Origami & Folding
what one sheet can do
Concept

Genericity — where it appears

The property of holding for almost every member of a family, with the exceptions forming a set of measure zero. Flat-foldability is the opposite: it is the exceptional set, so every pattern in this subject was arranged.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

-3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding

Almost every pattern fails

Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.

flat-folding · Genericity folded 8 times, then unfolded33 interior vertices, all of degree 433 of 33 satisfy Kawasakithe folding is the reason, not the drawing45 creases drawn at random485 interior vertices, all of degree 40 of 485 satisfy Kawasakisame count, same sheet, nothing folded

The creases a sheet gives itself

A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss.

material · Crumpling 0.5°1%1°2%2°3%5°8%10°16%20°33%how far each sector would have to moveshare that would fold40,000 random four-crease verticesthe median vertex is 31.31° per sector from folding, the mean 33.91°none of them folds, and almost none of them nearly does either

A near miss is nearly as rare

Flat-foldability is a coincidence of measure zero, which is usually where the argument stops. Measure how far a random vertex is from folding rather than whether it does, and the answer is thirty-one degrees a sector — so the tolerance real paper has does not buy back anything at all, and a pattern that nearly folds had to start near one that did.

flat-folding · Genericity 102°60°78°120°one smallest sectorthe lemma constrains one pair4 foldable assignmentsof the 16 markings120°60°60°120°two smallest sectors equalthe lemma constrains nothing8 foldable assignmentsof the 16 markings

Where the lemma says nothing

The big-little-big lemma asks for a sector strictly smaller than both its neighbours, and the word doing the work is strictly. At a vertex whose two smallest sectors are equal the lemma has no opinion at all — and those are the vertices origami actually uses. The count of markings the conditions admit doubles, discontinuously, at exactly the angles everybody folds.

flat-folding · Sector angles the bar is the middle run of a hundred and twentysame pattern, same code — only the order the letters are tried in differsthe square patch2725 at best · 27 at the middle · 36 at worstthe elongated patch3432 at best · 34 at the middle · 39 at worstthe hexagonal patch4339 at best · 43 at the middle · 51 at worstthe triangular patch4539 at best · 45 at the middle · 53 at worstthe rhombille patch16684 at best · 166 at the middle · 48 of 120 unfinished at 20000an unfinished run is left out of the middle rather than counted as its budget

Four easy patches and one that is not

Run the same search a hundred and twenty times on each of five tessellation patches, changing nothing but the order the letters are tried in. Four of them answer in between twenty-five and fifty-three steps every single time. The fifth answers in eighty-four steps at best, a hundred and sixty-six in the middle, and does not answer at all in forty-eight runs of the hundred and twenty.

tessellation · Twists 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 the second-smallest sector at a typical vertexthe triangular patch at seven turns, with the same panels and the same creases at all of themturn 0.261.92°smallest sector 60.00° · next 61.92° · no lettering exists, proved by exhaustionturn 0.2160.71°smallest sector 60.00° · next 60.71° · no lettering exists, proved by exhaustionturn 0.215560.06°smallest sector 60.00° · next 60.06° · no lettering exists, proved by exhaustionturn 0.21660.00°smallest sector 60.00° · next 60.00° · a lettering existsturn 0.2260.00°smallest sector 59.52° · next 60.00° · a lettering existsturn 0.2560.00°smallest sector 56.10° · next 60.00° · a lettering existsturn 0.3560.00°smallest sector 46.15° · next 60.00° · a lettering existsthe verdict changes exactly where that sector passes sixty degrees and stops being the second smallest

Where a sector crosses sixty

Turn the twist polygons of a tessellation patch a hundredth of a radian further and the pattern goes from having no mountain-valley labelling at all to having one immediately. Nothing about its graph changes across the transition — the same eighty-three panels, the same hundred and forty-two creases, the same four labellings at every one of its sixty vertices. What changes is which sector at a vertex is the smallest one.

flat-folding · Genericity 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 region with no lettering

One turn angle at which a tessellation patch has no consistent lettering was found by sweeping a dial. Sweeping two dials finds nine patches with none, across three tilings, filling a corner of the parameter space — and never touching the square tiling, whose sectors have no sixty degrees to cross.

flat-folding · Genericity the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

An alternating sum of angles

Kawasaki's condition says the sectors round a vertex alternate to a straight angle. A glued band has no vertices and obeys a condition of exactly the same shape: the crease angles have to alternate to a multiple of a straight angle. Two different quantities, two different sheets, one arithmetic — and in both cases what is being said is that a product of reflections came back the right way.

flat-folding · Sector angles the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

How rare a band that folds is

Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.

flat-folding · Genericity

Named alongside it

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

Sector anglesKawasaki's theoremNecessary conditionAssignmentThe big-little-big lemmaFlat-foldabilitySearchTwistClosureCountingDegeneracyGluing

All concepts