Origami & Folding
what one sheet can do
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The figure library — page 7

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.

edge-vertex

5 essays — The vertices nobody checks · A cut is a licence · A ring and a line · The rim lies over less · Most of a patch is edge · 8 claims checked

A boundary vertex is a stripA vertex where creases meet the edge of the paper, drawn as the fan of paper it has and again as the one-dimensional crease pattern that fan is. The sectors come in a line rather than in a ring, so the four conditions the subject states at an interior vertex are not weakened there — they are about a different object, and the object this is has a decidable condition of its own.the edge of the paperMVM40°60°20°60°the same sectors, in a lineMVM40°60°20°60°this lettering folds4 of 8 letterings foldVMV MMV VVM MVMno vertex theorem applies here at all— the sectors do not close, and there is no cycle to alternate round

evidence-strength

5 essays — Nothing here is as old as it sounds · A record is not a proof · Two traditions and a merge · The cure was named first · When two of them are first attested together · 4 claims checked

What each claim rests onThe record grouped by what kind of thing its oldest source is, strongest first, with one dot per independent surviving source. The claims resting on a single source are marked, and they are a third of the table — which is the honest statement of how much of this field's dating is one document away from being unsupported.15 claims · 5 resting on one sourceartefacta surviving folded object, or a picture of one made at the timePaper is made in Europe1056The pajarita is folded in Spain1793manuscripta hand-written document that survivesFolded paper is used ceremonially in Japan1600Paper reaches Japan720one sourceprinteda printed book or paper with a publication datePaper folding is taught as geometry1838The conditions at a flat-foldable vertex1979The Miura fold1970The diamond pattern in a crushed cylinder1951The dashed-and-dotted diagram notation1954Any straight-line drawing, from one straight cut1998Paper is folded for amusement in Japan1680one sourceThe thousand cranes1797one sourceOne fold solves a cubic1936one sourcesecondarysomebody later reporting it, with no surviving primary sourcePaper is made in China105A five-pointed star from one straight cut1873one sourcea source is dated; it is not thereby rightthis ranks what a source can bear, not what it says

fujimoto

5 essays — Folding a strip into thirds · The kindergarten was a geometry class · One crossing, and then another · Exact is not accurate · Dividing a loop into n · 5 claims checked

Folding a strip into thirdsA guess, and then halving. Each fold moves the mark to the midpoint of one of the two pieces, and each fold halves the distance to the exact division — so the error falls geometrically from whatever the first guess was. The sequence of halvings is read off the binary expansion of the fraction rather than chosen, and the halving of the error is asserted rather than observed.guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

n-division

5 essays — One crossing, and then another · Cheap where it reaches · The numbers a fold reaches · Exact is not accurate · The grid a division makes · 5 claims checked

One crossing, and then another: 1/5 of the squareThe anti-diagonal is folded once. Crossing it with the line from the corner through the mark at one half gives one third; crossing it with the line through one third gives one quarter, and so on. Every crossing is exact, and the ladder costs one fold per rung after the first.1/25 folds to reach 1/5, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops

pleat-ratio

5 essays — The propagation that never had to work · Closing the loops is not folding · One number where the corners wanted four · Every twist writes an equilibrium · The sheet draws in crooked · 47 claims checked

The equation on an edge, at both of its endsEach edge of a tiling carries one equation relating the twist sizes at its two ends. The first pair of columns is the tiling as it is drawn; the second is the same tiling with every vertex moved a little. A regular polygon has one interior angle, so both ends of an edge read the same numbers and every ratio is one.the equation on an edge, at both of its endsas drawnevery vertex moved 12 per centratio − 1round a loopratio − 1round a loopthe square grid000.200.90the triangular grid000.170.75the honeycomb000.351.73the rhombille tiling2.0002.992.14the elongated triangular tiling000.171.27the rhombille's tiles are not regular, its ratios are three and a third, and they still multiply to one

sector-ties

5 essays — Where the lemma says nothing · The dial that decides nothing · A tie is not a decision · The order decides the count · Walking between two foldings · 3 claims checked

The count halves the moment the tie is brokenHow many mountain-and-valley assignments a degree-four vertex admits, over a family in which the two smallest sectors stay equal, and then at a vertex a tenth of a degree away from that family. The tied family holds twice as many throughout, and the fall is a step rather than a slope.8, with the tie4, without it0the two smallest sectors, kept equalfoldable assignments of one interior vertexthe dashed line is a vertex 0.1° off the family: the lemma wakes up and takes half of them

senbazuru-cuts

5 essays — The oldest book cuts the paper · Which cranes can stay joined · Nearly every cutting fails at one crane · The border is where the cranes come apart · The prediction held at eight and broke at ten · 30 claims checked

The oldest book cuts the paperThe connected cranes of 1797, as the sheet they are cut from: a grid slit along every internal line except at the lattice points, which are left uncut so the birds stay joined. The arrangement is one sheet and it is emphatically not uncut, and the slitting per crane grows with the size of the piece.3×3 — 9 cranes, 4 corner joinsone sheet, and cut2×2 4 cranes 4 sides of slit3×3 9 cranes 12 sides of slit4×4 16 cranes 24 sides of slit5×5 25 cranes 40 sides of slit6×6 36 cranes 60 sides of slitcranes − joins = 2n − 1the rule the subject is usually stated under is one sheet and no cuts; theoldest surviving origami book does not keep it

thickness-technique

5 essays — Getting thickness round a corner · Panels with somewhere to go · Paper through paper · Nowhere to put the error · Thickness round a closed loop · 4 claims checked

Thickness: chamferA cross-section through one fold in a panel of real thickness, driven until the two panels touch. The contact is tested on the actual outlines rather than judged by eye, so the angle underneath is the travel the technique buys — and every technique buys it by giving something else up.0°61°108°travel before contact110.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upmaterial at the crease, sothe panel is thinnest whereit is worked hardesta zero-thickness pattern says the panels meet along a line; nothing that is built does

what-paper-is-not

5 essays — Four things that are not true · One curve and one number · One cut short of falling apart · The fifth thing that is not true · The sixth thing that is not true · 2 claims checked

Four things the model assumesThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded

yoshimura

5 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 · 7 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

assignment-census

4 essays — How many assignments fold · Hardness is about the worst one · How little the conditions decide · The lettering that folds nowhere · 8 claims checked

What survives the local conditionsFor each pattern: how many mountain-and-valley assignments there are, and how many of them satisfy every condition at every vertex. The filter is severe and it is not a decision — what passes is still an exponentially large set, and every member of it still has to be checked globally.degree-4 vertex4 of 1625.0% · 4 creasespreliminary base112 of 25643.8% · 8 creasesmiura 2×28 of 1650.0% · 4 creasesmiura 3×232 of 12825.0% · 7 creasesmiura 3×3256 of 4,0966.3% · 12 creasesevery count enumerated, none estimatedthe share falls as the pattern grows, and the count still rises

attribution-table

4 essays — The name is not the date · The patterns nobody owns · Two kinds of claim · A file has no paper · 6 claims checked

The name is not the dateSix results, each drawn from the year of the earliest proof anybody can point at to the year of the publication its name comes from. Nobody was robbed — a field with no journal rediscovers things — but a reader who takes the name for the date acquires a history that is decades too late, every time.the alternating-angle conditionHusimi, 1979Kawasaki, 198910 yrmountains minus valleys is twoHusimi and Maekawa, 1979Justin, 19867 yrthe big-little-big lemmaJustin, 1986the lemma, 19948 yrone fold solves a cubicBeloch, 1936Huzita, 199155 yrthe diamond buckling patternYoshimura, 1951Yoshimura, 196918 yrthe bi-directional foldMiura, 1970Miura-ori, 199525 yr1940196019802000mean lag 21 years · longest 55proof