Origami & Folding
what one sheet can do
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.

Assumes Which vertices are the random ones and Almost every pattern fails.

19 min read 8 figures Flat is rareThe pattern is the object

Which vertices are the random ones put a question to a single vertex: every measurement of the form over 373 random degree-four vertices is a statement about a population nobody declared, four defensible populations disagree by factors, and the disagreement reached a sentence this collection had published as though it were general.

The same question can be put to a crease pattern, and there it is worse in a way that is easy to state. A vertex is a list of angles. A pattern is a list of angles and a shape — how many vertices, how they are joined, where the sheet’s edge falls — so there is one more thing for a construction to decide and nothing that decides it.

There is no way to pick a crease pattern at random. What there is, is constructions.

Four ways of making a crease pattern, four answersFour constructions that each produce crease patterns satisfying every vertex condition, asked the same four questions: what share of each pattern's vertices lie on the edge of the paper, how many layers deep the folded state gets at its worst point, how many times smaller the folded footprint is than the sheet, and how much crease length each unit of paper carries. They are not four samples of one population — no two of them produce the same patterns — and they disagree by factors rather than by margins.each row is an exhaustive count over the patterns that construction producedon the edgedeepest piletimes smallercrease densitythe printed patterns8 patterns62%19.415.5×7.9twist tessellations12 patterns52%10.02.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2
Fig. 1 Four ways of producing a crease pattern that satisfies every vertex condition, each asked the same four questions. The rows are not four estimates of anything; they are four different objects being measured.

The four, each written down on purpose

The printed shelf. The eight patterns this collection prints and a reader can fold: a preliminary base, a Miura, two twists, a Yoshimura, a fold-and-cut triangle, a tapered corrugation, a waterbomb tessellation. It is a list somebody assembled, and it is the population closest to a reader’s own paper.

Twist tessellations. One construction run over four tilings at three turn angles. Every member has degree-four vertices in a lattice and a central polygon per tiling vertex, and the family is generated by a rule rather than chosen.

Quadrilateral meshes. Developable grids with no two vertices alike, built so that every vertex is Kawasaki-exact by construction, and solved so that every loop closes. These are the patterns the rigid-folding half of this collection is written about.

Fold-and-cut patterns. The straight skeleton of an outline, with one interior vertex per skeleton node and a great many creases running out to the paper’s edge.

They are not four samples of one thing, and the check that they are not comes before any comparison: no two of them produce patterns that agree in interior-vertex count and pile depth together, so the four populations are disjoint by the measurements themselves rather than by assertion.

Four tilings, and the twists they forceFor each tiling: how many edges meet at a vertex, the polygon that puts one side on each of them, the ratio the side-matching condition forces between two unlike twists, and the two ends of the twist angle. Only the rhombille has two kinds of vertex, and only there does the ratio have anything to say.tilingverticessize ratiofloorceilingsquare grid4-gonone kind onlynone55.52°triangular grid6-gonone kind only12.37°55.52°honeycomb3-gonone kind only12.37°55.52°rhombille6-gon + 3-gon3.000 : 112.37°55.52°the ratio is what the pleat demands: two sides facing each other must be the same lengthon the rhombille that makes the hexagon's sides sit exactly three times further out than the triangle'sthe floor is a labelling that stops existing; the ceiling is the paper running out
Fig. 2 One of the four constructions, generating twelve of its members: a twist tessellation over each of four tilings. Every pattern in this population has a central polygon at every tiling vertex, which is a structural fact none of the other three shares.

What each construction is good for, which is why they differ

The four are not arbitrary and each exists because somebody needed it, which is the reason they disagree.

The shelf exists so a reader can fold something. Its selection criterion is that the folding teaches what the picture cannot, which favours patterns that do something dramatic — collapse to a thirtieth of the sheet, turn a square into a tube, rotate a polygon nobody pushed.

Twist tessellations exist because a construction generates them. Hand it a tiling and it returns a pattern that folds flat, with the twist polygons’ shapes forced by the tiling’s own angles — so the population is as large as the supply of tilings and every member is structurally like every other.

Quadrilateral meshes exist because rigid folding needs them. A mesh is the object whose closure conditions can be written down and solved, and the population is what a solver reaches rather than what a folder would draw.

Fold-and-cut patterns exist because a theorem produces them. Any straight-line drawing can be folded flat so that one cut releases it, and the construction that does it makes patterns whose creases nearly all end on the sheet’s boundary.

So the four populations are four purposes, and the properties they differ on are properties correlated with those purposes. A reader who wants to know what a crease pattern is like has to say which purpose they mean. That is a sharper requirement than it sounds: a pattern that folds flat and a pattern that folds rigidly are drawn from populations that barely overlap, and a sentence about one is rarely a sentence about the other.

What they disagree about

Four questions, each measured by machinery written for something else.

shelf twists meshes fold-and-cut
patterns in the population 8 12 6 5
share of vertices on the paper’s edge 62% 52% 67% 86%
layers at the deepest point 19.4 10.0 8.7 9.8
times smaller, folded 15.5× 2.7× 4.8× 1.3×
crease length per unit of paper 7.9 12.9 5.2 2.2

The spread between the extremes: 12.4 times on how much a pattern shrinks, 5.9 on how much creasing it costs, 2.2 on the deepest pile, 1.7 on how much of the pattern is edge.

Those are not margins. A reader given the first column and a reader given the fourth have been told different things about what a crease pattern is.

How far apart the four constructions areThe ratio between the largest and smallest answer, for each question asked of the four ways of making a crease pattern. A ratio near one would mean four measurements of one thing. None of them is near one.the ratio of the highest row to the lowest, per questionhow much of the pattern is edge1.7× — cut against twistshow deep the folded stack goes2.2× — shelf against mesheshow much smaller the folded state is12.6× — shelf against cuthow much creasing per unit of paper5.8× — twists against cut
Fig. 3 The ratio between the largest and smallest answer, per question. A ratio near one would mean four measurements of one thing; none of them is near one, and the largest is twelve.

The population this collection has always used is the extreme one

That is the uncomfortable half, and it is the same shape as the finding one rung down.

Almost every general statement here about how much a folded pattern shrinks, how deep its stack goes and what it costs in creasing has been made over the printed shelf — because those are the patterns that exist as objects, get folded, and have their numbers reported. The shelf is the highest of the four on shrink by a factor of three and the highest on pile depth by a factor of two.

It is high for a reason nobody chose deliberately and everybody would recognise: patterns get printed here when folding them teaches something, and a pattern that collapses dramatically teaches more than one that barely moves. The shelf is a curated population, curated for exactly the property it then gets measured on.

What each corrugation costsHow much smaller each pattern folds and how many layers deep it gets doing it, both measured off the folded state. The last column is the two multiplied together against the sheet they came from, and it is one everywhere, because the paper has nowhere else to be.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
Fig. 4 The shelf measured on its own, which is how this collection has usually reported these numbers. Every value here is a number about eight patterns somebody chose, and the choosing was done for reasons correlated with the measurement.

And one column runs the other way

The fold-and-cut population is the lowest on three of the four measures and the highest on the fourth: 86 per cent of its vertices are on the edge of the paper, against 52 per cent for twist tessellations.

That is structural rather than incidental. A fold-and-cut pattern is a straight skeleton plus the perpendiculars from every skeleton node to the outline, and a perpendicular is a crease that runs from an interior vertex to the paper’s edge by construction. So the population is mostly boundary because the construction makes creases that end at the boundary.

It matters because no vertex theorem applies at the paper’s edge, and because the outline of a folded object is mostly crease rather than raw edge — two facts about boundaries that a population of 86 per cent boundary vertices makes load-bearing. A population that is 86 per cent boundary is a population in which the conditions this whole subject is built on are evaluated at one vertex in seven — so a statement of the form every vertex of these patterns satisfies Kawasaki means something quite different there from what it means on a tessellation.

The sentence this one reaches

The rung below found a published sentence its measurement contradicted, and honesty requires the same treatment here.

Almost every crease pattern fails to fold flat is a sentence of exactly the kind this essay is about. It is measured over patterns drawn at random and it is correct there — a pattern whose creases are drawn without regard to the conditions fails at its first vertex. What the sentence does not say, and what a reader takes from it, is that valid patterns are rare among the patterns anybody would draw.

Measured over the four constructions above, every pattern in every population satisfies every condition at every interior vertex, because each construction was built to produce patterns that do. The rarity is a property of the drawing-at-random population and of nothing else, and the populations a designer, a tessellator or an engineer works in are populations of valid patterns exclusively.

Both statements are true and they are about different sets. The general one — almost every pattern fails — is the one this collection has repeated, and it is the one about the population nobody works in.

What each refusal spends, in the units it spends itThe work each of the five refusals does on six crease patterns, counted in the operations each test performs rather than in seconds. Four of them are polynomial in the size of the drawing; the search over orderings is refused outright on half of these.the four cheap tests are polynomial in the drawing; the fifth is notreading across a row is one pattern put to all fivecrease pairsverticespanelscreasessearch nodesthe square twist6649127,565the Miura fold703152438refusedthe waterbomb sheet2,850255276refusedthe Yoshimura3,655226586refuseda square patch3,486364984refuseda rhombille patch39,621126157282refuseda refused search is a pattern about which the expensive test says nothing at all, at full price
Fig. 5 The sentence this one reaches, priced. Each population is a construction with a cost, and what a rarity figure reports is a fact about the space the construction is drawing from rather than about any pattern in it.

Which theorem was checked, and how

Every population is exhausted, not sampled. Each construction is run to completion — every tiling, every angle, every seed, every outline — and every row reports how many patterns it is over. A construction that produced four patterns says four.

The populations are checked to be disjoint — no two of them produce patterns agreeing in both interior-vertex count and pile depth — before anything is compared, on the measurements themselves rather than on how they were built.

Every quantity is measured by code written for another purpose. The boundary share comes from the machinery that decides a vertex at the paper’s edge, the pile from the layer map, the shrink from the folded panels, and the crease density from the pattern’s own coordinates. None of them was written for this comparison, which is what makes the comparison a comparison.

A pattern too large to answer a question leaves the column empty rather than being guessed at, and the count of patterns each average is over is reported beside it — a column that quietly averages over whichever patterns happened to be small is a column about size.

Four ways of making a crease pattern, four answersFour constructions that each produce crease patterns satisfying every vertex condition, asked the same four questions: what share of each pattern's vertices lie on the edge of the paper, how many layers deep the folded state gets at its worst point, how many times smaller the folded footprint is than the sheet, and how much crease length each unit of paper carries. They are not four samples of one population — no two of them produce the same patterns — and they disagree by factors rather than by margins.each row is an exhaustive count over the patterns that construction producedon the edgedeepest piletimes smallercrease densitythe printed patterns8 patterns62%19.415.5×7.9twist tessellations12 patterns52%9.52.7×12.9quadrilateral meshes6 patterns67%8.74.8×5.2fold-and-cut patterns7 patterns86%10.41.2×2.2
Fig. 6 The same table at a coarser sampling grid. The numbers move in the third digit and the ordering does not, which is the check that the comparison is about the patterns rather than about the resolution the folded states were measured at.

Where the model stops

Four constructions is four. A fifth — patterns taken from published designs — would be different again and is the one a reader might most want; assembling it means reading crease patterns out of other people’s work, which this collection does not do.

Nothing here is a claim about sampling. There is no estimator, no error bar, no confidence statement and no distributional assumption anywhere in it. Every number is an exhaustive count over a stated finite population, and the argument is a comparison between populations that were each written down on purpose.

The averages hide their own spreads. Twelve twist tessellations at three turn angles differ among themselves — the shrink runs from 1.4 to 5.0 within that column alone — so a between-population ratio of twelve is not twelve times the within-population variation, and no attempt is made here to separate the two.

And no population is the right one. There is no such thing. What there is, is a requirement to say which one a number came from — the same conclusion the vertex version of this question reached, arrived at independently on objects one dimension larger.

Whether the gaps are larger than the spreads

The comparison is worth nothing unless the differences between populations are larger than the differences inside them, and that is checkable from the numbers already reported rather than needing a variance decomposition nobody would trust on twelve patterns.

Take the column where the populations disagree most. The shrink runs from 1.3 for fold-and-cut patterns to 15.5 for the shelf, a factor of twelve. Within the twist population alone it runs from 1.4 to 5.0, a factor of three and a half.

Those are not the same kind of number — one is a ratio between means, the other a range over individuals — so the useful comparison is the direct one. The shelf’s mean of 15.5 lies outside the twists’ entire observed range, three times above its top. It is not that the two populations have different averages with overlapping spreads; the least dramatic pattern the shelf could contain and still average 15.5 is well above the most dramatic twist tessellation the construction produced at any of its settings.

That is the shape the argument needs. Two of the four columns could be reconciled by saying the same thing about different individuals; this pair cannot, and the disagreement is between the populations rather than inside them.

The same reading does not hold everywhere and the table says so. Boundary share runs from 52 to 86 per cent, a factor of 1.7, and no within-population range is reported for it — so on that question the honest position is that the populations differ and by how much relative to their own spreads is not established here.

The population nobody measured

The fifth population — patterns taken from published designs — is named and set aside, and something can be said about where it would fall without reading a single one of them.

A published complex design is box-pleated, which fixes three of the four columns before any measurement. Its creases run along a grid and its diagonals, so every one of its interior vertices carries at least one pair of equal sectors — a forty-five degree grid admits no vertex free of them at any degree. Its crease density is very high, because the whole discipline exists to put hundreds of folds on one sheet. And its boundary share is low, because a design that uses its paper has most of its vertices in the middle of the sheet rather than at the rim.

Put those together and the fifth population is predictably the extreme one on the axis that decides what a pattern costs to check. The four measured here are extreme on shrink, on crease density and on boundary share in various directions, and none of them is systematically loaded with coincident sectors: a solved quadrilateral mesh has no two vertices alike by construction, and a mesh built that way has no ties at all.

So the population a reader most wants a number for is the one furthest from every population a number exists for, and it is furthest in exactly the direction that makes a general statement about difficulty go wrong. That is worth stating as a warning rather than as a result, because it is an inference from two constructions rather than a measurement of a third — and it is the strongest reason to assemble the fifth population that this essay can offer.

What the picture cannot show

A table of four rows cannot show that the rows are measurements of different objects rather than different measurements of one. That is exactly the reading it invites, and it is the wrong one — there is nothing the four are estimates of.

Nor can any figure show a population. Each row stands for five to twelve patterns and the figure has room for none of them, so the object being compared is invisible in every figure of this essay. That is unavoidable, and it is why the constructions are described in words.

Which refusal fires firstFive ways of saying no to a crease pattern, in order of what they cost, with every member of the four test populations recorded against the first one that refuses it. The cheapest test catches the most, the two in the middle catch nothing here because the cheapest had already caught their cases, and the most expensive is the only one that reaches the rest.the bar is how many of the 38 patterns each refusal is the first to catchtwo creases cross5one sweep over pairs of creasesa vertex condition fails0one pass over the verticesthe panels do not place0one walk over the panelsthe letters force a loop1one pass over the crease listno ordering exists6every ordering of the panels26 of the 38 are refused by none of these and are folded, undecided, or waiting on a search too large to run
Fig. 7 What the picture cannot show, and the only thing that makes the four comparable: every member of every population satisfies every condition at every vertex. The ladder sorts them by what catches them afterwards, which is where they differ.

The idealisation, named

Every pattern in every population is zero-thickness, unstretchable and creased along lines. The measurements are therefore about patterns as drawn, which is the right object for a comparison of constructions and is not the object a folder holds.

One consequence is worth naming because it affects a column. The shrink is measured at the fully folded state, which requires the paper to close completely; a real Yoshimura at sixty layers does not, so the 60× shrink is an upper bound that the shelf’s own patterns approach and do not reach.

The generalisation

A measurement over instances is a measurement over the process that produced the instances, and when the instances are structured objects the process has more to decide than a sampler of numbers does.

The version of that worth carrying is about how a subject accumulates its general statements. Nobody in this collection ever decided that its sentences about crease patterns would be sentences about eight printed ones. The patterns got printed for good reasons, the numbers got measured because the patterns were there, and the sentences got written in the ordinary way — and the population arrived by accretion rather than by choice, which is how most populations arrive.

The second half is the practical one. A curated population is curated for something, and the something is usually correlated with what will be measured on it. The shelf here was assembled for pedagogical value and is extreme on shrink; a library of test cases assembled for difficulty will be extreme on difficulty; a benchmark assembled from published examples will be extreme on whatever gets published. None of those is a mistake in the curation, and each of them is a reason to say which population a number came from.

How far apart the four constructions areThe ratio between the largest and smallest answer, for each question asked of the four ways of making a crease pattern. A ratio near one would mean four measurements of one thing. None of them is near one.the ratio of the highest row to the lowest, per questionhow much of the pattern is edge1.7× — cut against twistshow deep the folded stack goes2.2× — shelf against mesheshow much smaller the folded state is12.5× — shelf against cuthow much creasing per unit of paper5.8× — twists against cut
Fig. 8 The disagreement again, measured at a different resolution. What it says is not that any row is wrong: it is that a number quoted without a population is a number about a construction.

Where the ladder goes next

Two rungs follow directly.

The first is the one this rung makes possible: going back through the general sentences in this collection and marking which population each came from — starting with the ones about how much a corrugation costs and how deep a folded stack goes, both of which are measured over the shelf. That is bookkeeping rather than discovery, and it is the kind of bookkeeping that changes what a reader can trust.

The second is the question of cost, which is the other thing a population decides. If four ways of producing a pattern disagree by factors about what a pattern is like, they will disagree about what deciding one costs — and where that disagreement comes from turns out to be a single feature of the instances rather than their size.

Where this goes next

populationpassfoldhave a gapbranchcut twice at random51 vertices, 13 kinds9.68.020%0%whole multiples of 45°60 vertices, 1 kinds30.019.3100%69%whole multiples of 30°60 vertices, 13 kinds19.713.177%26%a named vertex, jittered60 vertices, 4 kinds8.08.00%0% Which vertices are the random ones Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general. -3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding Almost every pattern fails Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured. 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 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.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 11 that link here.

The objects this essay names

Each one links to every other essay that touches it.

Boundary vertexThe fold-and-cut theoremMeasurementQuadrilateral meshTessellationTypical instances