Origami & Folding
what one sheet can do
Series

Folklore of folding — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · history
  2. 10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed

    The star that was cut before it was proved

    Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.

    part 2 · history
  3. the stack under a single cuta clean cut taken as 0.6 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm10 points20 layers1.3 mm2.0 mm1.4 mm800 µm520 µm360 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm16 points32 layers2.1 mm3.2 mm2.2 mm1.3 mm832 µm576 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.6 mm of stack

    How many wedges the paper allows

    A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.

    part 3 · history
  4. 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.

    part 4 · history
  5. 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.

    part 5 · history
  6. 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.

    part 6 · history
  7. 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.

    part 7 · history

All series