Origami & Folding
what one sheet can do
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.
15 min read 6 figures Paper is not idealThe pattern is the object

A crease pattern is an exact statement about a mathematical object: a surface of zero thickness, inextensible, with creases that are lines, which holds whatever configuration it is put into.

Nobody has ever folded one.

Four things the model assumesThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded
Fig. 1 The four idealisations every crease pattern rests on, and what each one costs. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first.

The gap between the object the theorems describe and the object on the table is where most of the practical difficulty in this subject lives.

No thickness

The largest of the four, and the one with a literature.

A sheet has a depth, and folding it has to get that depth around a corner. One fold costs about one thickness of positional error; layers multiply it; and a complex design accumulates thirty or forty layers where the flaps meet.

Forty layers of ordinary paper is four millimetres, which for a model whose features are five millimetres wide means the geometry has stopped describing the object.

The responses are familiar to anybody who folds: very thin paper, laminated foil, and very large squares — all of which reduce the ratio of thickness to feature size without removing it. In engineering the responses are offset panels and membrane hinges, and they constitute a design discipline of their own.

What a flat sheet can becomeA sheet that cannot stretch can only take shapes with zero Gaussian curvature everywhere — cylinders, cones and the general developable surface. A sphere or a saddle curves in two directions at once and no amount of folding reaches one, which is why a paper globe has to be made of gores.cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible
Fig. 2 What inextensibility buys. The classification of reachable shapes is a consequence of the sheet not stretching, and wet-folding is the deliberate exception.

No stretch

The inextensibility assumption is the one that makes everything a flat sheet can become a finite list, and it is the most nearly true of the four.

Paper stretches by perhaps a tenth of a per cent under normal folding forces, which is genuinely negligible for the geometry. That is why the developability arguments work so well.

It becomes false deliberately in wet-folding, where dampening lets the fibres slide, and the sheet takes doubly-curved shapes that no dry fold could produce. Yoshizawa developed the technique in the mid-twentieth century and it is why his models look organic.

So this idealisation is excellent until somebody chooses to violate it, at which point a different set of shapes becomes available and none of the geometry applies.

A curved creaseOne curved fold in a flat sheet. The paper either side cannot stretch, so it is forced into a developable surface, and the sheet arrives at a doubly-curved shape that nobody put there. The flat-folding theorems say nothing about this case — they are statements about straight creases meeting at a point.the patternconcentric arcs, alternatingwhat the sheet doesa shape with no flat state at allthe curve is in the crease; the saddle is the paper refusing to stretch
Fig. 3 A crease drawn as a line. Real ones have a radius, which consumes angle and sets a floor on how small a feature the material will take.

Creases are lines

A crease is drawn as a line and is a curved region of finite radius.

The radius depends on the paper’s thickness and stiffness — roughly proportional to thickness for a given material — and it has two consequences.

It consumes angle: the two panels either side of a crease are not at 180° when fully folded but a little less, because the material has to turn through the radius. For a single fold this is invisible; for a stack it accumulates.

It limits how sharp a fold can be. Force a crease tighter than the material’s radius and the outer fibres break. That is the tearing anybody who has over-creased thick card has seen, and it sets a floor on feature size for a given paper.

There is a related effect that is not geometric: a crease weakens the sheet along its line, so a pattern with many creases in one region has a mechanical weak point there, which is why complex models tear where the paper is busiest.

The Miura foldA grid of identical parallelograms. The assignment is the whole trick: the horizontal creases alternate by row, and the vertical ones change assignment every time they cross a row, so each vertex ends up three of one and one of the other rather than two and two.at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 39.3 sheet-widths of foldingmountainvalleyraw edge
Miura fold — sheet 170×106.83 mm — 22 mountain, 16 valley, 1049.4 mm of crease
Fig. 4 A pattern with a great many creases. Every one of them relaxes, and a tessellation folded and set down opens measurably over the following hours.

Perfect memory

The fourth is the one folders complain about most and the theory ignores completely.

Paper relaxes. A model folded and set down opens slightly over the next few minutes, then continues opening slowly for days. The fibres were displaced elastically and they recover; the crease holds a permanent set, but not a complete one.

Every folder’s countermeasures are responses to this: creasing hard, creasing repeatedly, wetting the paper, using a bone folder, gluing the model, or accepting that it will need reshaping before being photographed.

The theory has nothing to say about it because it is a materials question rather than a geometric one, and it is a large part of why a folded model and its crease pattern are different objects.

The idealisations are not independent

Treating the four as a list suggests they can be relaxed one at a time. They interact, and the interactions are where the real behaviour lives.

Thickness and crease radius are the same phenomenon seen twice: a thicker sheet has a larger crease radius, so a model that suffers from layer accumulation also suffers from angle loss at every fold, and the two compound.

Stretch and memory interact through the fibres. A sheet that has been stretched — wet-folded — holds its shape far better than one that has not, because the fibres have moved rather than merely bent. So deliberately breaking one idealisation substantially fixes another.

Thickness and stretch trade off in the choice of material. Thin papers are thin because they have fewer fibres, which makes them weaker and more prone to tearing at a crease, so reducing the thickness problem worsens the crease-damage problem.

A designer choosing paper is navigating all of that at once, mostly by experience, and the theory offers no help because it has assumed all four away.

The sheet has a thicknessAn ideal fold brings two panels into contact along a line. A real one has to get a finite thickness around a corner, so the panels no longer meet where the pattern says — and every layer added makes the error worse. Accommodating it is the central problem of building origami out of anything.zero thicknesspanels meet exactlyfour layers of real materialeach fold has to clear the ones belowhinge offset to the surfacethe panel rotates about the right linea crease pattern describes a surface with no thicknessand everything anybody builds has some
Fig. 5 The first idealisation, in cross-section. The crease radius visible here is the third one, and the reason the panels do not meet is both of them together.
The preliminary baseBoth diagonals and both midlines of a square, with the assignment that folds flat. Eight creases meet at the centre in equal sectors, so Kawasaki is satisfied by any assignment and Maekawa is the binding condition — five of one and three of the other, never four and four.at the centre8 creases, all sectors 45°3 mountain, 5 valleydifference 2 — Maekawa holdsfour and four would fail,which is what most people drawfold every line, then collapse — the four corners meetmountainvalleyraw edge
Preliminary base — sheet 150×150 mm — 3 mountain, 5 valley, 724.26 mm of crease
Fig. 6 A pattern where none of the four matters. For a simple model in ordinary paper the idealisation is an excellent description of the object.

Which one fails first

The four do not fail equally, and knowing which binds is useful.

For a simple model in ordinary paper, none of them matters. A crane’s geometry is described perfectly by the idealisation.

For a complex model, thickness binds first and binds hard. That is why the response is thinner paper and bigger squares rather than anything geometric.

For a curved or organic model, inextensibility binds, and the response is to break it with water.

For an engineered structure, thickness binds again and there is no option of thinner material, so the geometry has to be modified instead.

For a tessellation, thickness multiplies by the row count, and a large Miura sheet is limited by how thick its packed state is allowed to be.

The idealisations are what makes it a subject

It is worth ending on the other side of this, because a list of things that are false reads as a complaint and is not.

The idealisations are what make the theorems possible. Kawasaki and Maekawa are exact because the sheet is exact. The circle-packing argument works because the paper does not stretch. Developability classifies the reachable shapes because the surface is smooth and inextensible.

Take any of them away and the clean results dissolve into approximations. The right way to read the list is not that the theory is wrong, but that it describes a limit which real paper approaches closely enough to be useful, and departs from in ways that are themselves worth studying.

Every one of the departures has a field attached: thickness accommodation, wet-folding, crease mechanics, paper science. The idealisation is what makes it possible to say where they begin.

What a simulation would need

A useful way to see how much is being assumed: consider what it would take to simulate a real fold.

The sheet would be an elastic-plastic shell with finite thickness, orthotropic because paper has a grain direction. Creases would be regions of reduced stiffness and permanent curvature, with a radius depending on the local thickness and the force applied.

Contact between layers would need friction, and the layer count at a point would change during the motion. Self-contact would have to be detected continuously, which for a model with forty layers is the dominant cost.

And the material would need a time-dependent response, because paper relaxes, so the answer at the moment of folding differs from the answer an hour later.

That is a serious computation, and simulations of this kind exist for single folds and small patterns. Nothing simulates a complex model, and the crease-pattern geometry remains the only practical description — which is a good argument for the idealisations rather than against them.

The theory earns the simplification

A last point in defence of a page listing four falsehoods.

The idealised sheet is not a convenient fiction; it is a limit that real paper approaches closely enough for the predictions to hold. A crane folded from the exact geometry comes out looking like a crane. A Miura pattern folded from the exact geometry collapses exactly as the kinematics say. A trisection construction lands where the arithmetic puts it, to within the width of a pencil line.

That is a strong record. The idealisations are wrong in the way that frictionless planes and point masses are wrong — false as descriptions and excellent as models, with well-understood corrections where they matter.

The reason to enumerate them is not scepticism about the theory. It is that knowing exactly which assumption fails, and where, is what lets somebody tell the difference between a design that will not work and a design that needs thinner paper.

Where the model stops

Four is a convenient number. There are others — that creases are perfectly straight, that the sheet is homogeneous, that folding is quasi-static — and the four chosen here are the ones that bite in ordinary practice.

Nothing is quantified. The figure states each idealisation and its consequence in words. Actual numbers for crease radius, relaxation rate and stretch are material-dependent and none is given.

Paper is not one material. Kami, washi, tissue-foil, glassine and card behave completely differently, and a statement about “paper” is an average over a large range.

No mechanics. Everything here is descriptive. The actual behaviour of a crease is an elastoplastic bending problem with fibre damage, and that is a serious subject this does not enter.

What a good folder does

The idealisations are false and folders work around them constantly, mostly without articulating it, and the workarounds are informative about which assumption is binding.

Against thickness: thinner paper, larger squares, and shaping the model so the thick regions are hidden inside. Experienced folders also plan the layer distribution, choosing which flap goes above which so the bulk ends up where it will not show.

Against crease radius: a bone folder, and creasing in the right direction. A crease made by folding and pressing is sharper than one made by scoring, and the difference is visible in the finished model.

Against memory: creasing repeatedly, wetting locally, and in the limit gluing. Competition models are frequently glued, which the tradition regards as a compromise and everybody does anyway.

Against stretch: exploiting it. Wet-folding is the deliberate version, and even in dry folding a small amount of stretch is what lets a slightly-wrong pattern go together.

None of that is in any book of theory, and all of it is what the difference between a good fold and a poor one consists of.

The idealisation is a choice, not an oversight

A last observation, because a list of false assumptions can read as a criticism of the theory and should not.

Every one of the four could be relaxed. There are models of crease mechanics with finite radius, shell simulations with thickness, and elastoplastic treatments of paper. They exist, they are used where they are needed, and none of them has produced anything like Kawasaki’s theorem.

The clean results come from the clean assumptions. A theory of folding that included thickness from the start would have no exact theorems in it at all, because the conditions would be inequalities depending on material parameters rather than equations in angles.

So the idealisation is doing what idealisations do: buying exactness at the cost of scope. The right response is not to abandon it but to know its range — which is what this page is for.

Which assumption a discipline drops

A useful way to organise the neighbouring fields: each is characterised by which idealisation it abandons first.

Computational origami keeps all four and gets the theorems — flat-foldability, the design algorithms, the complexity results.

Engineering origami drops zero thickness and keeps the rest, which is why its literature is dominated by hinges and accommodation techniques rather than by geometry.

Curved-crease work keeps inextensibility and drops the assumption that creases are straight, which trades the combinatorial theory for differential geometry.

Wet-folding and paper science drop inextensibility, which trades the classification of reachable shapes for a much larger space and no theorems at all.

Crumpling physics drops the assumption that the folding is deliberate, and studies the statistics of a disordered crease network instead.

Five subjects, five different assumptions abandoned, each with its own methods and almost no shared results. The idealised sheet is the point they all radiate from.

The correction terms have their own fields

A final observation about the shape of the subject.

Each of the four idealisations, when relaxed, opens a research area with its own methods and its own results. Thickness gives engineering origami and accommodation techniques. Stretch gives wet-folding and paper mechanics. Crease radius gives the mechanics of a fold. Memory gives creep and relaxation in fibrous materials.

None of those areas produces theorems like Kawasaki’s, and that is not a criticism. They produce measurements, models with parameters and design rules — which is what an engineering subject produces, and what a geometry does not.

The division of labour is clean. The idealised theory says what is possible in principle and does so exactly. The correction terms say what is possible with this paper at this size, and do so approximately. A designer needs both, and confusing them is the source of most of the disappointment when a beautiful pattern will not fold.

The test that settles it

There is a simple way to find out which idealisation is binding on a particular model, and it is worth knowing because the answer determines what to change.

Fold it. If it comes out and the layers are bulky, thickness is binding — use thinner paper or a larger square. If it comes out and springs open, memory is binding — crease harder, or wet-fold, or glue. If the creases tear, crease radius is binding — use a softer or thinner paper. If the flaps do not reach, the reference construction was inexact and nothing about the material is at fault.

Four symptoms, four causes, four different remedies, and they are routinely confused. A model that will not stay shut gets blamed on the design when it is a memory problem; a model whose flaps are short gets blamed on the paper when it is an arithmetic problem.

That diagnostic is most of what the list on this page is for.

The ladder from here

Later rungs: crease mechanics and the radius. Fibre damage and tearing. Relaxation, measured. Paper types and their properties. Wet-folding and fibre mobility. Thickness accommodation in engineering. Laminates and foil-backing. The mechanics of a folded stack. Simulation of real sheets. And the question of how far the ideal theory can be trusted for a given material and feature size, which is the question every designer is implicitly answering.

The theorems in this subject are about a sheet with no thickness that does not stretch, creases along lines and never forgets. That object is a good enough description of a piece of paper that a crane folded from the geometry comes out looking like a crane, which is a more remarkable fact than it usually gets credit for.

Where this goes next

zero thicknesspanels meet exactlyfour layers of real materialeach fold has to clear the ones belowhinge offset to the surfacethe panel rotates about the right linea crease pattern describes a surface with no thicknessand everything anybody builds has some The sheet has a thickness Every crease pattern describes a surface with no thickness. Everything anybody builds has some, and getting it around a corner is the central problem of turning origami into hardware. cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible What a flat sheet can become A sheet that cannot stretch can only take shapes that are flat in one direction at every point. Cylinders and cones are reachable; a sphere is not, and no amount of folding will get one. the patternconcentric arcs, alternatingwhat the sheet doesa shape with no flat state at allthe curve is in the crease; the saddle is the paper refusing to stretch A crease that curves Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.