Origami & Folding
what one sheet can do
Theme

The pattern is the object

A crease pattern is not a picture of a model. It is the model, written down — complete, checkable, and foldable by anyone who has the paper.
axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass Axioms and construction

Why the list stops at seven

The seven axioms are not seven useful folds somebody collected. They are every fold there is, and the proof is an exercise in counting degrees of freedom that takes about a minute.

 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 Axioms and construction

Dividing without measuring

A square can be divided into any whole number of equal parts by folding alone — exactly, with no ruler, and with no error to accumulate. The construction is one fold and a theorem nobody expected.

 compassfolding3242546276the first gap8496104111012413121461581681716186191820821122210232224825202612n across the top, φ(n) underneathcompass: φ(n) a power of two — Gauss, and Wantzel's proof that nothing else worksfolding: φ(n) with no factor above three, because one fold solves a cubic Axioms and construction

The heptagon a compass cannot reach

Which regular polygons a tool can build is a condition on a single number. The compass needs it to be a power of two; a fold needs only that it has no factor above three — and seven is the first place the two answers differ.

 taco-tacoallowedforbiddentwo folds at the same place may nest or stand clearthey may not interleavetaco-tortillaallowedforbiddena flat layer may pass outside a foldit may not pass through onea crease pattern can satisfy every vertex condition and still break one of these Flat-folding

Which layer goes on top

The mountain-valley assignment says which way each crease turns. It says nothing at all about which sheet ends up above which, and that second question is a different object with its own rules — and all of the difficulty.

 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 Flat-folding

One straight cut

Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.

 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 Designing a base

A flap costs a circle

A flap of a given length uses up every point of the sheet within that distance of it. Two flaps whose circles overlap are asking for the same paper twice — and that one observation turned origami design from an art into an algorithm.

 leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 71% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs Designing a base

Packing is the hard part

Once a subject is a set of circles, designing the model is fitting them into a square. That step has no general algorithm, no known optimum, and it is where every remaining difficulty in origami design now sits.

 riverwidth 0.3legarmheadlegtailthe check10 pairs testedtightest by 0.0800(leg and arm)the ruledistance on the sheetat leastdistance through the treetwo nodes, and an edge between themthe extra width is thebody the flaps hang fromthe discs are what each flap costs; the strip is what joins the two halves of the subjectand both are the same condition, read off different pairs of leaves Designing a base

What joins the flaps

Circles are the rule for flaps that all hang from the same point. As soon as two groups of flaps hang from different places, the paper between them has to be paid for too — and the payment is a strip whose width is the distance between them.

 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley Designing a base

Designing on a grid

Box pleating gives up the efficiency of a free packing and buys creases that land where they are supposed to. For a design with hundreds of folds that is not a compromise — it is the only thing that makes it foldable.

 the treeleg0.62leg0.62body0.34arm0.50head0.78the basethe axis0.620.620.340.500.78the flaps are the tree's edges, at the tree's lengths, all square to one lineso the base's shadow along the axis is the tree, and nothing else can be designed this waywhich is the restriction the circle argument quietly depends on Designing a base

Every flap on one axis

The tree method does not design a shape. It designs a base whose flaps all lie along a single line — and that restriction, which is almost never stated aloud, is what makes the circle argument true.

 4 corners, all alikesectors 90°, 90°, 90°, 90°two equal pairs, so no sectoris strictly the smallestthe assignment256 of 4096 fold6 mountain, 6 valleythe ring takes two lettersthe panels can be orderedwhat is checked4 interior verticesand not the tilinga twist of radius 0.17 sheet-widths12 creases, 4.70 sheet-widths of foldingmountainvalleyraw edge Tessellations

A square that turns

A twist is a small polygon that rotates as the sheet closes around it. The geometry is forced rather than designed — Kawasaki fixes one sector, the big-little-big lemma forbids a strictly smallest one, and what is left is the pattern Ron Resch was drawing in the 1960s.

 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 Curves and material

Four things that are not true

Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

 state 0state 1V M M V — the same pattern in both2 valid stackings, found by enumerationwhat a junction would addthree wires meeting, with the layer orders forced to disagree —which is a clause, and which is where the reduction gets its powernot drawn and not verified: nothing here decides layer order in two dimensions Flat-folding

The gadgets that make it hard

Flat-foldability is NP-hard, and the proof is a construction rather than an obstruction: a machine for turning any satisfiability problem into a sheet of paper that folds exactly when the problem has an answer.

 what the packing gives5 discs, 4 contactsno two overlapping, checkedhinge creasesone per contact, perpendicularto the line of centresridge creasesalong the axial lines, dividingthe paper between the flapsthe packing is the hard part;this part is a constructioncorner discs of radius 0.28, the middle one 0.427 — every contact measured Designing a base

From a packing to a crease pattern

The circles say where the flaps are. They do not say where to fold, and the step in between is a construction rather than a search — two families of crease, both determined by the packing, neither of them visible in the picture of the discs.

 the shrink5 intermediate outlines drawneach edge moved inward by thesame distance — computed, not drawnthe tracesstraight, because every edge movesat one rate along its own normalhow far it can go0.5391 sheet-widthsfound by bisection on the outline'sown area, not by inspectionthe construction that always works — which is what universal means here Designing a base

The last free parameter

Once the packing is fixed, one number is left in the whole design: how far a leftover polygon can be shrunk before it stops being a polygon. Everything else about the crease pattern has already been decided.

 startend8x³ + 4x² − 4x − 1legs 1.000, 0.500, -0.500, -0.125each turn a right angle3 real rootsx = -0.900969x = -0.222521x = 0.623490each ray lands on the end to 1.1e-16the launch angle's negative tangentis the root — which is the folda right-angled bounce off two linesat once is exactly what one fold does Axioms and construction

Where the cubic comes from

Folding solves cubics, and the usual explanation stops at the sixth axiom. The reason is older and better: a right-angled bounce along a path of coefficients is a root-finder, and one fold is exactly such a bounce.

 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 What it costs to know

The machine that may choose

Three restricted machines lose patterns that fold perfectly well. Give one of them a choice — any block of layers, top or bottom — and the loss vanishes: over a hundred and seventeen spacings, every flat folding of every strip became reachable. Being forced was the whole problem.

 mapflat foldingsand what it took2 × 284 cells, computed here2 × 3606 cells, computed here2 × 43208 cells, computed here3 × 31,3689 cells, computed here2 × 51,9800.6 s3 × 415,55254 s4 × 4300,608not reached herethe 1 × n case is the strip, and it is the only row of this table with a fast methodnobody has a formula for any entry, and nobody has proved there is none What it costs to know

The answer is bigger than the question

A twelve-square strip of stamps is twelve numbers of input and 146,376 objects of output. No algorithm writes that faster than it can be written, so 'efficient' has to be measured against the answer rather than against the question — and in folding that is the normal case.

 developableKawasakiMaekawabig-little-bigsectors that do not alternatefour creases turning the same waya small sector flanked by one lettera 4×3 Miura, every vertexthe last row passes all four tests at all 6 of its vertices, and passing is not a proofthe tests are conditions at a single vertex; whether the layers can be stacked is a condition on the whole sheetno arrangement of vertex tests decides that, which is what NP-hardness means when it is spelled out What it costs to know

What a checker cannot check

Every crease pattern on this site is run past four theorems before it is allowed onto a page, and passing all four proves nothing. The gap is not a bug to be closed: it is the NP-hardness result, arriving as a property of a hundred lines of code.

 4681012010203040sides of the polygoncreasescreases in the patternperpendicularsskeleton arcsa 12-sided outline needs 36 creases and one skeleton nodea convex outline is the cheap casea reflex corner splits the shrinking front, and this solver refuses those rather than guessing What it costs to know

What universality costs

The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.

 polygonsectors at the twist vertexKawasakitiles the plane3-gon60.0 · 60.0 · 120.0 · 120.0180.0° = 180.0°yes — 6 round a point4-gon90.0 · 90.0 · 90.0 · 90.0180.0° = 180.0°yes — 4 round a point5-gon108.0 · 108.0 · 72.0 · 72.0180.0° = 180.0°no6-gon120.0 · 120.0 · 60.0 · 60.0180.0° = 180.0°yes — 3 round a point7-gon128.6 · 128.6 · 51.4 · 51.4180.0° = 180.0°noevery one of these twists satisfies the local theorems and folds flat on its ownthe interior angle has to divide 360° for the twists to meet, which only 3, 4 and 6 dobeyond 7 sides the assignment search runs out — 21 free creases, and the enumerator refuses above 22 Tessellations

Which polygons twist

Twist tessellations come in three kinds — triangle, square, hexagon — and it is natural to read that as a fact about twists. It is not. A twist can be built around any regular polygon and every one of them folds; what stops at three is the tiling, and the tiling is a fact about the plane.

 quadrilateralconvexmolecule builtpentagonconvexmolecule builtL, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkingdart, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkinga convex polygon shrinks inward and stays a polygon; a reflex corner is a wall the shrink runs intoso a non-convex region is split into convex pieces first, and choosing the split is a searchwhich is where a construction that always works hands the difficulty to whatever comes before it Designing a base

The molecule that does not exist

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.

 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 Who found it, and when

Nothing here is as old as it sounds

Paper folding is described everywhere as an ancient art. The oldest surviving book of it was printed in 1797, the oldest reference to folding for amusement is from 1680, and the median claim in this subject is dated two centuries before anything that attests it.

 4×4 — 16 cranes, 9 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 Who found it, and when

The oldest book cuts the paper

The Hiden Senbazuru Orikata of 1797 is the earliest surviving book of recreational paper folding, and its famous connected cranes are made by slitting one sheet into a grid. The founding rule of the modern subject is younger than the tradition it claims to describe.

 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 Who found it, and when

The name is not the date

Kawasaki's theorem is in Husimi's book ten years before Kawasaki's paper. Maekawa's is Justin's too. The mean gap between a result in this field and the name it is known by is twenty-two years, and it runs in one direction.

 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 Who found it, and when

Fifty years in the wrong language

Margherita Beloch showed in 1936 that one fold solves a general cubic. The result was correct, published, and in a mathematics journal — and the subject that needed it did not find it until 1991. The cost of a paper nobody reads is measurable, and it is most of a century.

 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 Who found it, and when

A record is not a proof

Every other claim here can be re-derived from the figure that makes it, and a wrong one shows. A date cannot: it is checked once, by hand, against a record that is itself a survivor. This field is the one most likely to be wrong and least likely to be caught, and saying so is the only defence it has.

 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 Who found it, and when

What a dashed line can say

Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.

 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley Who found it, and when

Publishing the pattern instead of the sequence

A diagram sequence is one picture per step and a crease pattern is one picture. When designers began releasing patterns rather than diagrams, the cost of publishing a model fell by two orders of magnitude and the difficulty moved onto the reader — which is what made the complex era possible and what made most of it unfoldable.

 10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed Who found it, and when

The star that was cut before it was proved

Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes.

 a crease — concentrated at a pointdeficit 0.5236 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.5236 rad, none of it anywherecurved at every pointa 30° wedge removed, against Ω = 1 − 0.0400 r² · same total curvature, 0.5236 Folding nobody designed

A crease carries no curvature

A fold looks like the sharpest curvature a sheet could have, and intrinsically it has none at all. Developability — the first of the four conditions this site's checker runs — is exactly the statement that folding an uncut sheet creates no curvature anywhere, including at the creases.

 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° Folding nobody designed

A leaf packs by corrugating

A corrugation is the cheapest fold there is — parallel creases, no interior vertex to think about — and a leaf that uses one has to taper it, because a leaf is broad in the middle. Which direction the taper is allowed to run turns out not to be a matter of taste.

 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 Folding nobody designed

A sheet that routes itself

DNA origami folds one long strand into a shape by holding it against itself with a few hundred short ones. There is no sheet and no crease — what has to be designed is a route — and the first thing that can go wrong is a counting argument crease patterns already know under another name.

 geometrypacks tocorrugation8 panels at 0.42 rad40.8% — 2.5× smallerMiura6 × 4, 15 interior vertices16.6% — 6.0× smallerfan8 sectors about one point25.0% — 4.0× smallerroll8 turns12.5% — 8.0× smallerpacked area as a fraction of deployed, computed from each geometry — not measured from any animal Folding nobody designed

The same corrugation in four places

A leaf, a wing, a crushed cylinder and a solar array arrive at nearly the same fold, and none of them copied any of the others. Convergence stories are cheap; this one is checkable, because the constraint that forces it can be computed rather than admired.

 computed herenot established herethe curvature a growth field forcesthe growth solverthat any leaf grows that waythat a corrugation with tapered columns folds flatthe pattern librarythat a leaf's creases are where these arethe packed fraction of four folding geometrieswing-packingthe packed fraction of any wingthat one freedom needs one driverdof-censushow an insect actually deploys a wingthe surface a fold count fits in a boxsurface-in-volumethe surface area of any organwhich lattice shapes admit a single routethe routing modulethat any published design used this routeevery row's left side is a computation in this repository; every right side is a claim it does not make Folding nobody designed

The organism is not the model

Every figure in this field draws a fold this repository computed. Not one of them measures a leaf, a wing or a gut. That is the rule the field was built to, and it is worth stating as a table rather than as a preamble — because a field about living things is where a computed geometry is most likely to be read as an observation.

 20 panels, two coloursno crease has the same colour on both sidesall 12 interior vertices carryan even number of creasesthe colour is which side of the paperthat panel shows when the sheet is foldedmountainvalleyraw edge Flat-folding

The sheet has two sides

Read a crease pattern as a set of panels rather than a set of lines and a condition appears that no vertex theorem states: the panels take two colours, no crease has the same colour on both sides, and the colour is which face of the paper each panel ends up showing.

 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 Flat-folding

The paper is all still there

A folded sheet is smaller than it was and none of it has gone anywhere. How much smaller it is and how many layers deep it is are not two properties of a pattern — they are one number, and their product is the sheet.

 00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements Designing a base

Bringing the other side to the front

Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

 2345678910-0.15-0.1-0.0500.050.10.15discsshortfall of the symmetric search14.6%1.7%0.0%-0.0%5.6%0.5%10.8%10.4%4.7%symmetry: mirror · both searches at 90 restartsneither number is a proved optimum — this compares two searches Designing a base

When symmetry costs

Design software and designers both reach for symmetry, and for a good reason: it makes the search enormously easier. It is a heuristic and not a theorem, and how much it gives away can be measured — including the case where the optimum is symmetric about an axis nobody imposed.

 123456760%65%70%75%80%sheet, longer side to shorterfraction of the sheet claimedbest at 7.0 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square Designing a base

The square is a choice

Every packing on this site has been into a square, because origami paper is sold square. Hold the area fixed and vary the shape instead and the efficiency turns out to be spiky rather than smooth — with the same peak value at every proportion that is a ratio of two factors of the flap count, and nowhere else.

 two kinds of vertex, both forced16 of degree 490°, 90°, 90°, 90°9 of degree 690°, 45°, 45°, 90°, 45°, 45°40 mountain and 36 valley creases14.3 sheet-widths of foldingmountainvalleyraw edge Tessellations

The base that tiles

The waterbomb base is the first thing most people fold and the last thing they think about. Repeat it across a sheet and it becomes a tessellation with two kinds of vertex, an assignment that has to be searched for rather than remembered, and a folded state thirty-two times smaller than the paper.

 562 × 24 cells323 × 39 cells324 × 416 cells325 × 525 cellsrepeating rules that pass every condition, out of 51224 rules pass on the smallest patch and on none of the others Tessellations

A unit that folds is not a tessellation

Of the 512 repeating rules for the waterbomb tessellation, 56 pass every condition on a two-by-two patch and 32 pass on every larger one. The twenty-four that die were never foldable — the small patch simply contained one of the four kinds of vertex the pattern makes, and the failures were at the other three.

 patternhow much smaller it foldscreasingper sheet-widthpreliminary8.0 layers, 8 at the deepest8.0×4.81.65×footprint × depth = 1.004 of the sheetmiura8.1 layers, 16 at the deepest8.1×6.21.31×footprint × depth = 1.001 of the sheetyoshimura32.0 layers, 36 at the deepest32.0×11.82.71×footprint × depth = 1.000 of the sheetwaterbomb31.6 layers, 32 at the deepest31.8×14.32.22×footprint × depth = 0.992 of the sheettwist3.0 layers, 9 at the deepest3.0×4.70.64×footprint × depth = 0.995 of the sheetthe shrinkage is the pattern's, not the paper's — nothing here knows what the sheet is made of Tessellations

What a corrugation costs

Every tessellation this repository can fold, measured the same way: how much smaller it gets, how deep the stack becomes, and how much creasing was needed to buy it. The last column is the one nobody quotes and the one a folder feels.

 sidesshare of the sheet the largest one usestilt346.4%15.00°4100.0%45.00°567.4%9.00°669.6%15.00°772.9%6.43°882.8%22.50°975.1%5.00°1075.3%9.00°1176.3%4.09°1280.4%15.00°1376.9%17.31°1476.9%6.43°1577.3%3.00°1679.6%11.25°the 8-sided polygon is the peak, and every one of the 8 polygons after it does worse Axioms and construction

The biggest one that can also be folded

Which regular polygon uses a square sheet best, and which of them a fold can actually construct, are two questions with completely different pedigrees. Answered side by side over sixteen polygons, they turn out to agree — and the reason is that both are questions about the arithmetic of the same number.

 straight creasescurvature 1.4curvature 2.6the tangents are the same in every panelsectors 60.0°, 120.0°, 120.0°, 60.0°they sum to 360.0°, and alternately to 180.0° and 180.0°which is Kawasaki, on tangents rather than on lines Curves and material

Where curved creases meet

A curved-crease design looks like a smooth object and its constraints are not smooth. They live at the finitely many points where creases cross, and at each of those the conditions are about the creases' tangent directions — the curvature does not appear in them at all.

 sectors 80°, 55°, 100°, 125° in every one of them, and 4 assignments fold in every onelongest ÷ shortest 1.00footprint 0.806longest ÷ shortest 3.09footprint 0.911longest ÷ shortest 3.33footprint 0.623longest ÷ shortest 4.00footprint 1.782every one of them folds; their folded footprints differ by a factor of 2.86 Flat-folding

The lengths are free

Kawasaki reads angles, Maekawa counts letters, and the big-little-big lemma compares one sector with its neighbours. Not one condition in the subject mentions how long a crease is — so a single vertex is not a pattern but a whole family of them, every member folding, no two folding into the same shape.

 creases at 0.25, 0.50, 0.75, marked MMM3 legal stackings of 4 segments, read from the bottom of the pile up12341: 2 · 1 · 4 · 3assignment12342: 4 · 2 · 1 · 3taco-taco12343: 2 · 4 · 3 · 1taco-tacothe paper lands in the same place every time — only the order through the pile differs Flat-folding

More than one way to lie flat

A crease pattern with its mountains and valleys marked is spoken of as though it named a folded object. It does not. The legal stackings can be counted exactly in one dimension, the count is routinely more than one, and its size is a property of the pattern that nobody quotes.

 interior vertices, by number of creases meeting therenone3odd594evennone5odd326evennone7odd18even8 patterns, 92 interior vertices, and not one of them with an odd number of creases Flat-folding

Nothing meets at three

Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars.

 010203040-3-2-10how far open — degrees for the cut sheet, the same fraction of the motion for the foldPoisson's ratiocut into squares−1 everywhere, exactlya fold, slant 0.35-0.12 at the startand without limit at the endthe two cross onceand agree nowhere elsethe flat line is a finite difference of two measured widths, taken the same way as the curve beside ita material made of matter cannot change its Poisson's ratio as it deforms; a material made of geometry can Curves and material

Bought with holes

A Miura-folded sheet gets wider as it is pulled, and by how much depends on its panels and on how far it happens to be folded. A sheet cut into squares joined at their corners does the same thing and holds the value at exactly minus one, everywhere in its motion — the same property, bought with different geometry, and paid for in holes.

 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 Designing a base

Paying in paper

A feature added to a finished design costs exactly the paper inserted for it. Cut the crease pattern along a line, slide in a strip, and every existing crease continues across it unchanged — so the bill is the strip's width times the length of the cut, and there is no second term.

 1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it Designing a base

The corner is worth four times the middle

A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.

 the flat pattern6 columns · 5 rowsthe course, closedradius 0.16676 sides of 0.1667 close into a ring of radius 0.1667, in the sheet's own unitsthe circle through the corners is 4.72% longer than the ring, which is what a 6-sided polygonowes its circle Tessellations

The cylinder the pattern chooses

A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.

 24682468times shorter across the sheettimes shorter down itaccordionMiuraleaf corrugationYoshimurasquare twistwhat each one doesaccordion8.00× across, 1.00× downMiura2.73× across, 1.75× downleaf corrugation3.57× across, 1.22× downYoshimura4.00× across, 4.00× downsquare twist1.52× across, 1.52× downthe dashed line is a directionthe fold never touchesone direction untouched: 1 of 5 · equal both ways: 2 · neither: 2the single number a corrugation is usually quoted by is these two multiplied together Tessellations

A shrink is two numbers

How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.

 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 Tessellations

Any tiling makes a twist

A twist tessellation is usually drawn, admired and copied. It can be derived instead: hand the construction any tiling of the plane and it returns a crease pattern that folds flat, with the twist polygons' shapes forced by the tiling's own angles and nothing left to choose but how large and how turned.

 what the construction produced23 twists, 122 interior verticesturned 24.1° from the tiling's edgespleats 0.068 to 0.068 wide1.58× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge Tessellations

Where two twists share a pleat

Every twist tessellation the tradition draws has one size of twist, because every tiling it is drawn on has one kind of vertex. Hand the construction a tiling with two, and the pleat between a large twist and a small one turns out to fix their sizes exactly — three to one, and nothing else folds.

 24681012foldsinterior verticesfacetscrease lengththe median facet falls from 2.6e-1 to 4.0e-3 of the sheet Curves and material

Every facet is a layer

Fold a sheet at random as many times as patience allows, then count three things: the creases it carries, the facets they cut it into, and the layers in the stack. The last two are the same number, always, and it is one more than the first — so how deep a crumpled sheet folds can be read off the flattened pattern without folding anything.

 patternraw edge against crease, by lengthpreliminary base8 panels29.3% rawMiura, 5 by 420 panels20.8% rawsquare twist9 panels26.7% rawYoshimura, 6 by 565 panels6.7% rawthe shaded part is the sheet's own edge; the rest of the outline is creasemeasured with a step of 0.001 of the sheet, and checked across a tenfold sweep of itevery length here is summed over the layers, so a buried edge counts for nothing Flat-folding

The outline is mostly crease

The edge of a folded model is what a reader looks at, and almost none of it is the edge of the paper. Measured across five patterns, the sheet's own boundary accounts for between nothing and a third of the exposed edge; the rest is fold, and on a waterbomb tessellation the raw edge does not reach the outside at all.

 4 patterns, one folded profile1/122/124/127/1225311/123/126/128/1225312/124/125/127/1225312/125/127/128/122531foldedthe layer counts under each band are the same in every row, and so are the widths Flat-folding

The shadow does not name the pattern

A photograph of a folded model carries an outline and a thickness at every point of it, and that is the whole of what it carries. It is not enough. Crease patterns in genuinely different places fold to identical outlines with identical layer counts, and nearly a third of the folded objects a short strip can reach are reached by more than one pattern.

 50% of the stack, 9 foldsthe whole stack, 9 folds13 interior vertices of odd degree13 creases with a loose end before planarisingrefused by the first condition it is put pastno odd vertex anywhereno loose end anywhereand every vertex satisfies Kawasaki Curves and material

The crease that stops in the middle

A sheet folded flat at random writes a crease pattern that satisfies every condition in the subject, everywhere. Leave one layer behind on each fold — one layer out of a dozen — and it stops writing crease patterns at all: the creases stop in the middle of the paper, and a crease with a loose end is a thing no flat folded sheet can have.

 7 flaps, packed as tight as they will goradius0.174457630loose flaps1room to move0.1127the inner ring is where that flap's centre may sit; the algorithm reports one point of it and stops Designing a base

The flap nobody holds

An optimal packing is presented as an answer: here are the circles, here is where they go. For some numbers of flaps that is not what it is. The best arrangement of seven discs in a square leaves one of them free to wander over an eleventh of the sheet without changing the answer at all — and the algorithm reports one point of that region and stops.

 42:3:4:363:4:6:583:4:6:5125:7:11:8165:7:11:8248:11:17:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution Designing a base

Spelling a tree on a grid

Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.

 0.287267creases in the moleculewhere the corner sits7 creases6 creasesat the marked shape three of the polygon's edges vanish at the same instant, and either side of it they vanish one at a time Designing a base

The skeleton changes its mind

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. It does not fill it continuously. Slide one corner along its edge and the number of creases in the molecule sits at six, jumps, and sits at seven — so two designs a hairsbreadth apart have crease patterns that are not small variations on one another.

 of the markings that fold, how many folded objects each one makesmarkings that foldexactly one objectthe most any one makesthe preliminary base4 creases · four equal sectors, the first vertex anybody folds881a halved four-crease vertex4 creases · degree four with its two smallest sectors equal — the case the lemma is silent at661the waterbomb tessellation's odd vertex6 creases · degree six, and this site prints nine of them on one sheet18126a Yoshimura vertex6 creases · degree six with every sector equal, and twenty-two of them on the printed pattern30122the preliminary base's centre8 creases · degree eight, and the vertex at the middle of the first base anybody folds112164a vertex at no particular angles6 creases · degree six, drawn from the census and rounded to a tenth of a degree881at four creases the marking names the object; above it, it need not Flat-folding

One marking, many objects

A crease pattern with every mountain and valley written on it is spoken of as though it named a folded model. At four creases it does. At six it need not, and at the eight-crease vertex in the middle of the first base anybody folds, a single marking can be folded into four genuinely different objects — same creases, same letters, four answers.

 what the outline and the thickness leave open, and what the order closesprofilesambiguousthe order settlesand does not3 creases on 12ths69166103 creases on 16ths1844712354 creases on 12ths23371692the last column is patterns that fold to the same object, so no better photograph reaches them Flat-folding

The order does not name it either

A photograph of a folded model carries an outline and a layer count, and that is not enough to recover the pattern. Hand the observer the layer order as well — everything the object physically is — and most of the ambiguity goes. Most. What is left are pairs of genuinely different crease patterns that fold to the same object, which no better photograph reaches.

 what the construction produced7 twists, 60 interior verticesturned 17.2° from the tiling's edgespleats 0.123 to 0.123 wide1.76× smaller once the pleats are taken upevery vertex passes all four conditionsmountainvalleyraw edge Tessellations

The dial that decides nothing

Turn a twist tessellation's angle from one fence to the other and every measurable thing about it changes: the smallest sector goes from 88 degrees to under one, the pleats swallow a quarter of the sheet and then almost none of it, the folded footprint changes by a third. The number of ways it can be creased does not change at all — sixteen, at every angle tested — because the lemma reads which sector is smallest and never how small.

 every flat-foldable vertex whose sectors are multiples of 45°6 of them, to degree 8passfoldone objectthe most45·45·135·135866145·90·135·90444190·90·90·90888145·45·45·45·90·90302012245·45·90·45·45·90301812645·45·45·45·45·45·45·451121121643 of the 6 carry markings the conditions accept and the paper refuses Designing a base

The whole alphabet of a grid

Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.

 one graft, then a second across itstrips 0.3 and 0.3 wideas designedone stripand one across itthe first strip adds 0.300the second adds 0.390two separate bills would be 0.600the sheet gained 0.690the excess is 0.090and the strips cross over 0.090the crossing rectangle is 13.0 per cent of everything the two features cost Designing a base

The second term

A feature grafted into a finished design costs exactly the paper slid in for it, and the bill has one term. Add a second feature across the first and it has two: the rectangle where the strips cross is paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.

 every condition holds here6 creases4 creasesevery condition holds at the vertex on the paper — and one crimp later the smallest sector has the same letter on both sidesthe four conditions all hold · a stacking does not exist What it costs to know

A short reason to say no

When a folding question comes back yes it brings an object anybody can check. When it comes back no it usually brings nothing but the assurance that a search looked everywhere. At one vertex that is false: a refusal comes with a witness one or two steps long, out of a search space of a hundred and twelve, and the witness is a vertex the crease pattern does not contain.

 40°95°25°110°60°30°2 strictly smallest sectors, at 25° and 30°every vertex with the same shading admits exactly the same letterings Flat-folding

The order decides the count

Ask how many mountain-and-valley letterings a vertex admits and the answer looks as though it should depend on the angles. It does not. Three of the four conditions never see an angle at all, and the fourth asks only which sector is smallest — so the count is a function of a combinatorial arrangement, and a walk round the cycle that never looks at a vertex reproduces it exactly.

 what the observer is givenambiguities separated, of 71the outline alonewhat a silhouette carries0the outline and the colourwhat a photograph of duo paper carries, from both sides0the complete layer orderwhat taking the model apart carries69the middle row is what anybody can actually see, and it is the top row Flat-folding

Which side is showing

Two earlier rungs asked what a folded object records about the pattern that made it, first from its outline and then from its complete layer order. Neither observation is one anybody can make. A photograph of duo paper carries the outline, the thickness and the colour showing at every point — and over the whole census the colour separates nothing at all.

 10 mountains, 14 valleys3.95 wide, 2.10 deep13 mountains, 11 valleys1.98 wide, 1.79 deep Tessellations

The Miura folds two ways

One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.

 -0.15-0.00-0.12-3.53-0.46-2.69-0.09-0.300.44the sheet as a whole reports -0.175spread 3.97, 23× the sheet's Tessellations

Nothing to average over

A folded corrugation is reported with a Poisson's ratio, and both of this site's measurements of one were made on a sheet that repeats a single cell. On such a sheet every cell behaves the same way and the cell's number is the sheet's number. On a sheet with no repeating cell the cells run from −3.5 to +0.4 — some widening while others narrow — and the sheet's own figure describes none of them.

 patternfootprint with both colours over itlayers thereThe preliminary base8 panels, 4 one way up and 4 the other100%8.0The Miura fold24 panels, 12 one way up and 12 the other100%9.2The square twist9 panels, 5 one way up and 4 the other65%4.1The hexagon twist13 panels, 7 one way up and 6 the other72%4.1The Yoshimura pattern65 panels, 32 one way up and 33 the other100%60.0Fold and cut — the triangle7 panels, 4 one way up and 3 the other4%6.4The tapered corrugation28 panels, 14 one way up and 14 the other100%8.4The waterbomb tessellation52 panels, 26 one way up and 26 the other100%31.1 Designing a base

Decided before the design

A colour change brings the reverse side of the paper to the front, and the usual account is that the two-colouring of the panels decides which panels are available. Measured on the site's own printed patterns, availability is not the constraint: both sides lie over more than ninety-nine per cent of most folded footprints. The other side is not scarce. It is under eight layers of paper.

 23456789101105101520flapshow much the requirement costs, per cent14.61.70.0-0.05.60.510.810.44.72.8 Designing a base

The shapes the optimum has

Requiring a circle packing to be its own mirror image halves the number of coordinates a search has to find, so the same effort covers a much smaller space. Whether that helps depends on something the search cannot know in advance — whether the best packing was symmetric — and measured flap count by flap count the answer alternates without a pattern anybody could use.

 patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions54% / 46%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 11%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45° Curves and material

The fifth thing that is not true

Four idealisations underlie every theorem here and each has had an essay. There is a fifth and it has never been named, because it is invisible in exactly the way the others are not: paper has a grain, no theorem in the subject mentions a direction, and so nothing in the whole apparatus can tell a folder which way up to lay the pattern down.

 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 Flat-folding

The vertices nobody checks

Every figure on this site is gated on four conditions evaluated at every interior vertex, and the word interior has been carrying the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, not one of them has ever been examined, and the condition that decides them has been available since the second phase of the collection.

 a single vertex is always one piece; a pattern with more is notand the number of pieces is decided by the creases that never reach the edge of the paperThe preliminary base1 vertices inside the paper112 letterings admitted1 piece of 1120 creases buried2^0 = 1The square twist4 vertices inside the paper256 letterings admitted16 pieces of 164 creases buried2^4 = 16The hexagon twist6 vertices inside the paper4096 letterings admitted64 pieces of 646 creases buried2^6 = 64Fold and cut — the triangle1 vertices inside the paper30 letterings admitted1 piece of 300 creases buried2^0 = 1 Flat-folding

The creases that cannot move

One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.

 sectorsletterings that branchdecided by the choice22.5° 22.5° 157.5° 157.5°4/16022.5° 45° 157.5° 135°0/16022.5° 67.5° 157.5° 112.5°0/16022.5° 90° 157.5° 90°0/16045° 45° 135° 135°4/16045° 67.5° 135° 112.5°0/16045° 90° 135° 90°0/16067.5° 67.5° 112.5° 112.5°4/16067.5° 90° 112.5° 90°0/16090° 90° 90° 90°14/160 Designing a base

The other grid

Box pleating is drawn at forty-five degrees, and the twenty-two-and-a-half-degree grid is usually described as the same thing done finer. It is not a refinement, it is a different alphabet: five kinds of vertex become fifty-six, and the share of letterings whose decision needs a search falls from 60 per cent to 22. A finer grid is a larger vocabulary and a less ambiguous one.

 6 flaps at radius 0.1875883 contact graphs among the runs that agree about itone run5 contacts · 5 against the paper's edgeanother run4 contacts · 5 against the paper's edge Designing a base

Two packings, one radius

A packing search reports a number, and the number is not the design. What a crease pattern is built from is the graph of which discs touch which — and at five and six flaps, runs of the same search that agree about the best radius to four decimal places come back with contact graphs that are provably not the same graph. The answer an optimiser gives has not determined the pattern it is supposed to have found.

 counting rules and counting objects are different measurementsrepeating rulesnine binary choices, one per crease of the repeating unit512pass on a small patchevery vertex of a two-by-two patch satisfies every condition56pass on a larger oneand on a patch that contains all four kinds of vertex32folded objectscounted by comparing the folded panels, not the letters1 Tessellations

Thirty-two rules, one object

Five hundred and twelve repeating rules for the waterbomb tessellation, fifty-six that pass on a small patch, thirty-two that pass on one containing every kind of vertex. Fold all thirty-two and compare their panels: the same panels, in the same places, with the same areas, every time. The rules are thirty-two labels on one object, and a count of them has counted the labels.

 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%10.02.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2 What it costs to know

Four ways to draw a pattern

Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.

 what is recorded, against what is left to the folderThe preliminary base8 panels · 15 bits of orderThe Miura fold24 panels · 79 bits of orderThe square twist9 panels · 18 bits of orderThe hexagon twist13 panels · 33 bits of orderThe Yoshimura pattern65 panels · 302 bits of orderFold and cut — the triangle7 panels · 12 bits of orderThe tapered corrugation28 panels · 98 bits of orderThe waterbomb tessellation52 panels · 226 bits of orderthe pattern, as every format records itthe order of the panels, which none of them does Who found it, and when

The half no notation records

Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.

 the bar is the creases with an interior vertex at each enda folder holding one of these patterns is in one piece of the count on the right, and cannot leave it107 of 862 moves survive across the shelf · 0 touch a buried creaseThe preliminary base0 buried · 1 piecesThe Miura fold22 buried · 4,194,304 piecesThe square twist4 buried · 16 piecesThe hexagon twist6 buried · 64 piecesThe Yoshimura pattern48 buried · 2.81 × 10^14 piecesFold and cut — the triangle0 buried · 1 piecesThe tapered corrugation27 buried · 1.34 × 10^8 piecesThe waterbomb tessellation42 buried · 4.39 × 10^12 pieces Flat-folding

The pieces without the list

The letterings a pattern folds in fall into pieces no folder can cross, and the count was found by writing every lettering down — which stops at eighteen creases. The Miura has thirty-eight, the Yoshimura eighty-six, and the number of pieces can be read off the drawing without listing anything: four million and two hundred and eighty-one million million.

 the bar is every legal stacking; the dark part is the ones with no move out of thema swap is legal when the result is still a stacking — the two layers need not be joined by a crease3 segments2 of 6 isolated · 4 legal moves4 segments16 of 16 isolated · 0 legal moves5 segments34 of 50 isolated · 16 legal moves6 segments144 of 144 isolated · 0 legal moves7 segments366 of 462 isolated · 96 legal moves Flat-folding

Nothing slides past anything

A marked strip has several legal stackings and this site has counted them at length. Nobody asked whether a folder holding one can reach another by lifting a flap over its neighbour: five hundred and sixty of six hundred and seventy-two stackings have no such move at all, and whether any exists depends on the parity of the segment count.

 The preliminary base: 8 symmetries, 112 letteringsthe bar is the share of letterings the symmetry carries to themselvesa quarter turnnone of 112 — this symmetry cannot be foldeda half turnnone of 112 — this symmetry cannot be foldedthree quarters of a turnnone of 112 — this symmetry cannot be foldeda mirror across the sheet12 of 112a mirror up the sheet12 of 112a mirror in one diagonal12 of 112a mirror in the other diagonal12 of 112 Designing a base

The symmetry the letters cannot keep

Every pattern in this subject is drawn symmetric and the symmetry is always quoted of the drawing. A folded object is a drawing and a lettering together, so a symmetry survives only if the letters keep it — and the preliminary base loses every rotation while the square twist, drawn with the same eight, loses the other half.

 the pale bar is the solid sheet, the dark one the sheet with a holelower is better: it is the average share of a flap's disc that has to be paid forflap 0.060.9293 against 0.9496flap 0.10.8826 against 0.9137flap 0.150.8296 against 0.8714flap 0.220.7586 against 0.8159flap 0.30.6760 against 0.7496 Designing a base

A hole is cheap paper

A flap claims every point of paper within its own length of its tip, but only the paper that is actually there — so an edge is half price and a corner a quarter. A hole in the middle of the sheet is more edge, and holding the area fixed, a square with a hole in it is cheaper paper than a solid square at every flap length measured.

 the pale bar is the published count, the dark one the objectsneither operation ever fixes a folding; doing both sometimes does, and that is why it is not a quarter2 stamps2 labelled · 1 objects · 2 fixed by doing both3 stamps6 labelled · 2 objects · 2 fixed by doing both4 stamps16 labelled · 5 objects · 4 fixed by doing both5 stamps50 labelled · 14 objects · 6 fixed by doing both6 stamps144 labelled · 38 objects · 8 fixed by doing both7 stamps462 labelled · 120 objects · 18 fixed by doing both8 stamps1392 labelled · 353 objects · 20 fixed by doing both What it costs to know

The count counts labels

One, two, six, sixteen, fifty, a hundred and forty-four: the oldest sequence in the subject counts foldings of a strip of numbered stamps. A folded strip of blank paper has no first stamp and no top side, and neither of those operations ever leaves a folding alone — so the count of objects is 1, 2, 5, 14, 38, 120, and it is not the count over four.

 the bar is the buried creases a crumple of that depth writestwo sheets at each depth, from two streams2 folds3 creases · 1 pieces3 folds8 creases · 2 pieces4 folds17 creases · 128 pieces5 folds22 creases · 512 pieces6 folds45 creases · 2.68 × 10^8 pieces7 folds61 creases · 1.09 × 10^12 pieces Curves and material

The decision a crumple has taken

A sheet crumpled at random satisfies every condition in the subject, because it just folded. It also wrote itself a lettering — one of very many the pattern admits — and it is now in a piece of that space it cannot leave: at six folds a crumpled sheet carries thirty-nine buried creases, which is half a million million million pieces, and every change it admits stays inside one of them.

 The square twist, one crease at a timethe bar is the number of pieces, and a cut anywhere reduces itbefore any cut16 pieces · 0 vertices releasedcutting a crease that reaches the edge4 pieces · 1 vertices releasedcutting a buried crease2 pieces · 2 vertices released Curves and material

A cut is not local

Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.

 a 3 × 3 patch: 42 creases, 18 of them buriedthe bar is on a log scale, because the two numbers differ by four orders of magnitudepieces the 3×3 patch has262,144pieces containing a repeating rule32, one eachevery one of the 32 rules is in a piece no other rule is in Tessellations

Thirty-two rules, thirty-two pieces

The waterbomb tessellation's surviving repeating rules fold to one object — same panels, same places, same areas. Put them in the space of letterings the patch admits and they occupy thirty-two different pieces of a quarter of a million, so no two of them can be reached from one another without unfolding the sheet.

 the bar is the pairs of panels that lie over one anotherThe preliminary base288 panels · 12 rules · an ordering existsThe Miura fold22824 panels · 228 rules · not decidedThe square twist369 panels · 48 rules · an ordering existsThe hexagon twist6613 panels · 96 rules · an ordering existsThe Yoshimura pattern205565 panels · 1187 rules · not decidedFold and cut — the triangle217 panels · 15 rules · an ordering existsThe tapered corrugation28228 panels · 351 rules · not decidedThe waterbomb tessellation92652 panels · 654 rules · not decideda pattern with no bar has no two panels over one another, and its order is not a question Flat-folding

No height to swap

A folded strip is a permutation of segments, and the smallest change a hand can make to it is a swap of two heights: 672 stackings, 560 of them isolated. A folded sheet has no height. Its layers are ordered by statements about which panels share ground, and on every printed pattern the search can finish, the answer is one stacking and no way out of it.

 the upper bar is the rim, the lower is the middlethe value is how many other panels an average panel of that kind lies overThe Miura fold18.0 · 21.016 at the rim, 8 away from itThe square twist8.0 · 8.08 at the rim, 1 away from itThe hexagon twist10.0 · 12.012 at the rim, 1 away from itThe Yoshimura pattern61.6 · 64.021 at the rim, 44 away from itThe tapered corrugation19.0 · 22.218 at the rim, 10 away from itThe waterbomb tessellation31.0 · 37.716 at the rim, 36 away from itthe difference is small and it has the same sign every time Flat-folding

The rim lies over less

A folded sheet's boundary is usually discussed as the place the theorems stop applying. It is also visible in the pile: a panel carrying a raw edge of the paper lies over fewer of the other panels than one that does not, on every printed pattern that has both kinds — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb tessellation, and never once the other way round.

 the bar is the draws whose letters do not contradict themselvesa loop of panels is a proof that no flat folded state exists, and it costs one passone square twist39 of 409 panels · 12 creasesone hexagon twist40 of 4013 panels · 18 creasesa small square tiling24 of 4049 panels · 72 creasesthe square tiling7 of 4049 panels · 84 creasesthe patch a propagation returns first is not a draw and has no reason to be among these Tessellations

The tiling the unit could not promise

Every twist on this site carries the same caveat: the unit is verified and the plane is not, because deciding a whole pattern is intractable. There is one thing about a whole pattern that costs a single pass over its crease list, and it says no. The square twist tiling was drawn with a lettering that contains a loop of twenty-eight panels, so the patch on this site had no flat folded state at all — and only seven of forty independent redraws avoid one.

 the bar is the share of the footprint showing the side that started face upthe rest of it shows the other side, and neither is chosen by anybodyThe preliminary base0.3%2 of 8 panels in viewThe square twist47.4%7 of 9 panels in viewThe hexagon twist90.8%10 of 13 panels in viewFold and cut — the triangle100.0%1 of 7 panels in view · 2 statesa panel out of view from above is not hidden — it is under the pile, and turning the sheet over shows a different set Designing a base

Which side arrives

A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.

 the bar is the number of foldings, on a logarithmic scaleboth routes give the number printed; a disagreement anywhere would be a defect in one of them2 × 122 letterings · 1 creases3 × 164 letterings · 2 creases4 × 1168 letterings · 3 creases5 × 15016 letterings · 4 creases6 × 114432 letterings · 5 creases2 × 288 letterings · 4 creases3 × 26032 letterings · 7 creases4 × 2320128 letterings · 10 creases3 × 31,368256 letterings · 12 creasesa strip of stamps is the one-row case, and the classical sequence 2, 6, 16, 50, 144 is the top of the table What it costs to know

The map counted from the layers

The classical map-folding counts are computed from a rule that never places a panel: work out which edge of the folded square each fold wraps around, and refuse the orderings that interleave two folds at one edge. Place the panels instead and order them by the general non-crossing rules, and the same numbers come out — 2, 6, 16, 50, 144, 8, 60, 320, 1368 — on nine sizes, by machinery that shares no line of code with the first.

 two tips 0.6 of a sheet width apartthe bar is how long a flap each can carrysolid paper0.300the discs meet halfwaya slot 0.1 wide0.35719% further than solid papera slot 0.2 wide0.40435% further than solid papera slot 0.3 wide0.46655% further than solid papera slot 0.4 wide0.50067% further than solid papera slot 0.5 wide0.55184% further than solid paperthe tips do not move; only the paper between them does Designing a base

Spending the cheap paper

A hole makes a sheet cheaper by the square inch, because a flap against its rim claims only half a disc. Whether a design can spend that was left open, because no packing search here could express a region that is not paper. It can now, and the mechanism turns out not to be the discount at all: across a hole, two flaps may overlap, and two tips three fifths of a sheet apart carry flaps of 0.500 rather than 0.300.

 the bar is millimetres of crease at the printed sizea pattern with more creases is not always a pattern with more folding in itThe preliminary base724 mm8 creases · printed at 150 mmThe Miura fold1,049 mm38 creases · printed at 170 mmThe square twist704 mm12 creases · printed at 150 mmThe hexagon twist916 mm18 creases · printed at 150 mmThe Yoshimura pattern2,380 mm86 creases · printed at 170 mmFold and cut — the triangle258 mm6 creases · printed at 150 mmThe tapered corrugation1,057 mm45 creases · printed at 160 mmThe waterbomb tessellation2,290 mm76 creases · printed at 160 mmsix point seven metres of crease on a sheet seventeen centimetres across Curves and material

How much line is on the paper

A crease pattern is described by its creases: how many, at what angles, in what arrangement. What a folder spends is length. The Yoshimura this site prints has eighty-six creases and 2,380 millimetres of folding on a sheet seventeen centimetres across; the Miura has thirty-eight creases and 1,049, and the fold-and-cut triangle has six and 258 — and the two counts do not rank the eight printed patterns the same way.

 the bar is sheet-widths of crease per layer of compactionshorter is a better exchange rate, and the order is nothing like the order aboveThe Yoshimura pattern0.2314.0 of crease · 60.0 layersThe waterbomb tessellation0.4514.3 of crease · 31.5 layersThe preliminary base0.604.8 of crease · 8.0 layersThe Miura fold0.676.2 of crease · 9.2 layersThe tapered corrugation0.796.6 of crease · 8.3 layersFold and cut — the triangle1.451.7 of crease · 1.2 layersThe square twist1.564.7 of crease · 3.0 layersThe hexagon twist1.886.1 of crease · 3.3 layersa corrugation pays less per layer than a base does, and the difference is not small Rigid folding

Fourth of eight, and still not chosen for it

A deployable is sold on compaction: large in use, small in transit. Measured, the pattern that actually gets built converts folding into compaction at 0.67 sheet-widths of crease per layer, which is fourth of the eight printed patterns — nearly three times worse than the Yoshimura, which nobody deploys, and nearly three times better than the hexagon twist, which nobody deploys either. The ranking does not pick out the pattern that flew from anywhere on the shelf, and that is the finding.

 the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly Flat-folding

The vertex the list does not have

Every condition this collection checks is asked at a vertex of a crease pattern, and a crease pattern is handed to the checker as a list of points and segments. A reader is handed ink. Read the same patterns the second way and eight printed sheets gain nothing at all — while four tessellation patches gain 12, 18, 12 and 5 vertices that nobody wrote down, every one of them a place where two creases were drawn across each other.

 the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge Flat-folding

Two creases that cross

A crossing is four creases at a point, so the four conditions of the subject apply to it — and three of them can be satisfied. It is developable at every angle, it satisfies the big-little-big lemma whenever its two lines carry different letters, and it satisfies Kawasaki's condition when the lines meet squarely. Maekawa's refuses it always, at every angle and under every lettering, because a crossing's four spokes belong to two creases and can only be four and none, two and two, or none and four.

 the same tessellation on the same square, cut out of the plane two waysassembled from whole unitsclipped from the plane12 crossings · panels 1.73 apart0 crossings · panels closemountainvalleyraw edge Tessellations

Cutting a patch out of a plane

A tessellation is infinite and a sheet is not, so every picture of one is a decision about where the paper stops. Assembling whole twist units on a square and running the outstanding pleats to the rim puts 12, 18, 12 and 5 creases across other creases on four of five tilings; generating the pattern over a larger region and clipping it puts none. The panels then place exactly — and what is waiting behind the repair is a different refusal that could not be asked about before.

 the bar is the share of the twists on the paper that the paper's edge cuts0.5 of the sheet86%1 whole · 6 cut by the edge0.42 of the sheet55%5 whole · 6 cut by the edge0.34 of the sheet59%7 whole · 10 cut by the edge0.28 of the sheet70%7 whole · 16 cut by the edge0.22 of the sheet37%17 whole · 10 cut by the edge0.18 of the sheet49%23 whole · 22 cut by the edgea patch is a picture of a tessellation, and the smaller the unit the less of the picture is edge Flat-folding

Most of a patch is edge

Between 34% and 91% of the vertices in the crease patterns drawn here sit on the edge of the paper rather than inside it, and on the tessellation patches — the figures that are meant to show what a repeating pattern looks like — it never falls below a third. A boundary is one unit deep whatever the unit is, so the share falls like one over the number of units across and reaches nothing at any size a page can carry.

 the bar is the proportion of the sheet the pattern asks for2 columns, 2 rows1.183(2 + tan 20°) / 2 = 1.1833 columns, 3 rows1.122(3 + tan 20°) / 3 = 1.1224 columns, 3 rows1.455(4 + tan 20°) / 3 = 1.4556 columns, 4 rows1.591(6 + tan 20°) / 4 = 1.5916 columns, 6 rows1.061(6 + tan 20°) / 6 = 1.0618 columns, 6 rows1.394(8 + tan 20°) / 6 = 1.394a square sheet needs a proportion of exactly one, which the counts and the slant have to be chosen for Tessellations

The paper a pattern asks for

A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.

 the cut line10 straight edgesthe pattern10 skeleton arcs0 perpendiculars1 interior vertexassignments that fold420 of 102410 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 Flat-folding

One cut for a star

The fold-and-cut construction here could reach a triangle, a pentagon and a house, and refused everything that turned back on itself, because shrinking an outline with a reflex corner needs an event the shrink did not implement. With split events it reaches a five-pointed star — ten creases through one point, four hundred and twenty letterings that fold, and every edge of the outline landing on one line to a part in 10^16.

 the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly Rigid folding

How deep is a crossing

A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

 the outline, shrunk — and the one corner that moves outwardwhat the shrink finds5 edges, 1 of them meeting at a corner that turns back3 skeleton nodes7 arcs traced by the cornersthe last of them forms at 0.181 of a sheet Designing a base

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

 the bar is the share of random drawings with at least one crossing in them2 segments23.1%0.23 crossings on average3 segments51.2%0.69 crossings on average4 segments73.5%1.36 crossings on average6 segments95.2%3.48 crossings on average8 segments99.4%6.53 crossings on average12 segments100.0%15.30 crossings on average20 segments100.0%43.76 crossings on averageevery crease pattern in this collection has none, and none of them was drawn at random What it costs to know

Drawn by the same hand

Two straight segments dropped on a square cross about 23% of the time; four of them cross 74% of the time; twelve cross with certainty, about fifteen times over. Every crease pattern in this collection's four test populations has none — not because the checkers were catching them, but because the same rules that drew the patterns were incapable of producing one, and nothing looked until a construction finally did.

 each number is the folding in that band against the paper in it — one is an even spreadthe rimband 2band 3band 4the middleThe preliminary base0.550.720.991.694.96The Miura fold0.431.410.751.941.67The square twist0.660.881.182.030.94The hexagon twist0.600.871.242.440The Yoshimura pattern0.791.011.091.211.77Fold and cut — the triangle00.201.313.446.79The tapered corrugation0.611.241.390.731.63The waterbomb tessellation0.890.931.260.951.34bands are equal in depth and not in area: 36% · 28% · 20% · 12% · 4% of the sheet, from the rim inward Curves and material

Where the length sits

A pattern's folding length is a total, and a total says nothing about where the work is. Divide each printed sheet into bands by distance from its own edge and the answer separates the patterns by kind: a traditional base carries five times its share of folding in the middle 4% of the paper, a tessellation carries between 0.9 and 1.4 everywhere, and a twist unit carries none at all at its centre. On all eight, the outermost band carries less than its share.

 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 Who found it, and when

A file has no paper

The field's interchange format is three arrays — where the vertices are, which pairs of them an edge joins, and a letter for each edge — and that is exactly the object every computation on a crease pattern starts from. A list of edges cannot say that two of them must not cross, because crossing is a property of the drawing and the list has no drawing in it. So a pattern that no paper could carry is a perfectly well-formed file, and four of this collection's own were.

 the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge Who found it, and when

The reader decides the junction

Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

 the bar is the share of draws whose letters agree among themselvesa draw that disagrees is a proof that the pattern has no flat folded state with those lettersthe preliminary base200 of 2008 panels · 8 creases · 0 contradict themselvesthe square twist198 of 2009 panels · 12 creases · 2 contradict themselvesthe Yoshimura190 of 20065 panels · 86 creases · 10 contradict themselvesthe Miura fold181 of 20024 panels · 38 creases · 19 contradict themselvesa square twist patch26 of 20049 panels · 84 creases · 174 contradict themselvesa hexagonal patch2 of 20077 panels · 142 creases · 198 contradict themselvesa rhombille patch0 of 200157 panels · 282 creases · 200 contradict themselvesthe sampler returns solutions rather than a uniform draw over them, so these are shares of what it found Flat-folding

A proof in one pass

Deciding whether a crease pattern has a flat folded state is hard, and the search that decides it gives up at twenty-four panels. One line of the same machinery does not search at all: each crease says which of the two panels it joins lies above the other, and a circle in what those statements demand is a proof that no folded state exists. It costs one pass over the crease list, and on a tessellation patch of a hundred and fifty-seven panels it answers in milliseconds.

 the bar is the letterings that pass every condition at the vertexnone of them forces a loop, because the one lettering that would is the one Maekawa forbidsdegree 48 pass · 0 loop16 letterings · 8 admissible · the alternation fails Maekawa alonedegree 630 pass · 0 loop64 letterings · 30 admissible · the alternation fails Maekawa alonedegree 8112 pass · 0 loop256 letterings · 112 admissible · the alternation fails Maekawa alonechecked at equal sectors and at a skew of 0.18 radians, so the count is not a fact about a symmetry Flat-folding

The loop a vertex cannot close

A crease pattern's letters can contradict themselves, and the contradiction is never local. Enumerate every mountain-valley labelling of a single interior vertex at degree four, six and eight — a hundred and fifty pass every condition the subject has — and not one of them sends its panels round in a circle. The one labelling that would is refused by Maekawa, alone: Kawasaki holds on it and so does the big-little-big lemma.

 shaded is every panel that lies on some loop49 panels · 1 tangle · biggest 3535 panels on some loop — 71.4% of the patch52 of 84 arcs run inside it, so one cut removes one of them Flat-folding

The loop is not the tangle

A search that finds a contradiction in a pattern's letters reports the first circle it meets, and on a tessellation patch that is eight to twelve panels of forty-nine. It reads as a local fault. Decompose the same arrows a second way and the set of panels that lie on some circle is thirty-five of forty-nine on the square patch and ninety-nine of a hundred and fifty-seven on the rhombille — which is why the smallest available repair does not reach it, and cannot be tried on most of the creases at all.

 the bar is how many circles of that many panels were found726 circles, from 6 panels to 32, over every pattern family measured here4 panels0round one vertex — Maekawa forbids it5 panels0odd — the two-colouring forbids it6 panels21129.1% of the circles measured7 panels0odd — the two-colouring forbids it8 panels21129.1% of the circles measured9 panels0odd — the two-colouring forbids it10 panels8211.3% of the circles measured11 panels0odd — the two-colouring forbids it12 panels9513.1% of the circles measured13 panels0odd — the two-colouring forbids it14 panels304.1% of the circles measured15 panels0odd — the two-colouring forbids it16 panels314.3% of the circles measured17 panels0odd — the two-colouring forbids it18 panels172.3% of the circles measured19 panels0odd — the two-colouring forbids it20 panels141.9% of the circles measured22 panels81.1% of the circles measured24 panels152.1% of the circles measured26 panels71.0% of the circles measured28 panels20.3% of the circles measured30 panels20.3% of the circles measured32 panels10.1% of the circles measuredthe empty rows are not rare cases — they are lengths that cannot occur, and each has its own reason Flat-folding

A contradiction is even

A crease pattern's letters can demand a circle of panels each of which lies below the next, which is a proof that the sheet has no folded state. Every such circle found here — one thousand one hundred and forty-nine of them, across every family of patterns this collection draws — has an even number of panels in it, and none has four. Both facts are theorems rather than observations, and they come from opposite ends of the subject.

 the bar is the share of draws that agree with themselvesthe rows are ordered by panel count, which is the only thing changing along them49 panels26 of 200square · 84 creases · 26 of 20062 panels5 of 200elongated · 106 creases · 5 of 20077 panels2 of 200hexagonal · 142 creases · 2 of 20083 panels0 of 200triangular · 142 creases · 0 of 200157 panels0 of 200rhombille · 282 creases · 0 of 200a zero is a zero of the draws taken and not a proof that no consistent lettering exists Tessellations

Letters that agree get rarer

Two hundred letterings drawn independently from a square twist tessellation patch, and twenty-six of them have letters that do not contradict themselves. On the next patch up it is five, then two, then none, then none. What the share falls with is not the size of the patch and not the angle of its twist: it is the number of independent closed chains its panels form, which is Euler's relation on the drawing and is fixed before a single letter is chosen.

 each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsa loop of 8 panels — no order existsthe arrows are the whole of the test — nothing here asks which panels lie over which Tessellations

The ring is the loop

The square twist's central polygon is four creases enclosing one panel, and a lettering that gives all four the same letter has no folded state. That was established by enumerating the orderings of nine panels. It can now be read off the crease list in one pass, because the eight panels the letters send round in a circle are exactly the ring — the twist's own defining feature, contradicting itself.

 every curve is one construction grown, and the axis is the same for all threethe Miura, grownthe Yoshimura, growntwist patches00.2500.5000.7501255075100125independent closed chains of panelsshare of letterings that agree with themselvesthe horizontal axis is read off the drawing before any letter is chosen, and it is the number of interior vertices Tessellations

A corrugation agrees with itself

A Miura fold of forty-eight panels and a twist tessellation patch of forty-nine have almost exactly the same number of independent closed chains for their letters to contradict themselves round — thirty-five against thirty-six. Sixty-four per cent of the Miura's drawn letterings are consistent and thirteen per cent of the patch's. A Yoshimura at thirty-three chains manages ninety-three. The room to fail sets the scale; the construction decides where in it a pattern lands.

 two kinds of vertex, both forced16 of degree 490°, 90°, 90°, 90°9 of degree 690°, 45°, 45°, 90°, 45°, 45°40 mountain and 36 valley creases14.3 sheet-widths of foldingmountainvalleyraw edge Tessellations

The rule that breaks the count

The waterbomb tessellation has five hundred and twelve repeating rules for its letters and thirty-two of them fold. A hundred and twenty of the other four hundred and eighty send four panels round in a circle — the shortest circle a crease pattern can have — and every single one of those hundred and twenty has broken Maekawa's count at the very vertex the circle goes round. The theorem that closes the shortest circle, caught doing it, a hundred and twenty times.

 the bar is the mean share of redraws that agree with themselvesas the populations stand, every member is consistent and the refusal fires on none of themthe printed patterns96.7%8 of 8 could be asked · worst member 90%twist tessellations55.0%7 of 12 could be asked · worst member 7%quadrilateral meshes96.9%6 of 6 could be asked · worst member 82%fold-and-cut patterns100.0%7 of 7 could be asked · worst member 100%a member with no folded state has no letters to redraw and is counted as not asked rather than as passing What it costs to know

A population that cannot fail

Thirty-three crease patterns are kept here to run the checkers over, and every one of them has letters that agree with themselves. That is not a property of the patterns. It is a property of how they were made: each came from a construction that returns a lettering, so a test looking for letters that contradict themselves has nothing to fire on. Reletter the same thirty-three and the failure is available at once — on one member, four of sixty redraws.

 the letters a folding gives a sheet always agree — these are the ones it might have had insteadfolded from seed 7folded from seed 11folded from seed 2300.2500.5000.750110203040panels in the folded sheetshare of redrawn letterings that agreeeach point is one sheet folded a given number of times, and the horizontal axis is what that produced Curves and material

The letters a crumple was given

A sheet creased by folding it and folding it again arrives with a mountain-valley labelling that cannot be wrong, because a folding produced it. Nothing about the pattern protects it: reletter the same creases and the share of labellings whose letters agree falls from every one of forty at eight panels to eleven of forty at forty-one. The foldability of a crumple is a fact about its history, not about its drawing.

 the bar is the cuts after which the panels still place at allnone of them clears the contradiction, and no cut of a buried crease leaves a sheet that placessquare0 of 8484 cut, one at a time · 0 still place · 60 are buried and none of those doeselongated0 of 106106 cut, one at a time · 0 still place · 74 are buried and none of those doeshexagonal14 of 142142 cut, one at a time · 14 still place · 100 are buried and none of those doestriangular2 of 142142 cut, one at a time · 2 still place · 100 are buried and none of those doesa cut along a crease removes no paper — the two panels are still there and are no longer joined Curves and material

One cut removes one arc

A crease pattern whose letters contradict themselves has, in principle, an obvious smallest repair: cut one crease and the statement it was making goes away. Cut every crease of four tessellation patches in turn — four hundred and seventy-four cuts — and sixteen of them leave a sheet whose panels still land anywhere at all. A cut gives the paper a freedom, and a sheet with a freedom in it has no folded state to order.

 the cut line10 straight edgesthe pattern10 skeleton arcs0 perpendiculars1 interior vertexassignments that fold420 of 102410 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 Designing a base

A tree cannot argue

A molecule fills a polygon with creases taken from its straight skeleton, and a straight skeleton is a tree. So a molecule's panels have almost no closed chains for its letters to contradict themselves round — one to three, against thirty-six on the smallest tessellation patch. Two hundred and eighty independent letterings across seven outlines, including an L and a five-pointed star, and not one of them disagrees with itself.

 what is recorded, against what is left to the folderThe preliminary base8 panels · 15 bits of orderThe Miura fold24 panels · 79 bits of orderThe square twist9 panels · 18 bits of orderThe hexagon twist13 panels · 33 bits of orderThe Yoshimura pattern65 panels · 302 bits of orderFold and cut — the triangle7 panels · 12 bits of orderThe tapered corrugation28 panels · 98 bits of orderThe waterbomb tessellation52 panels · 226 bits of orderthe pattern, as every format records itthe order of the panels, which none of them does Who found it, and when

The file records no verdict

A crease pattern file records vertices, edges and letters. Every one of the square twist's two hundred and fifty-six admissible letterings makes a perfectly valid file, and two hundred and forty-eight of them describe an object that does not exist. The format has a field for the layer order — the one thing that would settle it — and nothing fills it in, so a file is a drawing rather than a claim, and the field exchanges them as though they were claims.

 each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsa loop of 8 panels — no order existsthe arrows are the whole of the test — nothing here asks which panels lie over which Who found it, and when

The first thing about layers

A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.

 a lettering of the patch that agrees with itselffound by testing the arcs while the letters were chosen, not after561 nodes · 246 backtracks · verified against a rebuilt folded sheet157 panels · 282 creasesits own lettering sends its panels round in a circle0 of 200 random letterings agree with themselvesthis one was found in 561 nodes and 246 backtracksit differs from the drawn lettering on 155 of 282 creasesthe drawing is the pattern; nothing here is a picture of the folded object Flat-folding

The lettering nobody could draw

Two hundred letterings drawn at random from the rhombille tessellation patch, and not one of them agrees with itself. Two thousand, and still not one. The patch was left as an open question — and it has an answer, found in five hundred and sixty-one steps by a search that tests the arcs while it is choosing the letters instead of after it has chosen them all.

 the bar is how many creases the found lettering writes differentlymeasured against the lettering the pattern's own construction producedthe square patch4545 of 84 creases · 31 of them buriedthe elongated patch6666 of 106 creases · 42 of them buriedthe hexagonal patch6767 of 142 creases · 45 of them buriedthe triangular patch8787 of 142 creases · 65 of them buriedthe rhombille patch155155 of 282 creases · 117 of them burieda buried crease has an interior vertex at each end, and no legal move ever changes one Flat-folding

One solution of a search nobody ran

A crease pattern arrives with its letters already on it, and they look like part of the drawing. They are not. Every construction here ends in a propagation, a propagation ends wherever its first guess took it, and the lettering that comes out differs from the one a search finds on between a half and three-fifths of the creases — on patterns whose own letters are perfectly good.

 the bar is the shortest crease in the pattern, on a scale of powers of tenthe hexagonal patch at four turns of its polygons, everything else heldturn 0.21.4e-2130 creases · every one carries an arc · 2.25 mm on a 160 mm sheetturn 0.357.9e-6142 creases · 12 of them carry no arc · 1.3 µm on a 160 mm sheetturn 0.57.5e-3154 creases · every one carries an arc · 1.20 mm on a 160 mm sheetturn 0.72.8e-2154 creases · every one carries an arc · 4.53 mm on a 160 mm sheetone turn of one patch drops four orders of magnitude below the others, and it is the turn this collection prints Flat-folding

Twelve creases a micrometre long

A patch this collection has drawn for a long time carries a hundred and forty-two creases and a hundred and thirty arcs, and nobody had asked what the other twelve were. They are fragments left where the clip caught a pleat almost exactly at a corner — between one and nine micrometres long on a printed sheet, at one turn angle out of four, and it is the turn the collection prints.

 the 16 repeating rules that fold, written outrows first, then the two column classes — and every one of them alternates down the columnrows · columns above|below20VV · MV|MV21MV · MV|MV22VM · MV|MV23MM · MV|MV24VV · VM|MV25MV · VM|MV26VM · VM|MV27MM · VM|MV36VV · MV|VM37MV · MV|VM38VM · MV|VM39MM · MV|VM40VV · VM|VM41MV · VM|VM42VM · VM|VM43MM · VM|VMfour ways of writing the rows times four ways of alternating the columns is sixteen, and there is nothing else Tessellations

Sixty-four rules, sixteen fold

The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.

 the bar is how many of a hundred random letterings agree with themselveson the orthogonal grid a box-pleated design is drawn on, at five sizes4 by 4949 interior vertices · 16 panels · found in 16 nodes6 by 67325 interior vertices · 36 panels · found in 37 nodes8 by 85849 interior vertices · 64 panels · found in 65 nodes10 by 103681 interior vertices · 100 panels · found in 100 nodes12 by 1215121 interior vertices · 144 panels · found in 145 nodes16 by 161225 interior vertices · 256 panels · found in 261 nodesevery interior vertex is a four-panel circuit, so the number of places a contradiction could sit is the number of vertices Designing a base

What a grid costs in circuits

Box-pleating puts every crease on a square grid, and a square grid is the shape with the most short circuits per panel that this collection draws. On the sixteen-by-sixteen grid a designer actually works on, one mountain-valley labelling in a hundred agrees with itself. A search still finds one in two hundred and sixty-one steps.

 each circle holds a crease shorter than a thousandth of the sheet12 fragments7.9·10⁻⁶ of a sheet · M5.9·10⁻⁵ of a sheet · M7.9·10⁻⁶ of a sheet · M5.9·10⁻⁵ of a sheet · V7.9·10⁻⁶ of a sheet · V5.9·10⁻⁵ of a sheet · Mand 6 more1018× to draw the longest142 creases and 60 interior vertices, 12 of the first and six of the second invisible Flat-folding

The crease the drawing cannot show

Twelve creases on a printed crease pattern are eight millionths of a sheet long. They are in every count the collection takes of that patch, they pass every theorem, and no printer resolves them and no hand folds them. They are also the only thing holding the folded sheet together.

 one row per pitch, with the printed setting markeda fragment is a crease shorter than a thousandth of the sheetpitchcreasesinterior verticesfragments0.32015466—0.33015466—0.33514260—0.34014260120.34513054—0.35013054—0.36013054—130 creases and 54 vertices is what deleting the fragments was expected to give, and one step of pitch gives it with a sheet that folds Flat-folding

A patch on a knife edge

The tessellation patch this collection prints has twelve creases nobody can see. Move the pitch of its tiling by five thousandths and they are gone — and so is a whole ring of twists. The patch sits exactly on the moment a ring of the pattern passes through the edge of the sheet, and the blemish is what that moment looks like.

 each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all Tessellations

The dial and the tiling that is not alike

Four of the five tilings a twist tessellation can be built on behave identically under every dial the construction has. The fifth has two kinds of vertex, and everything about it is different: it is the only one whose search has a tail, the only one whose shallow patches take minutes to draw, and the only one where a distance has to be solved rather than assumed.

 each mark is one crease, ranked shortest to longest10⁻⁵10⁻⁴0.0010.010.1a factor of 498, and no crease in itlength, as a fraction of the sheet's side142 creases, rankedthe 12 in magenta are drawn, counted, lettered and put through every theorem, and none of them is visible Curves and material

The shortest crease is not a crease

A crease pattern's density is usually quoted as total crease length over sheet area, which treats a metre of folding as a metre whether it arrives as one long line or ten thousand short ones. Reading the lengths individually instead finds twelve creases on a printed patch that are shorter than a wavelength of light.

 the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false Flat-folding

A loop that goes somewhere

Every crease says which of its two panels lies above the other, and a loop in those statements is a proof that the pattern has no flat folded state. On a sheet with no edge that sentence is false. The loops of a periodic pattern carry a lattice step each, and a loop that ends one cell to the right is not a contradiction — it is a stack of paper with no bottom layer.

 the impossible lettering, on ordinary patchessquare ×140 creases16 vertices · every condition holds · no forced loopsquare ×2144 creases64 vertices · every condition holds · no forced loopsquare ×3312 creases144 vertices · every condition holds · no forced looptriangular ×1116 creases48 vertices · every condition holds · no forced looptriangular ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×1116 creases48 vertices · every condition holds · no forced loophexagonal ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×3924 creases432 vertices · every condition holds · no forced loopelongated ×1184 creases80 vertices · every condition holds · no forced loopelongated ×2688 creases320 vertices · every condition holds · no forced loopthe bar is the crease count; the note is what the ordinary checks said Flat-folding

The lettering that was proved impossible

A search closed its whole tree on a glued square tessellation and reported that no mountain-and-valley assignment of it is consistent. Written onto ordinary patches of one, four and nine periods and handed to the four vertex theorems and a folded sheet rebuilt from scratch, the assignment it says cannot exist passes every check, on four tilings, up to fifteen hundred creases.

 one period of the square grid's twist tessellationa ring is where a crease leaves and returns on the far side40 crease pieces → 32 creases25 drawn panels → 16 panels16 vertices, every one interiorV − E + F = 0mountainvalleyraw edge Flat-folding

A sheet with no edge

A twist tessellation repeats, so a rectangle of it is a description of the whole plane rather than a piece of paper. Joining the rectangle's opposite sides makes that explicit and produces an object every gate in this collection can read: twenty-five drawn panels become sixteen, forty crease pieces become thirty-two, sixteen vertices are all interior, and the three counts add to nothing.

 panels with nothing below them, and where they aresquare ×1125 panels, 16 of them touching the edge · all 1 at the edgesquare ×2281 panels, 32 of them touching the edge · all 2 at the edgesquare ×33169 panels, 48 of them touching the edge · all 3 at the edgetriangular ×1369 panels, 39 of them touching the edge · all 3 at the edgetriangular ×25233 panels, 79 of them touching the edge · all 5 at the edgehexagonal ×1469 panels, 39 of them touching the edge · all 4 at the edgehexagonal ×27233 panels, 79 of them touching the edge · all 7 at the edgehexagonal ×310493 panels, 119 of them touching the edge · all 10 at the edgeelongated ×12105 panels, 48 of them touching the edge · all 2 at the edgeelongated ×23369 panels, 96 of them touching the edge · all 3 at the edgethe sheet these letters belong to has no such panel at all Flat-folding

The bottom layer is at the rim

A hundred and sixty-nine panels of folded tessellation, and three of them have nothing underneath. All three touch the paper's edge, and the same is true on every tiling at every size measured. Which panel is at the bottom of a stack turns out to be a fact about where the sheet was cut rather than about the pattern, and the pattern itself has no bottom at all.

 sliding the cut across one period of the square tessellation36 vertices at every position, and a different set of creases divided at each0102030cut at the start of a periodone period alongnodes; the axis starts at zero, and the whole spread is inside a factor of 1.32 What it costs to know

Where you cut hardly matters

Slide the same rectangle across one whole period of the same tessellation and every position gives a different patch: different creases divided, different half-panels round the edge, panel counts from forty-nine to sixty-one. The cost of lettering them runs from twenty-five steps to thirty-three. Whether a cut is made changes the answer by three orders of magnitude; where it falls changes it by a third.

 one similarity, five tilingslong: a cell of the flat sheet · short: where it lands foldedthe square grid ×0.41037344the triangular grid ×0.41037344the honeycomb ×0.41037344the elongated triangular tiling ×0.41037344the rhombille tiling ×0.41037344turned 36.62°, the same on every onethe scale is a property of the pleat, and the tiling does not enter it Tessellations

Folding it flat is one similarity

Where a cell of paper goes when a twist tessellation collapses is a scale and a turn: multiply the plane's lattice by 0.410373441 and rotate it by 36.62°. That is the answer on the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the rhombille alike, agreeing to eight figures — while the collection's other answer to how much smaller it gets gives those five tilings five different numbers.

 how much a cut adds to a crease count, and to a crease lengthsquare ×150.0% too many12 creases counted for 8 · length 12.675 a unit either waysquare ×225.0% too many40 creases counted for 32 · length 12.675 a unit either waysquare ×316.7% too many84 creases counted for 72 · length 12.675 a unit either waytriangular ×141.7% too many34 creases counted for 24 · length 16.938 a unit either waytriangular ×220.8% too many116 creases counted for 96 · length 16.938 a unit either waytriangular ×313.9% too many246 creases counted for 216 · length 16.938 a unit either wayhexagonal ×141.7% too many34 creases counted for 24 · length 17.691 a unit either wayhexagonal ×220.8% too many116 creases counted for 96 · length 17.691 a unit either wayhexagonal ×313.9% too many246 creases counted for 216 · length 17.691 a unit either wayelongated ×130.0% too many52 creases counted for 40 · length 14.654 a unit either wayelongated ×215.0% too many184 creases counted for 160 · length 14.654 a unit either wayelongated ×310.0% too many396 creases counted for 360 · length 14.654 a unit either wayrhombille ×125.0% too many60 creases counted for 48 · length 23.514 a unit either wayrhombille ×212.5% too many216 creases counted for 192 · length 23.514 a unit either wayrhombille ×38.3% too many468 creases counted for 432 · length 23.514 a unit either waythe length is exact because the two halves of a divided crease add back up Curves and material

A count is not a length

Cut a rectangle out of a tessellation and it reports fifty per cent more creases than the pattern has, then twenty-five, then seventeen — converging on the truth from above and never reaching it. The crease length per unit area it reports is exact at every size, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not.

 panels with nothing below them, and where they aresquare ×1125 panels, 16 of them touching the edge · all 1 at the edgesquare ×2281 panels, 32 of them touching the edge · all 2 at the edgesquare ×33169 panels, 48 of them touching the edge · all 3 at the edgetriangular ×1369 panels, 39 of them touching the edge · all 3 at the edgetriangular ×25233 panels, 79 of them touching the edge · all 5 at the edgehexagonal ×1469 panels, 39 of them touching the edge · all 4 at the edgehexagonal ×27233 panels, 79 of them touching the edge · all 7 at the edgehexagonal ×310493 panels, 119 of them touching the edge · all 10 at the edgeelongated ×12105 panels, 48 of them touching the edge · all 2 at the edgeelongated ×23369 panels, 96 of them touching the edge · all 3 at the edgethe sheet these letters belong to has no such panel at all Rigid folding

An order with no least element

Enumerating every way a folded pattern can be stacked works by building upward from a panel with nothing below it. The smallest square twist patch has exactly one such stacking and takes eleven thousand steps to find it. The pattern that patch was cut from has no panel with nothing below it at all, so the enumeration has nothing to start from — and the sheet is perfectly well stacked anyway.

 the same 2×2 glued cell, searched under two rulesa cycle is a contradictiona cycle whose steps add to zero isand what the loops dothe square gridnothing, in 359 nodesevery loop travels (2 directions)the triangular gridnothing, in 12,143455 nodesevery loop travels (2 directions)the honeycombnothing, in 9,6191,043 nodesevery loop travels (3 directions)the elongated triangular tilingnothing, in 9,123162 nodesevery loop travels (5 directions)the rhombille tilingunfinished at 200,000unfinished at 200,000—“nothing, in n” is an exhausted search: a proof that the pattern has no consistent lettering, which is false Who found it, and when

A test imported without its hypothesis

The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.

 the composition, and what it has to equal3 reflections, in order[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )=?the gluing map of a Möbius band[ 1.000 0 ][ 0 -1.000 ]+ ( -1.732, 1.000 )they agree to rounding, so the band foldsand both turn the paper the same way, so the parity is righton a disc the right-hand side is the identity, which is why nobody writes it down Flat-folding

Closure is not the identity

Walk a folded state from panel to panel, composing a reflection at every crease, and come back to where the walk started: the composition has to be the identity. That is the rule everybody states, and it is a special case. On a sheet whose edges are glued the walk does not come back to where it started, and what the composition has to equal is the gluing map.

 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge Flat-folding

The triangle a strip becomes

A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.

 the Miura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters22182016panels1510128V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says Flat-folding

Half a rim

A rectangle of tessellation cut out of the plane has four edges; glued into a torus it has none. Gluing one pair and leaving the other gives the middle of the scale — the same drawing, the same vertices, the same conditions asked of them, and exactly half the rim. What the rim costs turns out to be measurable per edge rather than only at the ends.

 letters saved by gluing, and the two halves of itthe grid ×121 across + 1 along = 2 · 4 letters cut, 2 gluedthe grid ×242 across + 2 along = 4 · 12 letters cut, 8 gluedthe Miura ×132 across + 1 along = 3 · 7 letters cut, 4 gluedthe Miura ×264 across + 2 along = 6 · 22 letters cut, 16 gluedthe Yoshimura ×164 across + 2 along = 6 · 12 letters cut, 6 gluedthe Yoshimura ×2128 across + 4 along = 12 · 36 letters cut, 24 gluedone comparison says the rim costs something; four say the price is per edge Flat-folding

The rim adds up

What one glued pair of a cell's edges saves in free letters is what the other pair saves, and gluing both saves the sum. That is a rate rather than an observation, it is the form of the claim two objects could never support, and it is what makes 'the rim costs four letters a cell' a statement about tessellations rather than about one drawing.

 the Yoshimura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters36283224panels29202416V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says Flat-folding

Euler counts the gluing

Vertices minus creases plus panels comes to one on a rectangle of paper and nought on any gluing of it. That is the cheapest check that an identification did what it says, it costs three counts already being made, and it is what found a crease running exactly through the corner of a cell — a case the corner search could not see and no other check would have noticed.

 the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge Flat-folding

A grid that will not close

Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.

 the period cell of the Miuraone period, with its neighbours round it2 interior vertices in the cell7 crease pieces drawnperiod 1.000 × 2.000one column wide and two rows high, because the zigzag returns after twothe cell is a rectangle of ordinary paper until somebody says its edges are one edge Tessellations

The corrugation that closes on itself

A Miura cell crosses one crease per period in one direction and four in the other. So one of its two directions has a parity condition that half the sizes fail, and the other direction cannot fail at any size — the same sheet, the same drawing, and two gluings that behave completely differently.

 nodes of search per panel, as the rim goesthe grid ×1 cut1.004 nodes · 4 panels · 4 lettersthe grid ×2 cut1.009 nodes · 9 panels · 12 lettersthe grid ×2 cyl x1.177 nodes · 6 panels · 10 lettersthe grid ×2 cyl y1.177 nodes · 6 panels · 10 lettersthe grid ×2 torus1.506 nodes · 4 panels · 8 lettersthe grid ×3 cut1.0016 nodes · 16 panels · 24 lettersthe Miura ×1 cut1.006 nodes · 6 panels · 7 lettersthe Miura ×1 cyl y1.255 nodes · 4 panels · 6 lettersthe Miura ×2 cut1.0015 nodes · 15 panels · 22 lettersthe Miura ×2 cyl x1.1011 nodes · 10 panels · 18 lettersthe Miura ×2 cyl y1.0813 nodes · 12 panels · 20 lettersthe Miura ×2 torus1.2510 nodes · 8 panels · 16 lettersthe Miura ×3 cut1.0028 nodes · 28 panels · 45 lettersthe Miura ×3 cyl y1.0425 nodes · 24 panels · 42 lettersthe Yoshimura ×1 cut0.9110 nodes · 11 panels · 12 lettersthe Yoshimura ×1 cyl y1.139 nodes · 8 panels · 10 lettersthe Yoshimura ×2 cut0.9728 nodes · 29 panels · 36 lettersthe Yoshimura ×2 cyl y1.0024 nodes · 24 panels · 32 lettersthe Yoshimura ×3 cut0.9653 nodes · 55 panels · 72 lettersthe Yoshimura ×3 cyl x1.0042 nodes · 42 panels · 60 lettersthe Yoshimura ×3 cyl y0.9445 nodes · 48 panels · 66 lettersthe Yoshimura ×3 torus1.0036 nodes · 36 panels · 54 lettersfewer panels to divide by, and the same argument to settle Tessellations

One node per panel, with the rim gone

A rectangle of repeating pattern cut out of the plane costs exactly one node of search per panel, on every family and at every size. Take the rim away and the total falls and the cost per panel rises, because the letters that were removed were the ones that could not be wrong.

 the folded period of the Yoshimuradrawn periods across the top123456the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyesa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation Tessellations

The turn a column costs

The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.

 the folded period of the Yoshimuradrawn periods across the top1234567the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyes240°yesnoa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation Tessellations

The period nobody measured

Every repeating pattern in this collection has its drawn period recorded, because a drawing cannot be generated without one. Its folded period is recorded nowhere, and on one of the families measured the two differ by a factor of three — which means the number that has always been quoted is the wrong one for anything about the folded object.

 what each sheet costs, per panel — a square twistcut out ×10.5565 nodes on 9 panels · 12 lettersglued across ×10.6674 nodes on 6 panels · 10 lettersglued along ×10.6674 nodes on 6 panels · 10 lettersglued both ways ×10.7503 nodes on 4 panels · 8 letterscut out ×20.52013 nodes on 25 panels · 40 lettersglued across ×20.55011 nodes on 20 panels · 36 lettersglued along ×20.55011 nodes on 20 panels · 36 lettersglued both ways ×20.5639 nodes on 16 panels · 32 letterscut out ×30.61230 nodes on 49 panels · 84 lettersglued across ×32.02485 nodes on 42 panels · 78 lettersglued along ×30.57124 nodes on 42 panels · 78 lettersglued both ways ×317.361625 nodes on 36 panels · 72 lettersthe letters go down as the rim goes and the cost per panel goes up What it costs to know

Half the slack

Gluing one pair of a cell's edges removes half the free letters and costs almost nothing. Gluing the second pair removes the other half and costs three orders of magnitude. The letters go linearly and the search does not, and the reason is that the last free letter is worth more than all the others.

 the Yoshimura, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices8888free letters36283224panels29202416V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says What it costs to know

Which pair is glued

A cell's two cylinders have the same Euler number, the same amount of rim and the same name. On a symmetric drawing they have identical counts of letters, panels and vertices — and searching them costs twenty-four nodes one way and eighty-five the other. Half the rim is a description of the topology and not of the object.

 the grid, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices4444free letters1210108panels9664V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says What it costs to know

One population, four sheets

A population of patterns is a way of asking what is typical, and it has always been a population of drawings. Put the same drawings on four different sheets and the verdicts move — not because the drawings changed but because the sheet did, which means a population has two halves and only one of them was ever chosen.

 the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge What it costs to know

A map with no edges

Counting the ways a rectangular map folds is the oldest open problem in the subject, and every version of it assumes the map has an edge. Join the map's opposite edges and the question changes shape: half the sizes have no folded state at all, and the ones that do have no bottom layer to count from.

 the pieces that are one panelleft and right edges identified — 9 pieces, 6 panels9 pieces on the drawing6 panels on the sheet10 creases, 4 verticeskeeps the sidetwo pieces of one shade are one piece of paper, a cell apart What it costs to know

The tube a map makes

Join one pair of a map's edges and the result is a tube — a real object, foldable in the hand, and neither the strip's problem nor the torus's. It has one loop that cannot be shrunk instead of two, it keeps its bottom layer because it keeps half its rim, and half its sizes are refused by a parity the flat map does not have.

 a square twist, 2×2 cellsone drawing, four sheetscutoutgluedacrossgluedalonggluedboth waysvertices16161616free letters40363632panels25202016V − E + F1000the vertex row is the control: identifying edges can neither make nor destroy a vertexand Euler's number is the cheapest check that the gluing did what it says Tessellations

A tessellation on a cylinder

A twist tessellation has been drawn here as a patch and as a torus, and never as anything in between. Gluing one pair of a cell's edges gives the family its first sheet with exactly two edges — the shape every folded tube actually has, and the only object in the collection that can say whether the rim's cost is linear in how much rim there is.

 the case the corner search cannot seeedges clear of every vertex, and a crease through a corner anywaythe corner is where four edges meeta crease piece ending there has no partneron any one of themand Euler's count comes out −1the cure is a nudge along a gap the vertex search had already cleared Tessellations

The seam that is not a symmetry

Gluing a cell's edges looks like a symmetry of the drawing and is not. It is an instruction about which points of the paper are the same point, the drawing has to agree with it along the whole of a glued edge, and a rectangle that is not a period of the pattern does not glue at all — which turns out to be the only real restriction on which cylinders exist.

 the period cell of the Miuraone period, with its neighbours round it2 interior vertices in the cell7 crease pieces drawnperiod 1.000 × 2.000one column wide and two rows high, because the zigzag returns after twothe cell is a rectangle of ordinary paper until somebody says its edges are one edge Tessellations

A metamaterial with no edge

A folded metamaterial's properties are quoted per unit cell, because a material is supposed to be the same everywhere and a cell is supposed to stand for the whole of it. Every cell this collection has measured has been cut out of a patch, with a rim round it — and a rim is the one place a repeating material is not like itself.

 panels with nothing below them, and where they aresquare ×1125 panels, 16 of them touching the edge · all 1 at the edgesquare ×2281 panels, 32 of them touching the edge · all 2 at the edgesquare ×33169 panels, 48 of them touching the edge · all 3 at the edgetriangular ×1369 panels, 39 of them touching the edge · all 3 at the edgetriangular ×25233 panels, 79 of them touching the edge · all 5 at the edgehexagonal ×1469 panels, 39 of them touching the edge · all 4 at the edgehexagonal ×27233 panels, 79 of them touching the edge · all 7 at the edgehexagonal ×310493 panels, 119 of them touching the edge · all 10 at the edgeelongated ×12105 panels, 48 of them touching the edge · all 2 at the edgeelongated ×23369 panels, 96 of them touching the edge · all 3 at the edgethe sheet these letters belong to has no such panel at all Flat-folding

A bottom layer on half a rim

The bottom of a folded stack lives at the paper's edge, which is why a sheet with no edge has an order with no least element. A cylinder has half a rim, so it has a bottom — and the count of panels that could be it falls with the rim, which makes the claim a measurement rather than a boundary case.

 ruling out the square cell's loops, one direction at a timewhat is left splits248 arcs go, 24 remaindirection (1, 0)102 arcs go, 10 remainwhat is left splits010 arcs go, 0 remaindirection (-1, 0)102 arcs go, 10 remainwhat is left splits010 arcs go, 0 remainthe bar is how many arcs are still in play after the step Flat-folding

The arc that arrived twice

Which of two panels a crease calls its near one is decided by the order a face walk happened to number them, and the mirrored record is the same relation. Except on one sheet, where it is not — and that sheet turned out to be the one whose folded state comes back the other way up, which is how a duplicate in a graph became a diagnosis.

 one rectangle, glued four waysa disc, two cylinders and a torus — from one drawing4 edges lefta disc2 edges lefta cylinder, across2 edges lefta cylinder, alongno edges lefta torusthe same rectangle and the same creases in all four, and nothing in the drawing says which is whichmatching arrowheads mean the two edges are one edge of the paper Flat-folding

The drawing does not say what is glued

One crease pattern, four sheets, four different answers to whether it folds — and nothing in the drawing distinguishes them. The identification is data the picture cannot carry, and the picture is the object this collection has been treating as complete.

 the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge Designing a base

A grid glued

Box pleating is the designer's grid: every crease on a line, every angle a right angle or forty-five degrees, and a whole design method built on the convenience of it. Roll the grid into a tube and half the column counts stop folding, on a pattern whose whole selling point is that it always works.

 the impossible lettering, on ordinary patchessquare ×140 creases16 vertices · every condition holds · no forced loopsquare ×2144 creases64 vertices · every condition holds · no forced loopsquare ×3312 creases144 vertices · every condition holds · no forced looptriangular ×1116 creases48 vertices · every condition holds · no forced looptriangular ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×1116 creases48 vertices · every condition holds · no forced loophexagonal ×2424 creases192 vertices · every condition holds · no forced loophexagonal ×3924 creases432 vertices · every condition holds · no forced loopthe bar is the crease count; the note is what the ordinary checks said Designing a base

The symmetry a gluing adds

A patch of a tessellation has whatever symmetry its outline allows — a few reflections, a rotation or two. Glue its edges and it acquires translations, and a lettering of the glued sheet has to be invariant under them. That is a much stronger requirement than a lettering of the patch, and it is why one answer covers every patch at once.

 Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubepackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable Rigid folding

The tube that gets built

Every folded structure that leaves a laboratory is a sheet joined to itself — a boom, a stent, a bellows, an airbag, a packed antenna. The mathematics has been done on flat rectangles for the whole history of the subject, and the object is a cylinder, which is a different sheet with different counts and a condition the rectangle does not have.

 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 Who found it, and when

No format has a gluing

A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.

 050100150200250300350400020406080100120140160sides, up to npolygons reachablea fold — 155a compass — 40both sets computed by division to 400 · 4 Fermat primes and 13 Pierpont primes below it Axioms and construction

How many polygons a fold reaches

The heptagon is what the extra axiom buys and it is one polygon. What it actually buys is a density: to a thousand sides a compass reaches fifty-two regular polygons and a fold reaches two hundred and seventy-five, and the ratio between them is still widening. The compass has five usable primes in the whole of arithmetic and may use each once; a fold keeps acquiring new ones and may repeat the factor of three as often as it likes.

 the same census, on four sheetsshare of the sheet used, at each polygon's own best rotationsidessquareA series3 : 26 : 5346.4%40.8%38.5%48.1%4100.0%70.7%66.7%83.3%567.4%51.4%48.4%60.5%669.6%61.2%57.7%72.2%772.9%53.5%50.5%63.1%882.8%58.6%55.2%69.0%975.1%54.4%51.3%64.1%1075.3%57.4%54.2%67.7%1176.3%54.8%51.6%64.5%1280.4%56.8%53.6%67.0%best after the square: square 8 · A series 6 · 3 : 2 6 · 6 : 5 6each polygon at its own best rotation on each sheet · the square uses all of a square and 70.7% of the next sheet along Axioms and construction

The square is in the answer

The largest regular polygon a square sheet holds is not increasing in the number of sides, and the octagon's win is the striking part: it uses 82.8% of the paper against the twelve-gon's 80.4% and the hexagon's 69.6%. Run the same census over rectangles and the octagon's advantage is gone — on every proportion tried the hexagon leads, and the order among the even-sided polygons reverses outright.

 the bases at which a crossover is possiblethe backbone turns 360° over a turn's worth of bases, and a crossover needs it facing the neighbourhoneycomb — three neighbours6 exact, 0 near071421283542square — four neighbours2 exact, 4 near4.3°4.3°4.3°4.3°07142128354210.5 bases to a turn · a mark is a base whose backbone faces a neighbour to within 5°, and the number under it is the miss Folding nobody designed

The helix chooses the lattice

A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.

 seven constructions on five sheetswhere each construction's point lands, as a fraction of the sheet, against the squaresquareA series, tallA series, wide3 : 2, talldouble, widehalve it, edge onto edgea fold along the stretch1/2, 0the samethe samethe samethe samea third, from two crossing linescrossings only1/3, 1/3the samethe samethe samethe samea fifth, by repeated crossingscrossings only1, 1/5the samethe samethe samethe samea third, by Fujimoto's halvingsfolds along the stretch0.333, 0the samethe samethe samethe sameHaga's fold, corner to midpointa fold across a slant1, 2/31, 2/7off the paper1, 1/4off the papera corner halved, edge onto edgea fold across a slant1, 11, 0.7070.707, 11, 2/31/2, 1a corner onto the opposite cornera fold across a slant1, 0off the paper3/4, 0off the paper5/8, 0a stretch along the edges keeps crossings, midpoints and folds along the edges; it does not keep a fold across a slant Axioms and construction

A stretch keeps crossings

A rectangle is a square stretched along its edges, and a stretch along the edges keeps straight lines straight, crossings as crossings, midpoints as midpoints and the fraction a point divides a segment into. So a construction made only of those — halve an edge, cross two lines — lands at the same fraction of every rectangle, and four standard constructions do. A fold across a slanted line is a reflection the stretch does not keep, and every construction that uses one — Haga's, a corner halved, a corner brought to its opposite — returns a different point on some rectangle, or none.

 the lines a graft could useevery gap between vertex columns, and every gap between vertex rows, triedpatternvertical lineshorizontal linesgraftsThe preliminary base0 of 20 of 2nowhereThe Miura fold6 of 130 of 4one wayThe square twist0 of 60 of 6nowhereThe hexagon twist0 of 42 of 11one wayThe Yoshimura pattern0 of 120 of 5nowhereFold and cut — the triangle2 of 82 of 8both, in margin onlyThe tapered corrugation7 of 150 of 4one wayThe waterbomb tessellation0 of 80 of 8nowherea 4 by 4 grid4 of 44 of 4both waysthe grid, one square creased diagonally3 of 43 of 4both waysa line is admissible when every crease it crosses is square to it; the entry is admissible of the lines between vertex columns or rows Designing a base

A graft needs a square line

A strip can be slid into a finished crease pattern only along a line every crossed crease meets square, because only such a crease continues across the strip as itself. Tried on every line between the vertex columns and rows of eight printed patterns, four take no strip in either direction and three take one in a single direction. The eighth, the fold-and-cut triangle, appears to take strips both ways, and every line it admits runs through blank margin. No printed pattern takes a strip across its creases in both directions — and one diagonal crease in a plain grid removes exactly the row and the column it sits in.

 which tool the best polygon on each sheet needseach polygon at its own best rotation; the rank counts every polygon from three sides to twenty-foursheetbest polygonbest a compass buildsbest only a fold buildsbest no fold buildssquare8-gon · 82.8%8-gon21-gon · 7th · 77.9%23-gon · 6th11 : 108-gon · 75.3%8-gon14-gon · 4th · 72.6%22-gon · 9th6 : 56-gon · 72.2%6-gon14-gon · 5th · 66.6%22-gon · 9thA series6-gon · 61.2%6-gon14-gon · 5th · 56.5%22-gon · 9th3 : 26-gon · 57.7%6-gon14-gon · 5th · 53.3%22-gon · 9th2 : 16-gon · 43.3%6-gon14-gon · 5th · 39.9%22-gon · 9th3 : 16-gon · 28.9%6-gon14-gon · 5th · 26.6%22-gon · 9ththe square itself is left out; a fold builds an n-gon when the totient of n has no prime factor above three Axioms and construction

Every even polygon beats every odd one

Crossed with what each tool can build, the census of the largest regular polygon a sheet holds gives the same verdict on every proportion from a square to three to one: the best polygon is one a compass already builds, and the best polygon only a fold can build places fourth at best. The ranking itself stops moving at a proportion of 1.1284, where the hexagon overtakes the octagon. On every longer sheet only the short side holds a polygon, each polygon's share is a fixed constant divided by the length, and the constant — its area over the square of its least width — comes down to the circle's π⁄4 for even polygons and climbs up to it for odd ones. So every even polygon beats every odd one.

 the bar is the worst shortfall in hidden length, as a share of what the rim hidesa cap of 90°, straight tucks started at evenly spaced radii and at the best radii for the same count2 starts, evenly spaced12.3%2 starts, best spaced8.5%31% smaller3 starts, evenly spaced5.8%3 starts, best spaced3.7%36% smaller4 starts, evenly spaced3.3%4 starts, best spaced2.0%38% smaller8 starts, evenly spaced0.8%8 starts, best spaced0.5%40% smaller16 starts, evenly spaced0.2%16 starts, best spaced0.1%41% smallerthe best radii give every stretch between starts the same worst error, which crowds them toward the rim Curves and material

Crowd the tucks toward the rim

Straight tucks started at several radii follow a sphere's hidden length in a broken line, and evenly spaced starts leave a worst shortfall that falls as the square of their number. Evenly spaced is not the best spacing. A sphere's hiding bends hardest near the rim, so the best starts crowd outward — on a hemisphere, four of them at 0.36, 0.60 and 0.80 of the radius — and leave 38 per cent less error than four evenly spaced. As the count grows the saving closes on 42 per cent, a limit set by the square root of how the sphere's hiding bends; on a shallow dish it approaches five ninths. To follow a hemisphere within one per cent takes six rings of tucks instead of eight, and within a tenth of a per cent eighteen instead of twenty-four.

 the bar is the fewest joins that hold every crane in one pieceslack is how many merges those joins could make and do not: three a join, against n² − 13 × 34 joinsfloor 3 · 1 above · slack 4 · 1 way4 × 45 joinsfloor 5 · at the floor · slack 0 · 1 way5 × 510 joinsfloor 8 · 2 above · slack 6 · 50 ways6 × 612 joinsfloor 12 · at the floor · slack 1 · 1 way7 × 718 joinsfloor 16 · 2 above · slack 6 · 1,018 ways8 × 821 joinsfloor 21 · at the floor · slack 0 · 1 way9 × 928 joinsfloor 27 · 1 above · slack 4 · 308 ways10 × 1034 joinsfloor 33 · 1 above · slack 3 · 7,076 ways11 × 1142 joinsfloor 40 · 2 above · slack 6 · 2,068,604 ways12 × 1248 joinsfloor 48 · at the floor · slack 1 · 689 waysa slack of nought is a perfect tree of fours, every join merging four pieces that were separate until then Who found it, and when

The prediction held at eight and broke at ten

A join in a slit grid of cranes merges at most four pieces, so n² cranes need at least ⌈(n² − 1)⁄3⌉ joins, and exhaustion found four and six by six meeting that floor in exactly one way. The guess was that eight by eight would too, with twenty-one joins. Counted a join at a time, it does — twenty-one, one way. Ten by ten does not: it needs thirty-four against a floor of thirty-three, and has 7,076 ways to spend them. The grids that meet the floor with no merge to spare are four and eight by eight among every size to fourteen, and sixteen by sixteen by construction, because a perfect tree of joins on a grid twice as wide is four perfect trees and one join in the middle. The even grids were never the pattern; the doublings are.

 012340200400600800clearance above the basesurface, as a multiple of the basewalls: 2c ⁄ τplies: c ⁄ 4τeight times lesssheet thickness 0.01 · both lines are straight and their ratio is eight everywhere, so no clearance makes the stack competitive Folding nobody designed

Standing up beats lying down by eight

A level that fills its clearance with plies lying flat makes every ply share the clearance, and the sharing costs a factor of four. A level that fills the same clearance with walls standing on the base gives every member the whole height and charges them only for footing. The ratio is exactly eight, at every clearance and every thickness — and it is the difference between a cost charged against depth and a cost charged against the space beside it.

 02468101214010203040506070clearance above the basesurface, as a multiple of the basewalls stop at 40.0sheet 0.01, channel 0.05plies keep risingthey cross at 9.40the comb saturates at twice the reciprocal of its channel's share, and the stack does not saturate at all Folding nobody designed

The channel grows with what it feeds

A comb of standing walls beats a stack of plies by eight because its members do not share the clearance that pays them. Supply takes that back, and asymmetrically: a wall's channel has to be sized for the surface the wall carries, so it grows with the wall's height and is charged against the pitch, while a ply's channel is a constant charged against the clearance. The comb then saturates at twice the reciprocal of the channel's share, the stack does not saturate at all, and the two cross at a clearance the model gives in closed form.

 a body with four legs and a tail — one internal edge of 0.8circles of radius m·ℓ, and every pair's requirement drawn — the ones through the body ask for more than the two circles do Designing a base

Every pair, not every circle

A uniaxial base is designed by packing a circle for each flap, and the circles are not the condition. The condition is that every pair of the subject's extremities be separated on the sheet by the distance between them through the tree — which for two flaps meeting at one point is the sum of their lengths, and for two flaps across a body is more. Circles are the case with no body in it, so a design read off circles alone is promised a base sixteen to thirty per cent larger than the sheet can give.

 a body of three segments — two internal edges, 0.6 and 0.9the segments are the pairs at their limit — 4 of 8 of them measured through the body rather than around it Designing a base

What the condition does not decide

A tree of seven leaves imposes twenty-one separations and eight of them bind. The rest are slack, the eight pin six of the seven leaves against the sheet's own edges, and the seventh can be moved half a per cent of the sheet for nothing. The requirement that looks quadratic is doing linear work, and what it leaves undecided is the part a designer is actually choosing.

 what lengthening each edge of a body with wings, legs, a head and a tail coststhe scale falls from 0.2651 by this much per unit of extra length, measured by re-solving the arrangementchest–head0.0786a flap, 5.9% of the scale per 0.2rump–ll0.0471a flap, 3.6% of the scale per 0.2chest–rump0.0429the body, 3.2% of the scale per 0.2chest–wl0.0361a flap, 2.7% of the scale per 0.2chest–wr0.0264a flap, 2.0% of the scale per 0.2rump–tail0.0123a flap, 0.9% of the scale per 0.2rump–lr0.0120a flap, 0.9% of the scale per 0.2an edge no tight pair passes through is an edge the design can spend freely, and the condition says which Designing a base

The price of a limb is not its length

Lengthening an edge of a subject's tree costs the design some of its scale, and the amount can be measured by re-solving the arrangement. It is not proportional to the edge, and it is not the body that is dearest. On a bird whose wings are twice its legs, the head — nine tenths of a unit against the wings' one and six — costs three times what a wing costs, and two edges of a lizard cost nothing at all.

 200400600800100002004006008001000120014001600mean years ahead of the evidenceresamplingsquoted: 3575%: 15095%: 591spread 13320,000 resamplings of the fifteen entries · five per cent of them fall below 150 and five per cent above 591 Who found it, and when

The interval is wider than the number

The field's most-quoted statistic — how far ahead of its evidence a popular date runs — is the mean of eight positive gaps, and it is quoted as three hundred and fifty-seven years. Resampling the fifteen entries puts ninety per cent of its weight between a hundred and fifty and five hundred and ninety-one. The interval is wider than the number, the middle gap's interval is a fifth as wide, and the difference is the same two documents the leave-one-out found.

 how many independent sources survive for each claimthe record states this per entry and nothing has ever read the column as a distribution1 source5paper-japan, recreational-folding, senbazuru, beloch, one-cut-star2 sources5paper-china, pajarita, yoshimura, yoshizawa-notation, fold-and-cut3 sources3paper-europe, vertex-conditions, miura-ori4 sources2ceremonial-wrapping, froebela claim resting on one source is one document away from disappearing; five of the fifteen are in that position Who found it, and when

A question the record is too small to answer

Five of the fifteen claims rest on one surviving source and five on two, and those two counts are exactly what an estimate of the claims that left no source at all is made of. Applied, it says two and a half are missing. Its ninety-five per cent interval runs from fifteen to thirty-one, re-reading a single entry's source count moves it by a fifth, and its independence assumption is false in the one way documents actually fail — which is what makes computing it worth more than declining to.

 what one document would dono date is invented here — each row asks what the arithmetic would say if one turned upclaimdatedevidenceits gapif a source 200 years earlier turned upceremonial-wrapping12001600400357 → 332 (-25)recreational-folding7001680980357 → 332 (-25)senbazuru9001797897357 → 332 (-25)pajarita15001793293357 → 332 (-25)paper-china1051050nothing changespaper-europe11501056-94nothing changesbeloch19911936-55nothing changesyoshimura19691951-18nothing changesvertex-conditions19891979-10nothing changesyoshizawa-notation19611954-7nothing changesmiura-ori19951970-25nothing changespaper-japan610720110357 → 392 (+35)one-cut-star1776187397357 → 394 (+37)fold-and-cut1922199876357 → 397 (+40)froebel183718381357 → 408 (+51)the statistic is 357 as the record stands; a positive number in the last column is a discovery that makes the field look worse Who found it, and when

Some discoveries would make it worse

The field's headline statistic averages the claims dated ahead of their evidence. So a document found for a claim that was nearly right removes a small number from a mean of large ones and the average overrun goes up: finding a source two centuries earlier for the kindergarten entry would take the figure from 357 to 408. The four claims where a discovery helps can take twenty-five years off each, and no single document at any date can bring the number to 250.

 the bar is the size of the smallest shape the tests pass that no route reacheseach row adds one more cheap test to the ones above itsquares: the colour count, the ends and the cuts9 helices64 shapes of that size pass and have no routesquares: and the steps the ends force11 helices68 shapes of that size pass and have no routesquares: and the colour of every forced end11 helices32 shapes of that size pass and have no routesquares: and the colour count on a stretch a cut has fenced off12 helices12 shapes of that size pass and have no routehoneycomb: the colour count, the ends and the cuts12 helices18 shapes of that size pass and have no routehoneycomb: and the steps the ends force15 helices12 shapes of that size pass and have no routehoneycomb: and the colour of every forced end16 helices6 shapes of that size pass and have no routehoneycomb: and the colour count on a stretch a cut has fenced off16 helices6 shapes of that size pass and have no routeevery shape of every smaller size is either refused by the tests or routed by the search Folding nobody designed

A test that only knows one lattice

The cheapest argument that refuses the smallest shape no cheap test could refuse was read off that shape: cut at one helix, find the piece with no end in it, and count the colours of the stretch the route is then forced to cross. Added to the census it refuses every one of the square lattice's thirty-two eleven-helix survivors and pushes the smallest survivor to twelve, where twelve placements of two shapes survive out of half a million. On the honeycomb it refuses none of the six at sixteen. A test inherits the lattice of the witness it was read off, and the staircase is two staircases.

 how deep into the pile the machine has to be allowed to reach before it reaches every stateone layer is the patient machine and the full depth is the machine that may choose3 equal stamps22 of a possible 3 · 12 states4 equal stamps33 of a possible 4 · 32 states5 equal stamps44 of a possible 5 · 100 states6 equal stamps55 of a possible 6 · 288 statescreases at .20 .55 .7022 of a possible 4 · 8 statescreases at .15 .40 .50 .8522 of a possible 5 · 16 statescreases at .13 .31 .62 .7844 of a possible 5 · 24 statescreases at .40 .50 .62 .7233 of a possible 5 · 12 statescreases at .08 .24 .28 .35 .7255 of a possible 6 · 48 statesthe even strips are the ones that need the most, and they are the ones the machine that takes everything does best on What it costs to know

The easiest strip needs the deepest reach

The patient machine and the machine that may choose are the two ends of one number: how many layers of the pile a machine is allowed to hold. At one it reaches four states whatever the strip; at the pile's full depth it reaches everything. In between it is a machine nobody has defined, and measuring where completeness arrives inverts these essays' own ordering — the evenly creased strip, which the machine that takes everything folds perfectly, needs the deepest reach of all, and one uneven strip is complete at two.

 every degree a fold reaches up to 200, as a square root count against a cube root counta point at (a, b) is the degree 2 to the a times 3 to the b, and a tower to it takes a + b steps0123456701234square rootscube roots13927812618541624123610882472164814432966419212825 of the first 200 degrees, and they are the lattice points under a line of slope minus log 2 over log 3the pale points are the degrees a compass reaches as well Axioms and construction

Twos and threes run out

A fold reaches a number exactly when the degree of its equation is a product of twos and threes, which sounds like a large set because it is infinite and because it is so much larger than the compass's. Counted, the reachable degrees are the lattice points under a straight line, so there are about half a log-squared of them: twenty of the first hundred, a hundred and forty-two of the first million. The share falls from a fifth to one part in seven thousand, and the factor by which folding beats the compass rises at every decade without ever settling.

 how many extension steps the shortest tower to each polygon takesa polygon of n sides needs the degree of two cosine of a turn over n, which is Euler's totient halved3 sides0degree 1 · square roots only, so a compass reaches it4 sides0degree 1 · square roots only, so a compass reaches it5 sides1degree 2 · square roots only, so a compass reaches it6 sides0degree 1 · square roots only, so a compass reaches it7 sides1degree 3 · 1 cube root8 sides1degree 2 · square roots only, so a compass reaches it9 sides1degree 3 · 1 cube root10 sides1degree 2 · square roots only, so a compass reaches it12 sides1degree 2 · square roots only, so a compass reaches it13 sides2degree 6 · 1 square root and 1 cube root14 sides1degree 3 · 1 cube root15 sides2degree 4 · square roots only, so a compass reaches it16 sides2degree 4 · square roots only, so a compass reaches it17 sides3degree 8 · square roots only, so a compass reaches it18 sides1degree 3 · 1 cube root19 sides2degree 9 · 2 cube roots20 sides2degree 4 · square roots only, so a compass reaches it21 sides2degree 6 · 1 square root and 1 cube root24 sides2degree 4 · square roots only, so a compass reaches it26 sides2degree 6 · 1 square root and 1 cube root27 sides2degree 9 · 2 cube roots28 sides2degree 6 · 1 square root and 1 cube root30 sides2degree 4 · square roots only, so a compass reaches it32 sides3degree 8 · square roots only, so a compass reaches it34 sides3degree 8 · square roots only, so a compass reaches it35 sides3degree 12 · 2 square roots and 1 cube root36 sides2degree 6 · 1 square root and 1 cube root37 sides3degree 18 · 1 square root and 2 cube roots38 sides2degree 9 · 2 cube roots39 sides3degree 12 · 2 square roots and 1 cube root40 sides3degree 8 · square roots only, so a compass reaches itthe pale bars are the polygons a compass reaches, and they are not the cheap ones — 4 of the one-step polygons need a cube root Axioms and construction

Gauss's polygon is the expensive one

Which regular polygons a fold reaches is a condition on the factorisation of Euler's totient, and every polygon that passes it also has a height — the number of extension steps the shortest tower to it takes. Read that column instead of the verdict and the field inverts: the heptagon, which no compass reaches, costs one step; the seventeen-sided polygon that made Gauss famous costs three, the most on the list; and the polygons a compass finds easy are the ones a folder pays most for.

 several strips each way, and what the sheet gainedwidths 0.14, 0.14, 0.14 across and 0.18, 0.18 down3 strips across, totalling 0.422 strips down, totalling 0.36charged separately: 0.780the sheet gained: 0.931the excess: 0.1512the two totals multiplied: 0.15126 rectangles where they crossthe crossing term is 16.2 per cent of everything the features cost, and it is one term Designing a base

Six rectangles and one term

Two grafted strips crossing leave one rectangle both features are charged for and neither uses. Three strips crossing two leave six, and the obvious budget adds them up. It does not have to: the six rectangles sum to the product of the two families' total widths, exactly, so a design with any number of features is priced by two numbers rather than by a double sum — and a family of strips in one direction alone carries no crossing term at all, however many of them there are.

 a 6 by 6 grid, and the lines it still admits a strip onthe same diagonals placed three ways — the count is of lines, not of creases05100123456diagonal creases in the patternlines a strip could be slid into, both directionsno two in a row or a columnall in one rowdropped at randoma diagonal costs the row and the column it sits in, so a design that keeps its diagonals in a few rows keeps its lines clear Designing a base

A design that keeps its lines clear

A strip can be slid in only along a line every crossed crease meets square, so a diagonal crease spends the lines it crosses — and the census of eight printed patterns found four taking no strip in either direction. What decides how fast a design spends them is not how many diagonals it has but which rows and columns they sit in: six diagonals on a six-by-six grid leave twelve clear lines when they share a row and none at all when no two do, from the same six creases and the same amount of paper.

 a 8 by 8 grid, spacing 0.125 — what each strip width does to itthe bar is how many times finer the grid becomes — one means unchanged0.1251x1 spacings — the grid is unchanged0.251x2 spacings — the grid is unchanged0.3751x3 spacings — the grid is unchanged0.06252xthe grid becomes 0.0625, 2 times finer0.15xthe grid becomes 0.0250, 5 times finer0.1325xthe grid becomes 0.00500, 25 times finer0.25xthe grid becomes 0.0250, 5 times finera width four per cent away from a spacing divides the grid by twenty-five; a width half a spacing away divides it by two Designing a base

The width is charged in grid

A grafted strip may be any width at all and the bill in paper is exactly width times length, with no second term — which is what the first of these essays established, and is a statement about area. The grid is a different property of the same pattern and it is not conserved: a strip whose width is not a whole number of spacings puts every vertex past the cut onto a finer grid, and a width four per cent away from a spacing divides the grid by twenty-five where a width half a spacing away divides it by two. A width nearly right costs far more than one plainly wrong.

 the length a pattern reports, and the length a sheet of paper hassix of the eight are built on a unit square, so on those the distinction does not arisepatternown widthraw lengthreportedon paperThe Miura fold6.3739.36,6791,049out by 6.37xThe tapered corrugation1.187.81,2431,057out by 1.18xThe preliminary base1.004.8724724the sameThe square twist1.004.7704704the sameThe hexagon twist1.006.1916916the sameThe Yoshimura pattern1.0014.02,3802,380the sameFold and cut — the triangle1.001.7258258the sameThe waterbomb tessellation1.0014.32,2902,290the samea builder working in cells returns a pattern several units across, and not dividing by that is the whole of the error Curves and material

A length needs a scale

These essays measure crease length, and a crease length is a length in the pattern's own coordinates. Six of the eight printed patterns are built on a unit square, so their coordinates are sheet widths and the distinction never arises. Two are not — a Miura laid out as six cells of unit width spans 6.37 — and on those two the shelf multiplied by the printed size without dividing by the width. The Miura's folding length was reported as 6,679 millimetres and is 1,049, and the same pattern's printable sheet has carried the right number all along.

 metres of crease a square metre: what each pattern asks for, and what each paper allowsthe pale bars are patterns and the dark ones are paperswhat foil-backed tissue allows641026 µm, a crease 0.16 mm acrosswhat washi allows416740 µm, a crease 0.24 mm acrosswhat kami allows238170 µm, a crease 0.42 mm acrosswhat copier paper allows1667100 µm, a crease 0.60 mm acrossThe waterbomb tessellation89printed at 160 mmThe Yoshimura pattern82printed at 170 mmThe tapered corrugation41printed at 160 mmThe hexagon twist41printed at 150 mmThe Miura fold36printed at 170 mmThe preliminary base32printed at 150 mmThe square twist31printed at 150 mmFold and cut — the triangle11printed at 150 mma crease occupies about 6 sheet thicknesses, so the closest two creases can be laid is that, and the ceiling is its reciprocal Curves and material

The density a paper allows

Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.

 the proportion at which the hexagon passes each polygonmeasured on the share curves, and computed as √3⁄2 + ½√(8K⁄3√3 − 1) with K the polygon's constantsidessearchedclosed formdegreeodd primesthe tool it needs71.0696431.069643243one fold81.1284411.1284418nonea compass91.0802961.080296123one fold101.1163331.11633316nonea compass111.0852961.085296405two folds at once121.1097491.1097494nonea compass131.0880641.088064483one fold141.1057721.105772243one fold151.0897621.08976216nonea compass161.1031871.10318716nonea compassthe sheet is one wide; each proportion is where the hexagon's share equals the polygon's, found two ways Axioms and construction

The crossing is as hard as the polygon

Lengthen a square sheet and the largest hexagon it holds turns, pressed against all four edges, until it overtakes the polygons held by the short side alone. Every one of those overtakings happens at a proportion with a closed form, √3⁄2 + ½√(8K⁄3√3 − 1), and the number that comes out is exactly as hard to mark as the polygon being overtaken is to build. The octagon's 1.1284 is a compass number of degree eight. The heptagon's 1.0696 has degree twenty-four and needs a fold. The hendecagon's 1.0853 has degree forty and needs two folds at once.

 the best rank a tool's polygons reach, on a few sheets and on all of thema proportion between the named sheets is where each tool's best polygon does bestthe polygonon seven sheetson every sheetits own sheetonly one fold builds it14 sides · 4th14 sides · 3rd1.0257only two folds at once build it22 sides · 9th22 sides · 5th1.0103no two folds build it47 sides · 12th46 sides · 11th1.0023ranks among every polygon from three sides to 48; the seven sheets run from a square to three to one Axioms and construction

The sheet a polygon fits exactly

A regular polygon with 4k + 2 sides has flat edges along one axis and corners along the other, so there is one sheet, 1⁄cos(π⁄n) long, that it touches on all four edges at once. On that sheet it is beaten only by the multiples of four with fewer sides, and so it ranks exactly (n − 2)⁄4. That puts the fourteen-gon, which only a fold builds, third rather than fourth; the twenty-two-gon, which needs two folds at once, fifth rather than ninth; and the forty-six-gon, beyond two folds, eleventh. Seven sheets from a square to three to one had missed all three.

 crease density, the closest two creases come without meeting, and the two multipliedthe closest approach is between creases with no vertex in common, measured as segmentspatternm a m²closest mmproductThe Yoshimura pattern8224.52.02The waterbomb tessellation8920.01.79The square twist3136.11.13The Miura fold3625.10.91The hexagon twist4122.10.90The tapered corrugation4118.60.77parallel creases give a product of exactly one; a pattern above one carries more length than its own spacing would suggest Curves and material

A paper limits spacing, not density

The density ceiling put a sheet's limit at parallel creases a crease-width apart, 1⁄w of crease a square metre, and claimed no arrangement carries more. Crossing families do: they meet at vertices, which is shared ground, and never come closer than w anywhere else. Measured over every pair of creases that do not share a vertex, density times closest spacing settles at about 1.9 for both the Miura and the waterbomb, so each can be folded nearly twice as fine as the density bound said — 262 cells a side on copier paper for the Miura rather than 139, and 141 for the waterbomb rather than 74.

 the crease each pattern has least room on, once the ground near its ends is creased twicea narrow sector pushes the overlap out along both creases, as one over the sine of the anglepatternnarrowest sectorreach, both endsthat crease, mmroom to shrinkThe tapered corrugation65.9°2.1020.416×The waterbomb tessellation45.0°2.4128.320×The hexagon twist60.0°2.1525.520×The Yoshimura pattern60.0°2.3128.320×The Miura fold69.9°2.0626.722×The square twist90.0°2.0036.130×Fold and cut — the triangle58.2°1.1828.640×The preliminary base45.0°1.4175.088×reach is in band widths; room to shrink is the crease's length over that reach, on copier paper with a band 0.6 mm wide Curves and material

A vertex creases the paper twice

Two creases that meet share ground near the point, and their bands overlap out along each of them to w⁄sin θ for a sector angle θ below a right angle. Add the overlap at both ends of a crease, and a crease no longer than that is overlap from end to end — a crease only in the drawing. That third bound binds before the spacing does: the finest Miura on copier paper is 135 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 82 against 141. Both land within a tenth of the density bound, which had the wrong argument and nearly the right number.

 the shortest sequence of folds to each state, for the machine that may choose its blockfewest, mean and most over every state of every marking; the last column compares the machine that takes everythingstripstatescreasesfewestmeanmostall layers4 equal stamps32322.753the same5 equal stamps100433.404the same6 equal stamps288534.045the samecreases at .20 .55 .708333.003reaches nonecreases at .15 .40 .50 .8516444.004reaches nonecreases at .13 .31 .62 .7824444.004reaches nonecreases at .40 .50 .62 .7212444.004reaches nonecreases at .08 .24 .28 .35 .7248555.005reaches nonea fold uses at least one crease, so no sequence is longer than the crease count What it costs to know

A shallow machine pays in states, not folds

A machine allowed to take only a few layers of the pile at a time reaches fewer folded states, and the natural fear is that it also reaches the ones it does by much longer sequences. Walked breadth first, so that every state's shortest sequence is found, it does not. On unevenly creased strips every state takes exactly one fold per crease at every depth, because no two creases ever lie on one line. On strips of equal stamps a shallower machine needs one fold more for a minority of states and two more for eight of the 924 states at seven stamps — and never more than the crease count, which no machine can exceed.

 the pile 0 6 1 2 3 4 5 and its turns, which the all-layers machine cannot foldMVMVMMVVMVMMMMVMVMVVVMVMVMMMVMVMVVVMVMVMMM What it costs to know

Fourteen states are one pile

A machine that folds every layer at once reaches every folded state of a strip of six equal stamps and misses fourteen piles at seven. The fourteen are not fourteen things. Taking a pile's bottom stamp and putting it on top maps foldings to foldings, so the 462 piles of seven stamps fall into 33 classes of exactly fourteen, and the missed piles are one whole class: the pile 0 6 1 2 3 4 5 — an accordion of five stamps with the last stamp wrapped round it and slid into the fold that holds the first — seen from each of its seven stamps. At eight stamps the machine misses 64 piles, and they are exactly the piles that leave that one when an end stamp is removed.

 spending the bird's length where the arrangement prices it lowesta price is the scale lost per unit of length added to every edge of a group, re-solved at every stepstepscaledearestpricecheapestpricetight pairs00.2651head0.092tail-0.000510.2753head0.102wings0.034520.2748legs0.081wings0.012530.2781body0.072wings0.037640.2817head0.081wings0.054350.2778tail0.074head0.000560.2780head0.076tail0.0026a price holds for as long as the dearest and the cheapest group stay the same groups Designing a base

A price holds until the arrangement moves

Every edge of a subject's tree has a price — the scale lost per unit of extra length — and the obvious use of a price list is to spend a fixed total of limb where it is cheapest. Done a tenth of a unit at a time, re-pricing at every step, it works and then stops: the bird's scale rises 6.3 per cent in four steps and no further. But the prices do not hold while it happens. The bird's free tail stops being free after the first tenth, and its legs nearly treble in price without being touched. The lizard's prices hold for four steps, because its arrangement keeps the same three pairs at their limit for four steps. A price is a statement about which pairs are at or near their limit, and it lasts as long as they stay there.

 the bird rounded to a grid two waysunits in the order body, wings, head, legs, tail; the drawn tree unrounded has size 0.2651gridnearesterrorsizecheap wayerrorsize4 units1 4 2 2 316%0.26751 4 2 2 423%0.28076 units2 6 3 3 511%0.27241 6 3 3 545%0.27848 units3 8 5 4 715%0.26312 8 4 4 720%0.2782size is the scale times the sheet length one unit of the subject's own length receives; error is the worst limb's Designing a base

Rounding in the cheap direction

A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.

 the layer-order field for the printed patterns, filled in where it can bepairs is how many signs the field holds; varying is how many of them differ between folded statespatternpanelspairsstatesvaryingto recordThe preliminary base82810noneThe Miura fold24228not listed24 panelsThe square twist93610noneThe hexagon twist136610noneThe Yoshimura pattern652055not listed65 panelsFold and cut — the triangle721261.00 bitsThe tapered corrugation28282not listed28 panelsThe waterbomb tessellation52926not listed52 panelsa pattern past eighteen panels is not listed here, and those are the patterns anybody folds Who found it, and when

The field is empty where it would say nothing

The interchange format for crease patterns has a field for the layer order and nothing ever fills it in. Filling it in where the folded states can be listed — four of the eight printed patterns, and Miura patches to twelve panels — finds that the preliminary base and both twists have exactly one folded state, so every one of the field's signs follows from the crease pattern and the field would record nothing a reader could not compute. The fold-and-cut triangle has two states. The Miura is different: every patch with three or more columns has several — three, six and eleven on the three-by-two, three-by-three and four-by-three — so on the pattern that gets built the field carries information from six panels up, and the field's size had been measured as log₂ of the panels' orderings, which on the preliminary base is fifteen bits for an object that has zero.

 tests needed for each layer of compaction, pattern by patterna test campaign grows with the hinge count, so the pattern with fewest hinges per layer is cheapest to trustpatternhingeslayersper layertests per layerby testsby lengthThe preliminary base88.01.002981st3rdThe Yoshimura pattern8660.01.434272nd1stThe waterbomb tessellation7631.52.417203rd2ndThe square twist123.03.981,1854th7thThe Miura fold389.24.121,2285th4thFold and cut — the triangle61.25.051,5056th6thThe tapered corrugation458.35.401,6117th5thThe hexagon twist183.35.541,6508th8thtests are consecutive successes demonstrating 99 per cent for the whole pattern at 95 per cent confidence Rigid folding

The pattern cheapest to trust

Demonstrating that a one-shot deployment will open takes a number of successful tests proportional to its hinge count, so the pattern that needs fewest tests for what it delivers is the one with fewest hinges per layer of compaction. That criterion is a count nobody computes, and computed on the printed shelf it ranks the patterns differently from crease length per layer: the preliminary base is first, at exactly one hinge per layer, and the square twist rises from seventh to fourth. As patterns are refined the difference sharpens. The waterbomb settles at 2.47 hinges a layer and the Yoshimura at 1.47, but the Miura climbs without levelling — 1.32 at two cells a side, 5.69 at eight — so every finer Miura costs more tests for each layer it adds, and the pattern that gets built is the only one of the three that gets dearer to trust as it gets finer.

 four vertices round one panel, as a chain and as a loopa combination survives the loop only if going round it brings every fold angle back to where it startedthe faceopen chain of fourclosed loopthe same at every anglea Miura face164yesa face with no two vertices alike, seed 11161yesdriven at 0.3, 0.6, 1, 1.4, 1.8 radians · a combination counts when every crease is folded and the loop closes Folding nobody designed

A loop takes choices away

A chain of four sprung degree-four vertices has sixteen combinations of branches, each a resting state, and switches between them only through the flat sheet. Close the chain into a loop round one panel and the combinations must agree when the fold angles come back round. On a face whose four vertices all differ, one combination survives; on a Miura face, four. The count is the same at every angle the face is driven to, and two surviving assignments at the same driven angle are never closer than one and a half times that angle — so they separate as the face folds and meet only when it is flat. A loop does not create the junction a region would need to switch on its own. It removes choices and leaves the switch as global as before.

 do the pleat equations close round every loop, as drawn and under three linear mapstrivially means every edge's equation is one at both ends; otherwise the largest disagreement round a loopas drawnshearedstretchedgeneralthe square gridcloses, triviallycloses, triviallycloses, triviallycloses, triviallythe triangular gridcloses, triviallycloses, triviallycloses, triviallycloses, triviallythe honeycombcloses, triviallycloses, triviallycloses, triviallycloses, triviallythe rhombille tilingclosesoff by 1.76off by 2.18off by 2.96the elongated triangular tilingcloses, triviallyoff by 3.14closes, triviallyoff by 4.77the maps: a shear of 0.3, a stretch of 1.5 along one axis, and the matrix [1.3, 0.4; −0.2, 0.9] Tessellations

Closing the loops is not folding

The twist construction propagates one equation along every edge of a tiling, and it can only work where the equations agree round every loop. Asked which irregular tilings pass, a linear map gives a clean answer: the square grid, the triangular grid and the honeycomb pass under every shear and stretch tried, because each edge has a half-turn symmetry that makes its equation exactly one at both ends, and a half-turn survives any linear map. The rhombille passes only as drawn. But passing is not folding. On every one of those images — including the ones whose loops close exactly — the construction produces a pattern that fails the angle condition at every turn tried. The loops were a necessary condition all along, and the construction needs something the tilings' images do not give it.

 does the construction fold, with one side distance a vertex and with one a sideat a turn of 0.42 radians; the second construction's pleats run at -0.5 radians from their own edgesas drawnshearedstretchedgeneralone a vertex / one a sidethe square gridfolds/foldsno/foldsno/foldsno/foldsthe triangular gridfolds/foldsno/foldsno/foldsno/foldsthe honeycombfolds/foldsno/foldsno/foldsno/foldsthe rhombille tilingfolds/foldsno/foldsno/foldsno/foldsthe elongated triangular tilingfolds/foldsno/foldsno/foldsno/foldsthe maps: a shear of 0.3, a stretch of 1.5 along one axis, and the matrix [1.3, 0.4; −0.2, 0.9] Tessellations

One number where the corners wanted four

The twist construction gives a vertex a single side distance, and every account of these patterns does the same — it is what rotate-and-shrink means. The conditions never asked for it. Written out, the corner condition is one linear equation per pleat crease in the distances taken one per edge, so a degree-four vertex carries four unknowns against two independent equations. Given them back, the twelve sheared and stretched tilings that refused to fold all fold.

 each side of the twist, divided by the edge it facesone vertex of the the square grid under [1.3, 0.4, -0.2, 0.9]1.0520.3871.0520.387the weightsthe same edges, scaled by them, end to endthe 4 weighted edges close to 3e-16 of their own total length, and the two ends of every edge agree to 3e-15 Tessellations

Every twist writes an equilibrium

Divide each side of a twist polygon by the length of the edge it faces. The polygon closing says those numbers, weighted onto the edges, balance at the vertex; the pleat matching says the two ends of an edge agree on the number. Together they are a positive equilibrium stress — the thing a tiling has when it is the plan of a spider web — and the construction has been writing one at every vertex without being asked for it.

 how much more the folded sheet draws in one way than the otherthe ratio of the two principal factors of the collapse, fitted to the twists' positions before and after foldingas drawnshearedstretchedgeneralthe square grid1.00004.41391.00003.5417the triangular grid1.00001.00001.00001.0000the honeycomb1.00002.81795.38555.0597the rhombille tiling1.00002.81795.38555.0597the elongated triangular tiling1.00004.41391.00003.5417one is a similarity — the folded sheet is the flat one scaled and turned, with no direction preferred Tessellations

The sheet draws in crooked

Every twist tessellation measured here has collapsed by a similarity: the folded sheet is the flat one scaled and turned, the same way in every direction. The patterns that exist on sheared and stretched tilings do not. Ten of the fifteen images fold by a map with two different principal factors, up to five and a third to one — and the prediction that said which ten, made from the weights the pattern writes on its edges, is wrong in both directions.

 each family's dial, run, in the plane of the two directional factorsmeasured on the folded state of every pattern, not predicted from its rule1234561234along the patternacross itthe accordionthe square twistthe Miura foldthe Yoshimuraa point on the lower edge leaves one direction alone; a point on the dashed diagonal draws in equally both ways Tessellations

The plane the five points were in

Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.

 the cross factor of a leaf corrugation against its zigzag anglemeasured off the folded state's own extent, at every angle drawnno change0.600.680.760.840.921.00zigzag angle, radians0.981.001.02below one between 0.7 and 0.88 radians, at worst 0.98555 — and the areal factor never falls below 1.360 Tessellations

The direction that gets longer

A shrink factor below one is a direction in which the folded sheet is bigger than the flat one, and the leaf corrugation has one. The cause is not the taper and not the angle: a corrugation's folded extent across its own creases is a constant of the cell, the same number at two rows and at ten, so the cross factor is the sheet's height divided by a fixed length — a straight line through the origin that crosses one at a definite row count.

 the reach, the member length and the member count, against the anglesurface counted as a multiple of the base, lengths as multiples of the clearance90°60°30°10°1°the angle the members stand atthe reach: 200flat, at every anglehow long each member ishow many of them there areclearance 1, sheet 0.01 · the reach is 200 at every angle; the two factors move by 57 Folding nobody designed

The angle the eight does not know

A comb's members are always drawn standing square to the base, and nothing has asked why. Lean one to an angle and it must be longer to reach the same clearance, which is more surface; it also takes more of the base to stand on, which is fewer members. The two are reciprocal and cancel exactly — the surface a comb holds is the same number from a right angle down to one degree, where each member is fifty-seven times the clearance long and there are two of them where there were a hundred.

 two linings of one tube, and the ceiling neither passessurface per unit length of tube, from a sheet 0.01 thickradiuslayersfins at bestthe ceilinglayers over fins0.516078.51572.040016353146282.020022526125725132.01004100785027100532.0050the ceiling is the tube's cross-section over the sheet thickness, twice over, and no arrangement of flat sheet passes it Folding nobody designed

In a tube the standing members lose

Members standing across a clearance beat layers lying along it by eight, and every drawing of that argument has a flat base under it. Curve the base into a tube and the ranking inverts: radial fins converge, so the room they need is the room at their tips, and their best arrangement fills exactly half the cross-section. Concentric layers fill all of it. The eight becomes a half, and the half is exact.

 the cost of one lettering, by size and by how the cell is gluednodes of search, under one fixed branch orderperiodsfree lettersa discone cylinderthe othera torustorus over discthe square grid1×11254430.62×240131212131.03×384262428532.04×4144456244116926.05×52207066209292741.8the honeycomb1×1341291080.72×2116403439952.43×324676285386418655.1the triangular grid1×13412101080.72×2116373134166845.13×324691570524!12000131.9the rhombille tiling1×160226218160.72×2216!12000!120001009!120001.0a plus sign is a search that ran out of budget rather than out of possibilities; the free letters are the cut sheet's What it costs to know

Each drawing has its own threshold

Gluing a cell's edges was measured once, at one size, and found to cost three orders of magnitude — which cannot tell a threshold from a slope, nor say whether a cut sheet has one further out. Swept from one period to five on four tilings, every sheet starts at about a third of a node per free letter and every drawing leaves that behaviour at a size of its own: four periods on the square grid, three on the honeycomb, two on the triangular grid and two on the rhombille, where even the cut sheet crosses.

 the other fault a drawing can have, counted and measureda stub is a crease with a free end; the distance is how far from the rim it stoppedpatcheswith a crossingwith a stubstubsdistinct depthsshallowestdeepest120761566270.27 mm35.1 mm0.27 mm35.09 mm66 stubs27 depthsdistances at the 150 mm these patterns print at; the scale is logarithmic because the range is a factor of 128 Rigid folding

A stub is never alone

A crossing is a crease running past another and it has a depth. A stub is a crease that simply stops, and it has one too — how far from the rim it stopped, which is also how much shorter than a crease it is. Measured across a hundred and twenty drawings: sixty-six stubs, from 0.27 mm to 35 mm at printed size, every one of them paired with another at exactly the same distance, and not one on a drawing that did not already have a crossing.

 creases to mark each fraction of an edgeportable: every rectanglesquare onlyA-series sheet only0123456creases11112112313124131234515/2/3/4/5/6/7/8/9/10/11/12the portable column is one number for every rectangle; each sheet's column is true of that sheet alone Axioms and construction

What the square saves

A construction made only of crossings, midpoints and folds along the edges lands at the same fraction of every rectangle, so its cost is one number for all of them. Counted crease by crease against every fold the first four axioms allow, that portability costs about a crease and three quarters a fraction up to twelfths — and the square is not the cheapest sheet to give it up for. An A-series sheet, where Haga's fold goes silently wrong, marks two sevenths in two creases; the square needs three, and any sheet at all needs five.

 the bar is the footprint, split by how deep the other side liesone layer downtwo layers downnot under this point at allThe preliminary base8 panels · 1 stateThe square twist9 panels · 1 stateThe hexagon twist13 panels · 1 stateFold and cut — the triangle7 panels · 2 statesa 2 × 2 Miura patch4 panels · 1 statea 3 × 2 Miura patch6 panels · 3 statesa 3 × 3 Miura patch9 panels · 6 statesa 4 × 3 Miura patch12 panels · 11 statesthe letter fold3 panels · 2 statescounted over every folded state and from both faces, so nothing here is one lucky pile Designing a base

One sheet down

A colour change has been priced by how deep the pile is — eight layers over every point of the preliminary base, six on a small Miura. Ordered, the piles say something else: wherever the other side of the paper lies under a point, it is the next sheet down on every pattern with one folded state, and never more than two down on any. And relettering the same creases cannot reach it. Of 112 letterings of the preliminary base that fold, every one shows either the printed face or the whole face turned the other colour; the square twist's eight only turn its face round.

 11.051.11.151.20.850.90.9511.05the sheet's length, with its width oneshare, over the share on a square6 sides · +7.7%10 sides · +2.6%14 sides · +1.3%22 sides · +0.5%each curve is one polygon's share over its share on the square; the dot is its own sheet Axioms and construction

Turning is uphill all the way

A regular polygon of 4k + 2 sides on a sheet a little longer than a square cannot lie flat: it turns, pressed against all four edges, until the sheet is exactly its own. Its share on the way has a closed form, and the closed form's slope is proportional to h² − 1 for every such polygon — flat on the square, rising all the way to the own sheet, and falling after it. So the own sheet is exactly the peak, the gain from the square to it is the average of one and the sheet's length, and the rank the census measured for polygons of this kind, (n − 2)⁄4, is now a theorem.

 234567024681012columns (rows, for the two-column patches)folded statestwo rows high: 1, 3, 5, 7, 9, 11three rows high: 6, 11two columns wide: 1, 1, 1, 1, 1a strip two rows high has 2c − 3 states; every patch's states are one choice with that many answers Who found it, and when

One choice with eleven answers

A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.

 drawings carrying each fault, of those drawna junction is split in two: the fault, and the rim ending the reading also calls a junctionextended, of 120clipped, of 120a crossing760two creases pass through each othera stub150a crease stops in the middle of the papera crease ending on a crease00the junction as a faulta crease ending on the rim120120the junction as the reading counts ita fragment25a crease too short to seeevery stub is on a drawing with a crossing; every clipped fragment is on a drawing with nothing else wrong Rigid folding

Two faults, not four

A drawing departs from its crease list in four named ways — a crossing, a stub, a junction and a fragment — and a checker tests for all four. Counted side by side on two hundred and forty drawings, two of the tests find nothing the others do not. No crease anywhere ends on another crease: every junction the reading finds is a crease meeting the rim, which is where creases are meant to end. Every stub is on a drawing that already has a crossing. What is left is two independent faults, and the clipped construction, which makes no crossings, still makes the second — creases a thousandth of a millimetre long, in pairs.

 123456780.40.50.60.70.80.91how many lengths of finshare of the ceilingtips equally spacedheights halvingtwo thirdsradius 1, sheet thickness 0.01 · the share of the ceiling 2πR²/τ that fins of m lengths hold Folding nobody designed

The wedge belongs to one length

Radial fins inside a tube reach at best half of what any lining of sheet could hold, because converging fins leave empty wedges behind their tips. Tapering the fins cannot help: the tip already sets the count, and a fin cannot be thinner there than the sheet it is made of. Fins of several lengths can. Counted along the radius they are a staircase under a straight line, and the staircase with m steps is best with its steps equally spaced, where it holds exactly m ⁄ (m + 1) of the ceiling. The factor of two belonged to fins of one length, not to fins.

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