The condition that is not flat-foldability

The family the Miura belongs to answered one question and left a larger one standing. Every mesh in it has a crease family running straight through each vertex — draw a fan of lines, draw rows that reflect in each line they cross — and that assumption is what turns the rigid-folding condition into a statement about a table of cosines having rank one. It was named at the time as an assumption rather than as a fact, and the meshes in which both families bend are the larger class.

This is that class, and reaching it changes the answer rather than extending it.

Developability is free, and Kawasaki costs one direction #

Draw any quadrilateral mesh in the plane — a grid of points, joined into rows and columns, with the points wherever they like. Every interior vertex is developable immediately, because the four sectors round a point of a plane sum to a full turn whatever the drawing. So developability is not a condition on a planar mesh at all; it is a fact about planes.

Kawasaki is a condition, and it is a sharp one. At an interior vertex the four sectors, taken anticlockwise, are east-to-north, north-to-west, west-to-south and south-to-east, and Kawasaki asks the first and third to be supplementary. Three of those directions are fixed once the row through the vertex and the row below it are drawn. The fourth is therefore determined: the crease going north has no freedom of direction at all.

What it does have is freedom of length. The vertex to the north may sit anywhere along that ray, and the mesh is flat-foldable at every vertex whatever distance is chosen. So the flat-foldable quadrilateral meshes on a given first row form a family with one free number per vertex, and every member of that family passes every condition this site has ever checked.

The family, and the one member of it that folds #

Build that family through the Miura. Take the ordinary Miura grid; take one interior vertex; slide it along the ray Kawasaki forces at the vertex below it. That preserves the condition at the vertex below by construction. It disturbs the sectors at two other vertices, and both of those are repaired by turning their northward creases, which land on the sheet’s edge and are therefore free.

The result is a one-parameter family, and the worst Kawasaki residual anywhere in it is zero — not small, zero, to the last bit a double holds — at every setting of the parameter.

Only one member folds. At the Miura the four vertices round a face agree about their shared crease to about 6 × 10⁻¹⁴ radians, which is arithmetic noise. Five per cent of a panel away they disagree by 0.041 radians; at fifteen per cent by 0.124; at forty per cent by 0.356. The departure is first order in the displacement, which is the same shape of failure an undirected perturbation produces — arriving now for the one direction the subject’s own tests cannot see.

That is the finding. The flat-folding conditions are not what buys the motion. They are necessary — a mesh that fails them at a vertex has that vertex to answer for before anything else — and they are a long way from sufficient.

It is worth being precise about how strong the statement is. The family is one-dimensional and the meshes in it are three-by-three, so what has been shown is that in this family the rigid-foldable meshes are isolated points rather than an interval. A stronger claim — that flat-foldability never implies rigid-foldability on any quadrilateral mesh — would need a different argument, and is not made. What is made is the claim that the implication fails, which is enough to stop anybody reading the flat conditions as the rigid ones.

What the extra condition is #

It is a loop.

A degree-four developable vertex is a spherical four-bar linkage. Fix the fold angle at one of its creases and the other three follow, up to a discrete choice between two configurations. That is a local fact and it holds at every vertex of every mesh here, which is why no single vertex is ever the obstruction.

The four vertices round a quadrilateral face, however, share four creases in a cycle. Fix the fold angle on one of them, solve the first vertex, carry the answer to the second along their shared crease, solve that, and continue. After four solves the walk is back at the crease it started from, and the number it brings back need not be the number it set out with. That mismatch is the obstruction, it is measured in radians, and it is one equation per face.

Nothing at a vertex can see it. Every vertex in the family is developable, flat-foldable, and perfectly capable of moving on its own; the condition is about how four of them agree, and it lives on a loop rather than at a point. It is the rigid-folding counterpart of the sentence the flat-folding half of this site has been repeating since its second essay — and the counterpart is sharper, because here the local conditions are not merely insufficient for the global property, they are insufficient at the very first loop.

There is one more way to count the same thing. A mesh with V interior vertices has, in principle, one free fold angle and three closure equations per vertex, so 3V equations in however many creases it has. On a grid the creases number roughly 2V, so the system is over-determined by about V — and an over-determined system with solutions is a coincidence unless something arranges it. Flat-foldability at every vertex is part of what arranges it, and the loop conditions are the rest.

The class is not empty #

A negative result about a family is worth much less than a negative result plus a member. One equation with one free length in it has roots, and they can be found.

Take a mesh built the general way — a zigzag first row, free directions off it, Kawasaki-forced directions after that, and arbitrary lengths — and adjust one of those lengths until the loop residual crosses zero. Bisect. What comes back is a mesh whose four vertices have sectors like 67.2°/62.0°/112.8°/118.0° and 125.0°/90.4°/55.0°/89.6°, no two the same, no crease running straight through anything, and a loop that closes to 10⁻¹³ radians across the whole motion. Three independent meshes, at three seeds, all found the same way.

So the answer to what else moves is: a set of positive codimension, cut out of the flat-foldable meshes by one equation per face, and its members are found by solving rather than by drawing. That is a different kind of answer from the rank-one condition, which describes a family in closed form. It is weaker, and it covers the case the closed form was derived under an assumption to avoid.

Which theorem was checked, and how #

Three separate machines had to agree before any of this was written down.

The vertex solver is the closure of four rotations composing to the identity — the same closure the straight-family case is solved with — solved by Newton from a fixed lattice of starting points rather than a random spread. That detail is not tidiness. With random seeds the search found both configurations at most vertices and one at a few; the walk round a face then lost whichever chain the missing configuration belonged to, and the residual it reported moved around as the mesh was perturbed for reasons that had nothing to do with the mesh. The residual only became a monotone function of the slide once the search became exhaustive.

The control is the Miura, which two entirely different routes on this site solve — a posed parameterisation and a rank-one table — and neither is used here. The Miura sits inside the slide family at one particular length, and the walk built for the general case has to close on it, which it does at 6 × 10⁻¹⁴.

The family is checked before the residual is believed. Every member is required to have a worst Kawasaki residual below 10⁻¹², and the slide is rejected if it does not — because a comparison between a mesh that folds and a mesh that has stopped being flat-foldable would be worthless, and it is exactly the mistake this construction is shaped to avoid.

Where the model stops #

The measurement is on one face. A larger mesh has one equation per quadrilateral face with four interior corners, so a four-by-four grid has four equations and tuning a single length does not satisfy them; a coordinate descent over the free lengths on such a mesh reached 0.038 radians and stopped, which is a great deal better than the 0.32 it started at and is not a solution.

That is not a limitation of the argument — the argument is about whether flat-foldability implies rigid-foldability, and one face settles it — but it is a real limitation of the construction. The general many-face solve is a system of equations in the free lengths, the lengths outnumber the equations, and whether solutions exist for large meshes is a question this essay does not answer. It is named here rather than left implied.

Two further idealisations. The panels are rigid, which is the definition of the question and not an approximation to paper: paper bends, and a sheet that has no rigid motion may fold perfectly well in the hand. And the panels have no thickness, so nothing here says whether a mesh that closes its loops can be built; that is a separate subject with its own arithmetic.

How large the solution set ought to be #

The closing section leaves the dimension open, and the naive count is worth doing, because it turns two loose ends of this essay into one statement and it changes what the stalled four-by-four descent means.

Take a grid of $m$ by $n$ quadrilateral panels. Its interior vertices number $(m-1)(n-1)$, and the construction above gives each of them one free length once the first row is drawn — so the flat-foldable family through a given first row has $(m-1)(n-1)$ parameters. Its loop equations sit on the faces with four interior corners, which are the panels off the boundary ring: $(m-2)(n-2)$ of them, one equation each.

Subtract, and if the equations are independent the rigid-foldable meshes inside that family form a set of dimension

$$(m-1)(n-1) - (m-2)(n-2) = m + n - 3.$$

The family grows with the area and the solution set grows with the perimeter. On a three-by-three the two numbers are four and three; on a ten-by-ten they are eighty-one and seventeen; on a hundred-by-hundred, nine thousand eight hundred and one against a hundred and ninety-seven. So a rigid-foldable quadrilateral mesh is not a rare member of the flat-foldable ones in any fixed proportion — it is a vanishingly thin subset, and thinner the larger the sheet.

That count also explains the construction it came from. A set whose dimension is $m + n - 3$ is a set whose members are fixed by data along two edges of the grid, which is exactly what determining each new vertex from its neighbours does: choose a first row and a first column and everything else follows. The counting and the construction are the same statement, and neither is usually written beside the other.

Which makes the stalled descent a fact about the method #

The four-by-four coordinate descent reached 0.038 radians and stopped, and this essay reports it as a real limitation without saying what kind. The count says which kind.

A four-by-four panel grid has nine interior vertices, so nine free lengths, and four faces carrying loop equations. Nine less four is five. If the equations are independent there is a five-parameter set of rigid-foldable meshes in there, and the descent’s failure to find one is a failure of the descent — a stall on a residual surface with narrow valleys, from a start a third of a radian away — rather than evidence that the set is empty.

It also puts the one-face result in its proper place. The slide family is a line, and a line through a four-dimensional family generically meets a three-dimensional subset in isolated points. So the rigid-foldable meshes are isolated in this family and the rigid-foldable meshes form a three-parameter set are the same finding seen along different cuts, and the first does not license the stronger reading it invites — that rigid-foldability is rare in the sense of being sporadic. It is rare in the sense of being thin, which is a different and much more ordinary thing for a solution set to be.

None of that is a proof, because the independence of the equations is exactly what is not established. What the arithmetic gives is the number to check against, which the essay was previously carrying only as a question.

What the picture cannot show #

The crease patterns in the first figure are flat drawings, and the difference between them is a single vertex moved by a few per cent. A reader cannot see which of the three folds by looking, and that is the point: there is nothing to see. Every visible property — the angles, the closure of each vertex, the alternating sums — is identical in all three, and the one that differs is a number produced by walking a loop.

Nor is the motion drawn. A figure of the folded mesh would be a picture of one instant of one member, and what is being claimed is about all instants of all members.

The generalisation #

Stated without any of the machinery: flat-foldability is a condition at points, and rigid-foldability is a condition on loops, and no amount of the first implies the second.

That sentence has a shape the rest of mathematics recognises. A quantity defined at every point of a space, consistent with its neighbours, and yet failing to assemble into a global object because of what happens round a closed path, is the ordinary situation whenever a local rule is integrated. The crease pattern’s faces are the loops; the fold angles are what is being carried round them; and the mismatch on return is what obstructs the assembly.

This site does not develop that reading and does not need to. Every argument above survives with the word deleted and the loop of four vertices left standing, and the number that matters — 0.82 radians of disagreement per panel length — is a measurement on a walk rather than a class in a cohomology.

Who found it, and when #

Rigid origami as a subject is Tomohiro Tachi’s more than anyone’s, and the generalised Miura-ori — developable quadrilateral meshes built to fold with one degree of freedom — is his construction from 2009. That construction produces meshes by determining each new vertex rather than by choosing it, and what is measured here is what the determination is doing: the flat-folding conditions fix a direction and leave a length, and the length is where the rigid condition lives.

The spherical four-bar reading of a degree-four vertex is much older and belongs to kinematics rather than to origami; it is the same linkage a Bennett mechanism is built from, and that boundary is somebody else’s — nothing here counts a degree of freedom or a mobility, and every number in the essay is a closure residual or a fold angle.

Where the ladder goes next #

The loop condition has an immediate consequence that runs the other way. If one crease’s fold angle determines the whole of a mesh that folds, then deciding one crease decides the sheet — and the flat-folding version of that experiment, run on a tessellation, settles almost nothing. The same experiment, two questions, opposite answers, and it is the next rung.

Left open, and named: the many-face solve, and whether the set of rigid-foldable quadrilateral meshes is a manifold of the dimension the equation count suggests.