Origami & Folding
what one sheet can do
Concept

Typical instances — where it appears

The population an average is taken over, made explicit. There is no canonical way to draw a crease pattern or a vertex at random, several defensible constructions disagree about the same questions by factors, and a number quoted without its population is a number about the construction that produced it.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

populationpassfoldhave a gapbranchcut twice at random51 vertices, 13 kinds9.68.020%0%whole multiples of 45°60 vertices, 1 kinds30.019.3100%69%whole multiples of 30°60 vertices, 13 kinds19.713.177%26%a named vertex, jittered60 vertices, 4 kinds8.08.00%0%

Which vertices are the random ones

Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.

complexity · Typical instances each row is an exhaustive count over the patterns that construction producedon the edgedeepest piletimes smallercrease densitythe printed patterns8 patterns62%19.415.5×7.9twist tessellations12 patterns52%10.02.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2

Four ways to draw a pattern

Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.

complexity · Typical instances the four cheap tests are polynomial in the drawing; the fifth is notreading across a row is one pattern put to all fivecrease pairsverticespanelscreasessearch nodesthe square twist6649127,565the Miura fold703152438refusedthe waterbomb sheet2,850255276refusedthe Yoshimura3,655226586refuseda square patch3,486364984refuseda rhombille patch39,621126157282refuseda refused search is a pattern about which the expensive test says nothing at all, at full price

The cost is in the coincidences

How big an instance is, is what a hardness statement is about, and it is the weaker predictor of what deciding one costs. Hold the degree fixed and vary only how many of a vertex's sectors are equal: the work of deciding it rises by a factor of nearly three, against a factor of two for doubling the number of creases. The expensive instances are the ones a designer draws on a grid.

complexity · Hardness of folding the bar is the average number of distinct folded statesevery printed pattern on this site has exactly one, and none of its swaps is legal2 folds1.1724 of 24 measured · 2 of 71 swaps legal3 folds1.3824 of 24 measured · 2 of 164 swaps legal4 folds2.1619 of 24 measured · 6 of 335 swaps legal5 folds2.336 of 24 measured · 0 of 121 swaps legala refused row is a sheet with too many panels to search, and refusals are counted rather than dropped

The crumple keeps its options

Every crease pattern this site prints has exactly one folded state and not one of its thirty-nine available rearrangements is legal. A sheet creased by folding it at random four times has an average of 2.16 folded states, one of them has nine, and six of three hundred and thirty-five rearrangements are legal. The sheet nobody designed is the one with room left in it.

material · Crumpling the bar is the share of the population with a folded stateevery pattern in all four passes every condition at every interior vertexthe printed patterns4 of 80 cannot be placed · 0 cannot be ordered · 4 undecidedtwist tessellations2 of 125 cannot be placed · 2 cannot be ordered · 3 undecidedquadrilateral meshes2 of 60 cannot be placed · 4 cannot be ordered · 0 undecidedfold-and-cut patterns5 of 70 cannot be placed · 0 cannot be ordered · 2 undecidedundecided is a real answer here and is not rounded toward either side

The patterns a checker is tested on

This site keeps four populations of crease patterns and runs its checkers over them, which is what makes a claim about typical instances measurable rather than rhetorical. Asked whether the members actually fold, the populations answer: thirteen of thirty-three do, six place and cannot be ordered, five cannot be placed at all, and nine are past what the search will finish.

complexity · Typical instances the bar is how many of the 33 patterns each refusal is the first to catchtwo creases cross5one sweep over pairs of creasesa vertex condition fails0one pass over the verticesthe panels do not place0one walk over the panelsthe letters force a loop0one pass over the crease listno ordering exists6every ordering of the panels22 of the 33 are refused by none of these and are folded, undecided, or waiting on a search too large to run

The order the refusals come in

This collection can say no to a crease pattern in five ways, and they cost wildly different amounts: a sweep over pairs of creases, a pass over the vertices, a walk over the panels, a pass over the crease list, and an enumeration of every ordering of the panels. Run all five over the thirty-three patterns in the four test populations and the cheapest refuses five, the most expensive refuses six, and the three in between refuse nothing at all.

complexity · Hardness of folding the bar is the share of random drawings with at least one crossing in them2 segments23.1%0.23 crossings on average3 segments51.2%0.69 crossings on average4 segments73.5%1.36 crossings on average6 segments95.2%3.48 crossings on average8 segments99.4%6.53 crossings on average12 segments100.0%15.30 crossings on average20 segments100.0%43.76 crossings on averageevery crease pattern in this collection has none, and none of them was drawn at random

Drawn by the same hand

Two straight segments dropped on a square cross about 23% of the time; four of them cross 74% of the time; twelve cross with certainty, about fifteen times over. Every crease pattern in this collection's four test populations has none — not because the checkers were catching them, but because the same rules that drew the patterns were incapable of producing one, and nothing looked until a construction finally did.

complexity · Typical instances 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 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 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.

MeasurementDecision procedureNecessary conditionSamplingSearch costWorst-case analysisCrease patternCrimpingCrossingEvidenceFolded stateLayer order

All concepts