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.

dividing-the-square

9 essays — One fold at a time, and there are exactly seven of them · Dividing without measuring · Designing on a grid · Folding a strip into thirds · The eleven-sided one nobody can fold · The kindergarten was a geometry class · A schoolteacher's theorem · Two creases at once · What the grid settles · 1 claim checked

Dividing a square into 3, exactlyHaga's theorem. Folding one corner onto a point part-way along an opposite edge produces exact rational divisions elsewhere on the sheet — so a square can be divided into any whole number of parts by folding alone, with no measurement and no accumulated error.3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

fold-and-cut

9 essays — One straight cut · What universality costs · The oldest book cuts the paper · The star that was cut before it was proved · The symmetry the letters cannot keep · The vertex the list does not have · One cut for a star · The corner that splits the shrink · A tree cannot argue · 19 claims checked

One straight cut: the triangleAn outline, the straight skeleton computed by shrinking it, and the perpendiculars dropped from each skeleton node onto the edges that formed it. Folding along these carries every edge of the outline onto a single line, so one straight cut releases the shape. The mountain-and-valley assignment is found by search, because the obvious one fails Maekawa.the cut line3 straight edgesthe pattern3 skeleton arcs3 perpendiculars1 interior vertexassignments that fold30 of 646 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once
Fold and cut — triangle — sheet 150×150 mm — 2 mountain, 4 valley, 257.98 mm of crease

graft-strip

9 essays — Paying in paper · The second term · A graft needs a square line · Six rectangles and one term · A design that keeps its lines clear · The width is charged in grid · A spacing costs one plus the other side · A grid chosen before the features · The split that is worst on a rectangle · 58 claims checked

A strip slid into a finished patternA crease pattern cut along a line, with a strip of paper slid into the gap. Every crease that meets the cut continues across the strip at the same angle; every crease that does not is carried along at the length it had. The shaded band is the whole of what the new feature cost.the design as it stood20.00 of paper · 5 creases meet the cutthe same design, one strip wider21.80 of paper · the strip is 0.45 acrossthe band is 0.45 × 4 = 1.8000, and that is the entire difference between the two patternsall 44 creases away from the cut keep their length; the 5 that cross it are longer by 0.45 and by nothing elsea design grows by accretion because the arithmetic of growing it is this short

how-many-fold

9 essays — The smallest sector decides · Local is not global · How many assignments fold · The same vertex, found four times · A near miss is nearly as rare · Even is not enough · Where the lemma says nothing · How little the conditions decide · Walking between two foldings · 7 claims checked

How many assignments actually foldFor a fixed set of crease lines, the number of mountain-and-valley assignments that satisfy the local conditions, against the number of assignments there are. The valid ones are a small and shrinking fraction, which is the quantitative form of the claim that flat-foldability is rare.degree-4 vertex, 60/90/120/90°4 of 164 creases · 25.0% surviveone degree-6 vertex8 of 646 creases · 12.5% survivethe preliminary base112 of 2568 creases · 43.8% surviveand these are only the local tests — a pattern can pass every vertexand still collide once the layers stack, which is the hard part

map-foldings

9 essays — The gadgets that make it hard · The oldest open problem · More than one way to lie flat · Two directions that will not separate · The map that is not a rectangle · The count counts labels · The map counted from the layers · A map with no edges · The tube a map makes · 8 claims checked

How many ways a map foldsThe number of distinct flat foldings of a ruled rectangle, on a logarithmic scale. The filled bars were counted by exhaustive search during this build; the open one is Lunnon's published value, past what a build can reach. There is no formula for any of them, and the next term is not known.1 × 4161 × 5501 × 61442 × 282 × 3602 × 43203 × 31,3684 × 4300,608filled — counted here, by exhaustive search over stacking ordersopen — Lunnon's published count, quoted rather than computed

paper-thinness

9 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 · A proud hundredth buys almost nothing · 49 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

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

scaffold-route

9 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 · A corridor has two mouths · 46 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

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

yoshimura

9 essays — Patterns nobody designed · The sheet has a thickness · The cylinder the pattern chooses · How much line is on the paper · A corrugation agrees with itself · A twisted tube has two heights · Four clicks make sixteen lengths · Clamped, a stack counts in pairs · The Yoshimura folds flat at a few heights · 36 claims checked

The Yoshimura patternThe diamond pattern a thin-walled cylinder falls into when it is crushed axially. Nobody designed it — it is the buckling mode with the lowest energy, and it was named after the engineer who described it in aluminium tubes rather than after a folder. Drawn here from the construction that satisfies all three local conditions.found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter22 interior vertices · 26 mountain and 60 valley creasesmountainvalleyraw edge
Yoshimura pattern — sheet 170×122.69 mm — 26 mountain, 60 valley, 2380 mm of crease

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