Origami & Folding
what one sheet can do
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The figure library — page 4

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.

notation-coverage

9 essays — What a dashed line can say · The half no notation records · A file has no paper · The reader decides the junction · The file records no verdict · The first thing about layers · No format has a gluing · The field is empty where it would say nothing · One choice with eleven answers · 11 claims checked

What a dashed line can sayThe share of a strip's flat foldings that a sequence of simple folds can reach, over seeded random spacings with every assignment of each enumerated. A dashed line and a dotted line are exactly one simple fold, so this is the reach of the basic notation — and it collapses as the model grows, which is why the vocabulary acquired named symbols for the moves that are not simple folds.34560%20%40%60%80%100%creases in the stripreachable by simple folds72%30%17%13%the basic symbolsa dashed line — valleya dotted line — mountainan arrow — fold it nowand what they missreverse, squash, sink,petal — every one of thema move no dashed linecan ask forevery assignment of 68 seeded spacings

opening-monotone

9 essays — The bud chooses the pattern · Opening with nothing to pull · Four finders, one option · The census returns one · Four materials, four optima · The fold count sets the spring · How far open is a question about the grip · Springs that disagree do not offer a choice · Holding a fold moves the force · 39 claims checked

The opening has to run one wayA corrugation opening as a single parameter runs from packed to flat. The exposed span rises at every step, which is what lets growth alone drive the motion: there is no hand and no muscle in a leaf, so a pattern that had to narrow before it widened would have nothing available to narrow it.00.20.40.60.811.21.402468fold angle (radians)exposed spanθ = 0.06θ = 0.75θ = 1.449 panels · one parameter · the span rises at every step, so nothing has to reverse

pattern-population

9 essays — Four ways to draw a pattern · The cost is in the coincidences · The patterns a checker is tested on · The order the refusals come in · Drawn by the same hand · The refusal that reads the list once · A population that cannot fail · A population nobody chose · One population, four sheets · 12 claims checked

Four ways of making a crease pattern, four answersFour constructions that each produce crease patterns satisfying every vertex condition, asked the same four questions: what share of each pattern's vertices lie on the edge of the paper, how many layers deep the folded state gets at its worst point, how many times smaller the folded footprint is than the sheet, and how much crease length each unit of paper carries. They are not four samples of one population — no two of them produce the same patterns — and they disagree by factors rather than by margins.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%9.82.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2

rule-sweep

9 essays — Sixty-four rules, sixteen fold · The loop is in the rule · Where a rule can close a loop · Half the recipe is decoration · The leaf's rules are the Miura's · A corrugation never backtracks · The plant's pattern is not a hard case · A recipe needs degree four · Two sentences for the Yoshimura · 29 claims checked

Four corrugations, every repeating rule triedFor each of four corrugation families, how many of its repeating mountain-valley rules satisfy every condition at every interior vertex. The note records what refuses the rest: in all four families it is the counting theorem alone, with the angle condition and the smallest-sector lemma holding at every failing vertex.the bar is how many repeating rules fold flat at every vertexeach family's rules are every way of letting the letters depend on the row and column paritiesthe Miura fold1664 rules · 48 refused, all by the count · 38 of them close a loopthe tapered leaf1664 rules · 48 refused, all by the count · 38 of them close a loopthe Yoshimura pattern2664 rules · 38 refused, all by the count · 0 of them close a loopthe waterbomb tessellation32512 rules · 480 refused, all by the count · 120 of them close a loopthe counting theorem does all the refusing in all four families, and it is the oldest statement in the subject

table-cost

9 essays — The lettering that was proved impossible · One step per panel is a table size · Six creases and the same straight line · A knife edge nine decimals wide · The most decided vertex here · The edge was not what made it hard · The designer's grid is the dearest thing here · A count is not a length · Nothing grown was cut out of anything · 19 claims checked

One node per panel is a table sizeEach dot is one crease pattern: across, how many labellings the conditions at a vertex leave on average; up, how many search nodes it costs per panel. The dashed line at one is where the grid, the leaf, the Miura and the crumple sit exactly. The twist patches, whose vertices keep four labellings, are at half of it; the Yoshimura as it is normally drawn keeps thirty and is just below one.labellings a vertex keeps, against nodes a panel costs0.000.250.500.751.00481530the box-pleating gridthe tapered leafa crumple, deepeningthe waterbombthe Yoshimura, as drawnthe Yoshimura, tiltedthe twist patcheslabellings the conditions leave at a vertexthe dashed line is one node a panel, which four of these families sit on exactly

why-a-circle

9 essays — A flap costs a circle · Packing is the hard part · What joins the flaps · How much paper is wasted · Every flap on one axis · The corner is worth four times the middle · The flap nobody holds · A hole is cheap paper · A notch is not a hole · 1 claim checked

A flap costs a circleA flap of a given length, folded from a point on the sheet, uses up every point within that distance of it. Two flaps whose circles overlap are asking for the same paper twice, which is the conservation argument the whole design method rests on.Levery point within L is spentthe flapLL = 0.28 of the sheet's side, so the disc costs πL² = 24.6% of itthe circle is not a metaphor — it is the paper the flap consumesso designing a base is packing circles

crumple-pattern

8 essays — The creases a sheet gives itself · Every facet is a layer · The crease that stops in the middle · The decision a crumple has taken · The crumple keeps its options · The letters a crumple was given · Rare is not hard · A crumple has no tail · 4 claims checked

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.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

hinge-bound

8 essays — Nothing in a body folds on a line · How much surface fits in a body · Four materials, four optima · The number is the angle · The surface has to be supplied · The fold count sets the spring · How far open is a question about the grip · What a second deployment costs · 7 claims checked

Nothing in a body folds on a lineA hinge that cannot go below a radius consumes a fixed length of surface every time it is used, so the share of the sheet spent on hinges is proportional to the number of folds. Past some count the pattern is mostly hinge and a finer fold buys nothing, and where that happens is arithmetic.010203040506000.20.40.60.81foldsshare of the sheet lost to hinges50% of the sheet64 foldshinge radius 0.08 on a 10 unit sheet · (π − 2)ρ = 0.0913 lost per fold

holed-sheet

8 essays — A hole is cheap paper · A hole is an edge · Spending the cheap paper · A notch is not a hole · A cut that reaches the edge · The edge was there first · The corner premium, with no corners · A base needs an edge to point at · 10 claims checked

Where the cheap paper is, with a hole and withoutThe paper a flap of a given length can claim at each point of the sheet, dark where it claims least. A hole in the middle makes the paper around it as cheap as the paper at the sheet's own edge, and the two sheets hold the same amount of paper.a flap of 0.12 of the sheet's side, on two sheets of the same areadarker is cheaper: less of the flap's disc is paper that has to be paid forwith a holesolid, same areamean claim 0.8646mean claim 0.8959

kirigami

8 essays — What one cut buys · Bought with holes · One cut short of falling apart · A cut that removes no paper · A cut is a licence · One cut removes one arc · A cut is surgery · The cut that changes nothing · 13 claims checked

A wedge out, and the cone that closesA disc of paper with a 60° wedge marked for removal, and the cone the rest of it closes into when the two cut edges are brought together. Nothing is stretched: the paper that is left is exactly the paper that was there. What changed is how much of it surrounds the centre, and that fixes the cone's half-angle at 56.44° with nothing left to choose.the sheeta wedge of 60° marked for removal60°56.4°what it closes intoa cone of half-angle 56.44°83.3% of the turn is left, and the sine of the half-angle is that same fractionthe circles of latitude are the disc's own, arriving shorter than a flat sheet would needno fold can do this: folding moves paper about and never alters how much of it surrounds a point

paper-thinness

8 essays — The paper had to arrive first · Closer than a crease is wide · A sheet has a size as well · Which ceiling is binding · A sheet is as large as two arms · The paper that will not hold a crease · Eighty layers and the sheet decides the rest · The deepest point pays for the paper · 30 claims checked

The paper had to arrive firstStack thickness is the layer count times the sheet thickness, and a fold stops working when the stack approaches the smallest feature being folded. So the number of layers a design can reach is fixed by the paper rather than by the folder — and the complex tradition is downstream of paper thin enough to carry it.a fold stops working when the stack reaches 3 mm8 layers16 layers32 layers64 layers128 layersnewsprint65 µm520 µm1.0 mm2.1 mm4.2 mm8.3 mmcopier paper100 µm800 µm1.6 mm3.2 mm6.4 mm12.8 mmkami70 µm560 µm1.1 mm2.2 mm4.5 mm9.0 mmwashi40 µm320 µm640 µm1.3 mm2.6 mm5.1 mmfoil-backed tissue26 µm208 µm416 µm832 µm1.7 mm3.3 mmunryu tissue18 µm144 µm288 µm576 µm1.2 mm2.3 mmthickness measured across the sheet; the smallest feature is a folder's working figurerather than a constant of nature

scaffold-route

8 essays — A sheet that routes itself · Two things called folding · Two ceilings · The helix chooses the lattice · A row the route cannot leave · Every cheap test misses a shape · A domain too short to be unique · A test that only knows one lattice · 42 claims checked

One strand, through every helix, onceEach circle is one helix seen end-on and each step is a crossover to a lattice neighbour. The route is found by a search that never looks at the colouring; the colour counts are computed separately, and a shape whose two colours differ by more than one is refused before any search is run.a route the search found123456121110987131415161718242322212019helices 24colours 12 : 12a route is not forbiddenscaffold used 21%48 staples of 3224 helices · 1536 bases · 48 staples · colours 12 : 12