Origami & Folding
what one sheet can do
Concept

Senbazuru — where it appears

The practice of folding a thousand cranes, and the 1797 book that first recorded connected-crane patterns. That book cuts the paper, which makes the oldest printed origami source a kirigami source.

Named by 7 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 share of sound cuttings that hold the cranes in one piecesound means every crane keeps at least one join at its corners3 × 3100.0%2⁴ subsets of 4 joins · 1 states carried4 × 465.6%2⁹ subsets of 9 joins · 7 states carried5 × 564.6%2¹⁶ subsets of 16 joins · 23 states carried6 × 654.3%2²⁵ subsets of 25 joins · 52 states carried7 × 751.1%2³⁶ subsets of 36 joins · 123 states carried8 × 846.7%2⁴⁹ subsets of 49 joins · 294 states carried9 × 943.6%2⁶⁴ subsets of 64 joins · 714 states carried10 × 1040.5%2⁸¹ subsets of 81 joins · 1,758 states carried11 × 1137.7%2¹⁰⁰ subsets of 100 joins · 4,380 states carried12 × 1235.1%2¹²¹ subsets of 121 joins · 11,024 states carriedcounted one join at a time, carrying only which cranes on the frontier are already joined

The border is where the cranes come apart

Counted one join at a time rather than one subset at a time, the slit grid of connected cranes runs to twelve by twelve, where there are 2¹²¹ ways to keep some of the joins. Among the cuttings that hold every crane by something, the share that also hold together keeps falling — 54.3 per cent at six by six, 35.1 at twelve — and from eight by eight on it falls by the same factor at every size. A constant factor is the signature of the border: at six by six, 99.5 per cent of the sound cuttings that come apart do so through a stray piece touching the outermost ring of joins.

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 orikataThe counting problemLattice identityConnectivityAttestationDocumentary recordKirigamiLocalityOne sheet no cutsPrimary sourceBoundaryIndependent discovery

All concepts