on a grid twice as wide is four perfect trees and one join in the middle. The even grids were never the pattern; the doublings are.

history · Folklore of folding

Named alongside it

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

Hiden senbazuru orikataSenbazuruThe counting problemConnectivityKirigamiOne sheet no cutsLocality

All concepts