Origami & Folding
what one sheet can do
Concept

Connectivity — where it appears

Whether a cut sheet remains in one piece. It is a step function of how the cuts are placed rather than a gradient, so a slit array holds together and then falls apart all at once rather than weakening gradually.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

evenly spaced — 4 creasesany flat folding16 of 16some-layers16 of 16all-layers16 of 16one-layer2 of 16crimping only6 of 16uneven — 4 creasesany flat folding8 of 16some-layers8 of 16all-layers0 of 16one-layer2 of 16crimping only0 of 16a machine that takes fewer layers is weaker, not more patientthe paper is joined, so what it declines to hold it also cannot move

The patient machine is the weak one

A machine that folds one layer at a time sounds like a machine with more freedom, not less. It has less, and the reason is the most ordinary fact about paper there is: it is joined, so whatever a machine declines to hold it also cannot move.

complexity · Simple foldability cuts at 85% of the pitch0.15 of material between one cut and the nextcut length, as a fraction of the pitchligament6 piecesone piece at every length up to 97%and the count is worked out from the cuts, not measured off a picture

One cut short of falling apart

Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.

material · Kirigami the bar is how many moves survive the conditions at a vertexa move flips two creases meeting at one point, which is what pushing a vertex through doesthe square patch0216 pairs tried at each of two letterings · 0 legal · 0 leave the verdict alonethe elongated patch6270 pairs tried at each of two letterings · 6 legal · 6 leave the verdict alonethe hexagonal patch8360 pairs tried at each of two letterings · 8 legal · 8 leave the verdict alonethe triangular patch8360 pairs tried at each of two letterings · 8 legal · 8 leave the verdict alonethe rhombille patch16756 pairs tried at each of two letterings · 16 legal · 16 leave the verdict aloneevery one of them leaves the lettering on the side of the question it was already on

Every move leaves the verdict

The only change a folder can make to a lettering without breaking it is to push one vertex through, flipping two creases at once. Try every such move on five tessellation patches, from two different letterings each: nineteen of two thousand nine hundred and sixty-four survive the conditions, and not one of the nineteen turns a lettering that agrees with itself into one that does not, or the other way about.

flat-folding · Local moves what each sheet's shape costs in conditionsa square0 loopsχ = 1 · no loop that cannot be shrunka slit from the rim0 loopsχ = 1 · the same paper, topologicallyone hole1 loopχ = 0 · one parity conditiona cylinder1 loopχ = 0 · the same sheet as one holetwo holes2 loopsχ = -1 · two independent conditionsa torus2 loopsχ = 0 · two conditions, no rim at alla slit inward from the rim changes nothing, and a closed cut changes everything

The cut that changes nothing

A slit goes right through the material and leaves the sheet exactly the object it was. A closed cut removes almost no paper and produces a different sheet with a condition it did not have. Kirigami is made almost entirely of the first kind, which is why every result about it survives the distinction untouched.

material · Kirigami 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 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.

The counting problemSenbazuruHiden senbazuru orikataParityBoundaryKirigamiLattice identityLocalityThe all-layers simple foldAssignmentBuried creaseConfiguration space

All concepts