Origami & Folding
what one sheet can do
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The figure library — page 3

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.

tiling-twist

12 essays — Any tiling makes a twist · Where two twists share a pleat · Fenced at both ends · The propagation that never had to work · The dial that decides nothing · Four ways to draw a pattern · The tiling the unit could not promise · Cutting a patch out of a plane · Letters that agree get rarer · The ring is the loop · Where a sector crosses sixty · The dial and the tiling that is not alike · 15 claims checked

The the square grid's twist tessellationThe crease pattern the offset construction produces, with its assignment found by propagating the two vertex conditions rather than drawn on. Every interior vertex has four creases and passes developability, Kawasaki, Maekawa and the big-little-big lemma.what the construction produced9 twists, 36 interior verticesturned 24.1° from the tiling's edgespleats 0.118 to 0.118 wide2.20× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge
square twist tessellation — sheet 165×165 mm — 41 mountain, 43 valley, 2180.74 mm of crease

twist-unit

12 essays — A sheet with one freedom · A square that turns · Which polygons twist · The base that tiles · Any tiling makes a twist · The patterns nobody owns · The creases that cannot move · A cut is a licence · The symmetry the letters cannot keep · A cut is not local · The tiling the unit could not promise · The ring is the loop · 15 claims checked

The triangle twistA triangle twist: a triangle with a pleat running out from each of its 3 corners, drawn at 150 mm and carrying 5 mountain and 4 valley creases — 563 mm of folding on a sheet 150 mm across. As the sheet closes the triangle rotates, which is what gives the family its name. The sector angles are fixed by Kawasaki and the assignment is chosen for having a folded state rather than for reading well — and the unit is verified, while the tessellation it belongs to is not.3 corners, all alikesectors 60°, 60°, 120°, 120°two equal pairs, so no sectoris strictly the smallestthe assignment64 of 512 fold5 mountain, 4 valleythe ring takes two lettersthe panels can be orderedwhat is checked3 interior verticesand not the tilinga twist of radius 0.17 sheet-widths9 creases, 3.75 sheet-widths of foldingmountainvalleyraw edge
triangle twist — sheet 150×150 mm — 5 mountain, 4 valley, 562.66 mm of crease

attestation-gap

11 essays — Nothing here is as old as it sounds · The oldest book cuts the paper · Two traditions and a merge · Two kinds of claim · The cure was named first · When two of them are first attested together · One lost source and the story changes · The interval is wider than the number · A question the record is too small to answer · Some discoveries would make it worse · A resampled maximum can only fall · 36 claims checked

What is repeated, against what survivesEach bar runs from the date a claim is generally given to the year of the oldest surviving source that attests it. Nearly every bar points forward, which means the claim is older in the telling than in the record; the two that point backwards are the cases where the practice was published long before anybody proved it.Paper is made in ChinaPaper reaches JapanPaper is made in EuropeFolded paper is used ceremonially in Japan400 yrPaper is folded for amusement in Japan980 yrThe thousand cranes897 yrThe pajarita is folded in Spain293 yrPaper folding is taught as geometryOne fold solves a cubicThe diamond pattern in a crushed cylinderThe conditions at a flat-foldable vertexThe dashed-and-dotted diagram notationThe Miura foldA five-pointed star from one straight cutAny straight-line drawing, from one straight cutyear of the source500100015002000the date generally giventhe oldest source that says somedian overrun 201.5 years

folded-layers

11 essays — The paper is all still there · Bringing the other side to the front · Error is folded too · The base that tiles · The cylinder the pattern chooses · The creases a sheet gives itself · The creases that cannot move · The pile, not the panel · The rim lies over less · The property a patch does not have · Where the length sits · 16 claims checked

The pattern, and where its panels landEvery panel of the pattern drawn at the place folding puts it, at the same scale as the pattern itself. The outlines are left in so the layers can be counted; which panel lies above which is a separate question this construction does not answer.the patternthe panels, foldedsheet 12.000footprint 1.966 · 6.11 layers on average · 12 at the deepest1.966 × 6.11 = 12.007, which is the sheet

glued-band

11 essays — The seam carries a sign · The band that needs an odd number · Closure is not the identity · Parity is not enough · The triangle a strip becomes · An alternating sum of angles · How rare a band that folds is · A proof in no nodes at all · A crease with no vertex to belong to · The proportion a band asks for · Found by people not folding paper · 6 claims checked

3 creases on a Möbius bandA rectangular strip of paper with 3 creases across it and its two ends glued into a Möbius band. The arrows on the left and right edges show which way round the gluing carries one onto the other; on a Möbius band they point in opposite directions, which is the whole of the difference between the two sheets. The strip has 3 panels and no interior vertex at all, so every vertex condition in the subject is satisfied here without deciding anything.3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursoff by 4.000 of a widthand turn the paper the right waymountainvalleyraw edge

notation-coverage

11 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 · A line in the columns, a binomial in the rows · Two stacks shuffled in glued pairs · 22 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

rule-sweep

11 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 · Eight creases close the circle too · Half the waterbomb's rules have no stack · 64 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

vertex-kinematics

11 essays — What the vertex does on the way · The vertex is geared · One crease decides the sheet · Which crease to push · The Miura folds two ways · Two mechanisms at one point · A corrugation has one resting state · The wall is the flat sheet · A chain of vertices switches all at once · A loop takes choices away · A crease held the other way · 37 claims checked

What the vertex does on the wayThe four fold angles of a degree-four vertex against the parameter that drives it, solved from the spherical linkage the creases make. Setting one angle sets the other three, which is the single degree of freedom. Where the vertex can reach a flat state the signs split three to one throughout the motion, which is Maekawa's theorem holding all the way and not only at the end; where it cannot, they need not, and that is the difference the two theorems are about.050100150-100100how far the vertex is drivenfold anglesectors60° 90° 120° 90°crease 1: Mcrease 2: Vcrease 3: Mcrease 4: Mcrease 2 is the odd onethree agree, one does notKawasaki holdsand it reaches flatfound by the linkage,not by the theorem

leaf-corrugation

10 essays — A leaf packs by corrugating · The bud chooses the pattern · The same corrugation in four places · Where the paper stops · Twice as thick where it is thickest · A leaf ends its pattern · The taper decides nothing · Sixty-four rules, sixteen fold · The leaf's rules are the Miura's · The plant's pattern is not a hard case · 9 claims checked

A corrugation that tapers, and the direction it is allowed to taper inThe fold a corrugated leaf packs into. The column widths are free — the vertex angles do not depend on them — so the pattern can be broad in the middle and narrow at the ends. The row heights are not free, and tapering those instead is what anybody drawing a leaf would try first.18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

miura-folded

10 essays — One vertex, repeated · A sheet with one freedom · A material made of creases · What a flat sheet can become · No motor in the fold · Bought with holes · A shrink is two numbers · The vertex is geared · The family the Miura belongs to · The corrugation that curves · 1 claim checked

One degree of freedomThe same sheet at three points in its motion, solved rather than sketched: the vertex positions are the ones that keep every panel rigid and every edge its original length, and there is a single free number that sets them all. Both in-plane dimensions shrink together, which is what a negative Poisson's ratio means.nearly flatwidth ×0.91 length ×0.98ν = -0.22half closedwidth ×0.66 length ×0.88ν = -0.52nearly packedwidth ×0.45 length ×0.55ν = -2.93both dimensions shrink together — pulling it open in one direction opens it in the other

opening-monotone

10 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 · A crease held the other way · 42 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

beloch-fold

9 essays — Folding beats the compass, by exactly one degree · Why the list stops at seven · The heptagon a compass cannot reach · Where the cubic comes from · The eleven-sided one nobody can fold · Fifty years in the wrong language · The biggest one that can also be folded · Seven, and then twenty-two · The axiom that names two folds · 11 claims checked

Axiom 6 is a common tangentA fold that carries a point onto a line is tangent to the parabola with that point as focus and that line as directrix. Axiom 6 does it twice at once, so it asks for a line tangent to two parabolas — and two parabolas have three common tangents, which is why one fold solves a cubic and a compass cannot.the cubic8x³ + 4x² − 4x − 1its roots-0.900969-0.2225210.6234903 real common tangentsone fold for each rootand the fold gives cos 2π/7each curve is the set of folds that puts one point on its line; a line touching both does the two at oncea compass intersects circles and gets two answers; a fold touches parabolas and gets up to three