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.

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

machine-census

11 essays — The patient machine is the weak one · The machine that may choose · Four questions about one sheet · Hardness is about the worst one · Publishing the pattern instead of the sequence · The crease that stops in the middle · Where the machine catches up · Deciding is not making · The easiest strip needs the deepest reach · A shallow machine pays in states, not folds · Fourteen states are one pile · 28 claims checked

What each machine can reachFor three one-dimensional crease patterns, the number of mountain-and-valley assignments that fold flat at all, and the number each kind of folding machine can actually reach. Every bar is a separate search: the top one over stackings, the next three over sequences of folds, the last over rewritings of the segment lengths.evenly spaced — 4 creasesany flat folding16 of 16some-layers16 of 16all-layers16 of 16one-layer2 of 16crimping only6 of 16one short segment — 4 creasesany flat folding4 of 16some-layers4 of 16all-layers0 of 16one-layer2 of 16crimping only4 of 16uneven — 4 creasesany flat folding8 of 16some-layers8 of 16all-layers0 of 16one-layer2 of 16crimping only0 of 16a machine that takes fewer layers is weaker, not more patientthe paper is joined, so what it declines to hold it also cannot move

attestation-gap

10 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 · 28 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

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

vertex-kinematics

10 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 · 27 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

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

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

growth-curvature

9 essays — A sheet that grows cannot lie flat · Which way the disc curves · The excess does not choose its waves · A crease carries no curvature · What one cut buys · A leaf ends its pattern · The test measures the rim · Three marks see nothing · A cut reads a slope · 27 claims checked

Growth is a change of metric, and a metric decides a curvatureThe linear growth factor against radius, and the Gauss curvature that metric forces. Nothing about a material enters either panel: given how much each part of the sheet grew, the curvature is determined, and a curvature that is not zero is a sheet that cannot lie in the plane.00.20.40.60.8100.511.5radius on the flat sheetgrowth factor Ωhow much each ring grew00.20.40.60.81-2-112radius on the flat sheetcurvature Kclosed formthe curvature that forcesgrown more at the rim · K(0) = −4a = -2.40

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