Origami & Folding
what one sheet can do
How this site is made

The figure library — page 12

Every picture here is generated from code at build time. This is every generator, ordered by how many essays call it, each rendered at its defaults.

object-order

2 essays — The order does not name it either · Which side is showing · 3 claims checked

What the order settles, and what it leavesFor strips with a given number of creases on a given grid: how many distinct folded profiles there are, how many of those are reached by more than one crease pattern, and how many of those ambiguities survive being handed the complete layer order as well.what the outline and the thickness leave open, and what the order closesprofilesambiguousthe order settlesand does not3 creases on 12ths69166103 creases on 16ths1844712354 creases on 12ths23371692the last column is patterns that fold to the same object, so no better photograph reaches them

quad-family

2 essays — The condition that is not flat-foldability · One crease decides the sheet · 11 claims checked

One family, and the one member of it that movesThree quadrilateral meshes from a one-parameter family. One vertex has been slid along the ray that keeps Kawasaki's condition exact at every interior vertex, so all three satisfy every flat-folding condition this site checks. Under each is how far the four vertices round one face are from agreeing about the crease they share.flat-foldable at every vertex, all threeworst Kawasaki residual 0e+0 radiansone vertex moved -30 per centloop residual 2.4e-1the Miuraloop residual 5.8e-14one vertex moved 40 per centloop residual 3.6e-1every one of these is developable and flat-foldable at every interior vertex, exactly

reachable-states

2 essays — A state no motion reaches · Opening with nothing to pull · 3 claims checked

States the motion never reachesFor five degree-four vertices: how many mountain-and-valley assignments satisfy every local flat-folding condition, and how many of those a continuous rigid motion actually arrives at. The two numbers are computed by machinery that shares no code, and where they differ there is a folded state that exists and cannot be got to without bending a panel.60° / 90°all 4 reached30° / 120°all 4 reached45° / 45°2 of 8 reached50° / 70°all 4 reached80° / 55°all 4 reachedsectorseach square is one assignment the theorems allowfilled — a rigid motion arrives there · open — a flat state with no path to itthe gap opens where two sectors are equal, and nowhere else on this listbig-little-big has nothing to forbid there — the linkage still does

reference-reach

2 essays — How far from the nearest reference · The third fold cannot be listed · 6 claims checked

The count of references and the reach they buyFor no folds, one fold and two: how many reference points the sheet carries, and the radius of the largest disc that can be placed on it without covering one. The count grows by two orders of magnitude and the worst gap closes by less than one.the bar is the largest gap between anywhere on the sheet and a referencea folder who needs a crease somewhere in particular is asking about this and not about the countno folds0.7074 references · worst point 0.50, 0.50 · under 0.710one fold0.3549 references · worst point 0.25, 0.25 · under 0.3572 folds0.089565 references · worst point 0.93, 0.93 · under 0.092141 times the references for 8.0 times the reach, which is what a count hides

river-packing

2 essays — What joins the flaps · Paying in paper · 8 claims checked

Rivers, and what they separateOne disc per flap, and — where two groups of flaps hang off different nodes of the skeleton — a strip of paper between them whose width is the length of the edge that joins them. The rule is not that circles must not overlap; it is that any two flaps must be at least as far apart on the sheet as they are through the tree.riverwidth 0.3legarmheadlegtailthe check10 pairs testedtightest by 0.0800(leg and arm)the ruledistance on the sheetat leastdistance through the treetwo nodes, and an edge between themthe extra width is thebody the flaps hang fromthe discs are what each flap costs; the strip is what joins the two halves of the subjectand both are the same condition, read off different pairs of leaves

split-events

2 essays — One cut for a star · The corner that splits the shrink · 10 claims checked

Shrinking a shape that turns back on itselfAn outline with a reflex corner, drawn with the shrunk copies of itself that define its straight skeleton, and the track that corner takes. Every other corner moves inward; this one moves out, and where it reaches an edge on the far side the shrinking region breaks into two.the outline, shrunk — and the one corner that moves outwardwhat the shrink finds6 edges, 1 of them meeting at a corner that turns back3 skeleton nodes8 arcs traced by the cornersthe last of them forms at 0.150 of a sheet

twist-bounds

2 essays — Fenced at both ends · A no costs more than a yes · 5 claims checked

The band a twist tessellation lives inThe two ends of the twist angle, over how much of the available room the twists take up. Above the upper curve the pleats have negative width and there is no paper; below the lower one the pattern has no mountain-valley assignment at all, though every angle condition still holds. The band between them is where a twist tessellation exists.0°30°60°90°0.250.400.550.700.85how much of the room between two vertices the twists takeno paper leftno assignment existstwist angleboth curves are measured rather than plotted from a formula

two-counts

2 essays — Two things called folding · Two ceilings · 4 claims checked

Two things called folding, and only one of them has a local testThe number of states against the number of units, on a log scale. Both grow exponentially and that is not the difference. The difference is that a crease pattern's states can be filtered by four conditions checked one vertex at a time, and a chain's cannot be filtered by anything local at all.246810012345units (creases, or joints)log₁₀ statesa chain, 3 states per jointa sheet, two letters per creaseat one vertex, 4 of 16 assignments survive four local conditionsthe sheet's count has a local test that removes 75% of it · the chain's has none

wave-family

2 essays — The excess does not choose its waves · The container picks the member · 8 claims checked

The excess fixes the family, not the memberA strip whose rim has grown longer than its span has to put the extra length somewhere. Spending it in two waves and in eight are the same metric — identical arc length, identical excess — and the geometry has nothing to say about which. Ranking them needs a rule about the material, which is not a question this site answers.one excess of length, 4 ways to spend it2 wavesheight 0.0398∫κ² ds = 173 wavesheight 0.0266∫κ² ds = 395 wavesheight 0.0159∫κ² ds = 1088 wavesheight 0.0100∫κ² ds = 276amplitude × waves is constant, so these are one shape at four scalesexcess 6.0% of the span · every profile below has exactly that excess

allowance-motion

1 essay — The allowance is spent at the end · 5 claims checked

The allowance a solved mesh has, at each point of its foldHow far a quadrilateral mesh may be cut wrong before its closure fails, measured all the way along the motion rather than at the one fold angle a tolerance is normally quoted at. It falls by an order of magnitude between the flat sheet and the packed one.the height is the allowance in millimetres on a 150 mm sheetthe horizontal axis is the fold angle of the driven crease, in radians0.425 mm0.033 mm0.32.4fold angle of the driven creasea budget of 0.02 radians on a 150 mm sheet12.9 times less allowance at the closed endthe mesh is most forgiving where it is doing least

angles-not-lengths

1 essay — The lengths are free · 2 claims checked

One vertex, several sets of crease lengthsThe same two free sector angles drawn with the creases run out to wildly different lengths. Kawasaki, Maekawa and the big-little-big lemma read the angles and nothing else, so every one of these is the same vertex as far as any condition in the subject is concerned — and each folds into a different shape.sectors 80°, 55°, 100°, 125° in every one of them, and 4 assignments fold in every onelongest ÷ shortest 1.00footprint 0.806longest ÷ shortest 3.09footprint 0.911longest ÷ shortest 3.33footprint 0.623longest ÷ shortest 4.00footprint 1.782every one of them folds; their folded footprints differ by a factor of 2.86

contact-graph

1 essay — Two packings, one radius · 3 claims checked

Two packings, one radiusTwo arrangements of the same number of discs, each returned by a complete run of the same search, agreeing about the largest radius to four decimal places. The lines join discs that touch, and the crease pattern a designer builds is built from those lines. The two are not the same graph.6 flaps at radius 0.1875883 contact graphs among the runs that agree about itone run5 contacts · 5 against the paper's edgeanother run4 contacts · 5 against the paper's edge