Origami & Folding
what one sheet can do
Concept

Hiden senbazuru orikata — where it appears

The 1797 book of connected cranes, the oldest printed source the subject has. Its patterns are made by slitting a sheet, so the founding document of recorded paper folding is about cutting.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

Paper is made in ChinaPaper reaches JapanPaper is made in EuropeFolded paper is used ceremonially in Japan400 yrPaper is folded for amusement in Japan980 yrThe thousand cranes897 yrThe pajarita is folded in Spain293 yrPaper folding is taught as geometryOne fold solves a cubicThe diamond pattern in a crushed cylinderThe conditions at a flat-foldable vertexThe dashed-and-dotted diagram notationThe Miura foldA five-pointed star from one straight cutAny straight-line drawing, from one straight cutyear of the source500100015002000the date generally giventhe oldest source that says somedian overrun 201.5 years

Nothing here is as old as it sounds

Paper folding is described everywhere as an ancient art. The oldest surviving book of it was printed in 1797, the oldest reference to folding for amusement is from 1680, and the median claim in this subject is dated two centuries before anything that attests it.

history · Attribution 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 how many witnesses each claim has5 of 15 have one, and a loss removes them from the record rather than weakening themclaimsurviving sourcesafter one lossPaper is made in China21 leftPaper reaches Japan1nothing attests itPaper is made in Europe32 leftFolded paper is used ceremonially in Japan43 leftPaper is folded for amusement in Japan1nothing attests itThe thousand cranes1nothing attests itThe pajarita is folded in Spain21 leftPaper folding is taught as geometry43 leftOne fold solves a cubic1nothing attests itThe diamond pattern in a crushed cylinder21 leftThe conditions at a flat-foldable vertex32 leftThe dashed-and-dotted diagram notation21 leftThe Miura fold32 leftA five-pointed star from one straight cut1nothing attests itAny straight-line drawing, from one straight cut21 leftby kind of source: artefact 2 · manuscript 2 (1 single) · printed 9 (3 single) · secondary 2 (1 single)the average overrun falls from 357 years to 193 without them, and the median from 201.5 to 184.5 — the effect is two rows, not a tendencywitnesses counted from the record itself · Paper is folded for amusement in Japan and The thousand cranes are the two largest overruns and have one document each

One lost source and the story changes

Five of the record's fifteen claims rest on exactly one surviving document. Take those away and the field's most-quoted statistic — how far ahead of its evidence a popular date runs — falls from three hundred and fifty-seven years to a hundred and ninety-three, while the median hardly moves at all. The effect is not a tendency spread through the record; it is two documents, and both of them are single-witness.

history · Two traditions 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 the lattice points of two grids, coloured like a checkerboardringed: the point each corner crane touches, which must be a join5 by 5: 16 lattice pointsthe four ringed points are two coloursso no perfect tree can hold all four4 by 4: 9 lattice pointsthe four ringed points are one colourso a perfect tree may use that colour

No odd grid holds a perfect tree

A perfect tree of joins holds a slit grid of cranes with every join merging four pieces that were separate. Two joins side by side share two cranes and close a loop, so a perfect tree never has two — and so all its joins lie on one colour of the checkerboard the lattice points make. The four corner cranes each touch one lattice point, which must be a join; on a grid with an odd side those four points are two colours. So no odd grid holds a perfect tree, and seventeen by seventeen, the size the count was pointing at, is settled without counting. On one colour the search is small enough to finish: sixteen by sixteen has exactly one perfect tree, the doubled one, and twenty by twenty has none.

history · Folklore of folding the perfect trees on eight, sixteen and thirty-two, drawn by the joins they leave outdots: joins kept; crosses: points of the tree's colour cut, the middle row and column marked in the second colour8 by 8: 4 cuts for 12 loopsfour cuts round the middleevery other join kept16 by 16: 28 cuts for 84 loops12 on the middle row and column16 in the four quarters32 by 32: 140 cuts for 420 loops28 on the middle row and column112 in the four quarters

A perfect tree is the joins it leaves out

On the colour a perfect tree lives on, the lattice points and the cranes they share form a graph full of loops, and a perfect tree is that graph with some points cut out. Euler's count fixes how many: every cut removes three loops' worth, and the loops always number exactly three times the cuts a perfect tree needs — so a set of that many cuts, no two sharing a crane, leaves a perfect tree exactly when what remains is one piece. Searched that way, ten by ten fails in seven cases that fit on one page, fourteen in 761, and sixteen's one tree is four copies of eight's diamond of cuts with a cross of cuts along the middle.

history · Folklore of folding

Named alongside it

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

SenbazuruThe counting problemConnectivityLattice identityAttestationDocumentary recordKirigamiOne sheet no cutsParityPrimary sourceIndependent discoveryLocality

All concepts