Origami & Folding
what one sheet can do
Concept

Sampling — where it appears

Drawing members of a set that is too large to enumerate, in order to estimate a proportion in it. A sampler over solutions of a constraint problem is not a uniform draw over them, so a share it reports is a share of what it found.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

the bar is the share of draws that agree with themselvesthe rows are ordered by panel count, which is the only thing changing along them49 panels26 of 200square · 84 creases · 26 of 20062 panels5 of 200elongated · 106 creases · 5 of 20077 panels2 of 200hexagonal · 142 creases · 2 of 20083 panels0 of 200triangular · 142 creases · 0 of 200157 panels0 of 200rhombille · 282 creases · 0 of 200a zero is a zero of the draws taken and not a proof that no consistent lettering exists

Letters that agree get rarer

Two hundred letterings drawn independently from a square twist tessellation patch, and twenty-six of them have letters that do not contradict themselves. On the next patch up it is five, then two, then none, then none. What the share falls with is not the size of the patch and not the angle of its twist: it is the number of independent closed chains its panels form, which is Euler's relation on the drawing and is fixed before a single letter is chosen.

tessellation · Twists the bar is the mean share of redraws that agree with themselvesas the populations stand, every member is consistent and the refusal fires on none of themthe printed patterns96.7%8 of 8 could be asked · worst member 90%twist tessellations55.0%7 of 12 could be asked · worst member 7%quadrilateral meshes96.9%6 of 6 could be asked · worst member 82%fold-and-cut patterns100.0%7 of 7 could be asked · worst member 100%a member with no folded state has no letters to redraw and is counted as not asked rather than as passing

A population that cannot fail

Thirty-three crease patterns are kept here to run the checkers over, and every one of them has letters that agree with themselves. That is not a property of the patterns. It is a property of how they were made: each came from a construction that returns a lettering, so a test looking for letters that contradict themselves has nothing to fire on. Reletter the same thirty-three and the failure is available at once — on one member, four of sixty redraws.

complexity · Typical instances the letters a folding gives a sheet always agree — these are the ones it might have had insteadfolded from seed 7folded from seed 11folded from seed 2300.2500.5000.750110203040panels in the folded sheetshare of redrawn letterings that agreeeach point is one sheet folded a given number of times, and the horizontal axis is what that produced

The letters a crumple was given

A sheet creased by folding it and folding it again arrives with a mountain-valley labelling that cannot be wrong, because a folding produced it. Nothing about the pattern protects it: reletter the same creases and the share of labellings whose letters agree falls from every one of forty at eight panels to eleven of forty at forty-one. The foldability of a crumple is a fact about its history, not about its drawing.

material · Crumpling a lettering of the patch that agrees with itselffound by testing the arcs while the letters were chosen, not after561 nodes · 246 backtracks · verified against a rebuilt folded sheet157 panels · 282 creasesits own lettering sends its panels round in a circle0 of 200 random letterings agree with themselvesthis one was found in 561 nodes and 246 backtracksit differs from the drawn lettering on 155 of 282 creasesthe drawing is the pattern; nothing here is a picture of the folded object

The lettering nobody could draw

Two hundred letterings drawn at random from the rhombille tessellation patch, and not one of them agrees with itself. Two thousand, and still not one. The patch was left as an open question — and it has an answer, found in five hundred and sixty-one steps by a search that tests the arcs while it is choosing the letters instead of after it has chosen them all.

flat-folding · Forced order the bar is what the whole job costs if every attempt is stopped thereon the rhombille patch, read off 120 measured runsstop at 10051219% of runs finish by thenstop at 20053033% of runs finish by thenstop at 500105435% of runs finish by thenstop at 1000162442% of runs finish by thenstop at 2000263847% of runs finish by thenstop at 5000569749% of runs finish by thenstop at 100001060450% of runs finish by thenstop at 200001629160% of runs finish by thena run that never finished counts as above every cutoff, so the tail is read conservatively

Stopping is cheaper than finishing

A search whose cost varies by a factor of two hundred with nothing but the order of its guesses should not be waited out. Give up after a hundred steps, reseed and start again, and the whole job costs five hundred and twelve steps in expectation; run each attempt to twenty thousand and it costs sixteen thousand two hundred and ninety-one. Patience is thirty-two times more expensive than impatience.

complexity · Hardness of folding the bar is how many patterns the population holdseach one sampled forty times and then searched, to see whether the two methods ever disagreethe printed patterns80 never lettered by 40 draws · all 8 settled by search · worst 60 nodestwist tessellations70 never lettered by 40 draws · all 7 settled by search · worst 19 nodesquadrilateral meshes60 never lettered by 40 draws · all 6 settled by search · worst 6 nodesfold-and-cut patterns70 never lettered by 40 draws · all 7 settled by search · worst 14 nodesthey never do here — the patterns that separate them are not in any of these four

Four populations with nothing to separate

This collection keeps four standing populations of crease patterns to test its machinery against. Twenty-eight patterns, sampled forty times each for a lettering that agrees with itself and then searched for one — and on every single member the two methods return the same verdict in the same breath. The patterns that separate them are in none of the four, and the reason they are not is what the populations are for.

complexity · Typical instances the bar is how many nodes the search visitedone sheet crumpled deeper and deeper, its letters rechosen each time4 folds1716 panels · 34 of 40 random letterings agree · 1 backtracks5 folds1918 panels · 34 of 40 random letterings agree · 1 backtracks6 folds3435 panels · 15 of 40 random letterings agree · 0 backtracks7 folds3839 panels · 19 of 40 random letterings agree · 0 backtracks8 folds7271 panels · 11 of 40 random letterings agree · 2 backtracksthe share that agrees falls by more than half along this ladder; the search's cost tracks the panels and nothing else

Rare is not hard

Crumple a sheet deeper and the share of its labellings that agree with themselves falls from thirty-four in forty to eleven. The number of steps a search needs to find one of them does not move at all: it stays at about one per panel, with no backtracking, the whole way down. How often an answer turns up at random and how much work it takes to find one are different quantities, and a crumpled sheet is where they come apart.

material · Crumpling the bar is the largest gap between anywhere on the sheet and a referencea third fold specifies more folds than can be listed, so a sample of them is taken insteadnone of them0.089565 references, from two folds50 of them0.0763,498 references · 0.02% of the round100 of them0.0508,056 references · 0.04% of the round200 of them0.04723,480 references · 0.07% of the round400 of them0.02274,694 references · 0.15% of the round800 of them0.014270,882 references · 0.29% of the roundevery row is a lower bound on what the whole round would buy, because leaving folds out can only make the gap larger

The third fold cannot be listed

Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.

construction · Reference points the bar is how many DIFFERENT letterings 20 runs returneda coin at every choice1414 of 20 runs found onea constant, with the coin only on the creases no vertex constrains120 of 20 runs found onea constant at every choice120 of 20 runs found oneon the rhombille patch, 157 panels and 282 creases

One witness or forty

Taking the randomness out of a search made it three orders of magnitude cheaper in the worst case and cost it thirty-nine of its forty answers. The compromise everybody reaches for — randomise only the choices that cannot matter — recovers four of the forty on two patches and none on the other three, because the diversity was never where it looked.

flat-folding · Layer multiplicity each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all

A population nobody chose

Five crease patterns were measured over and over because somebody had drawn five. Ninety-six drawn from a stated grid of tiling, turn and pleat width say something the five could not: nine of them have no consistent lettering at all, and the phenomenon the collection had spent so long measuring belongs to the one tiling the grid leaves out.

complexity · Typical instances each point is one pattern: panels across, nodes up00100100200200one node per panelnodes visitedpanels2 by 2 to 16 by 16, and not one backtrack anywhere in the family

A corrugation never backtracks

As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.

tessellation · Miura each point is one pattern: panels across, nodes up00202040406060one node per panelnodes visitedpanels3 folds to 8 folds, and not one backtrack anywhere in the family

A crumple has no tail

The least structured crease pattern this collection can produce is a sheet folded at random and flattened. Its consistent letterings get rarer as it deepens — thirty-four of forty down to eleven — and finding one costs one step per panel from beginning to end, with no wrong guess anywhere. Disorder and difficulty turn out to be unrelated quantities.

material · Crumpling sliding the cut across one period of the square tessellation36 vertices at every position, and a different set of creases divided at each0102030cut at the start of a periodone period alongnodes; the axis starts at zero, and the whole spread is inside a factor of 1.32

Where you cut hardly matters

Slide the same rectangle across one whole period of the same tessellation and every position gives a different patch: different creases divided, different half-panels round the edge, panel counts from forty-nine to sixty-one. The cost of lettering them runs from twenty-five steps to thirty-three. Whether a cut is made changes the answer by three orders of magnitude; where it falls changes it by a third.

complexity · Typical instances the grid, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices4444free letters1210108panels9664V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says

One population, four sheets

A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.

complexity · Typical instances

Named alongside it

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

AssignmentSearch costCrease patternMeasurementSearchEnumerationConstraint propagationCrumplingFlat-foldabilityLayer orderLayer orderingPatch

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