Solved is not built

Solving every face at once found a developable quadrilateral mesh with no two vertices alike that folds rigidly, and a surface of them sixteen dimensions wide. That is a good answer to a question the rung before it left open, and it is worth asking immediately what it is worth to somebody who intends to make one.

The answer is a number in millimetres, and it is not a kind number.

Two ways of being right #

Error is folded too measured what happens to an angular error made at one crease as it propagates through the folds after it, and where an error goes took the sharper case: a displacement that leaves every flat-folding condition exactly satisfied and breaks the rigid folding anyway. Both are about an error in a pattern.

This is a question about two patterns, and about what makes each of them fold.

A Miura folds because its columns run straight through every vertex and its whole sheet repeats one vertex. Those are structural facts. Change the panel width, change the height, change the slant — the columns still run straight, the sheet still repeats one vertex, and what comes out is another Miura. There is no accuracy requirement at all, because there is no equation being satisfied to a tolerance; the property is combinatorial.

A solved general mesh folds because four numbers are zero. Move any of its twenty lengths and they stop being zero, at a rate the derivative can be asked for directly.

The rate, and the millimetres #

The derivative of the worst mismatch with respect to a length is 1.009 radians per unit length, where a panel is about one unit across. That is close enough to one to be worth stating as one: an error of a hundredth of a panel produces a mismatch of about a hundredth of a radian.

Turning that into millimetres needs the sheet’s size. The mesh spans about 8.85 of its own units, so on a sheet 150 mm across a unit is about 17 mm, and holding the closure inside 0.02 radians — a degree, roughly — needs every length correct to about a third of a millimetre.

The joint measurement is the one to take seriously, because a real sheet is not cut wrong in one place. Every length wrong by a random amount of the same size, twenty-four times over:

• 0.017 mm of error leaves 0.0021 radians of mismatch;
• 0.051 mm leaves 0.0058;
• 0.169 mm leaves 0.0199;
• 0.508 mm leaves no closure at all — the propagation cannot find a consistent set of fold angles anywhere.

That last row is not a large number that has been rounded up. It is the solver reporting that the mesh has stopped having a folded state: at half a millimetre of error on a 150 mm sheet, there is no assignment of fold angles that closes every loop, and the sheet is a rigid structure rather than a mechanism.

Read the other way round, that figure is the tolerance stated as a picture. The mesh on the left is not a mesh that was drawn badly; it is what the mesh on the right becomes when its lengths are wrong by a few per cent, and the two are indistinguishable to every test this site had before the previous rung.

Why the failure is total rather than gradual #

The four rows above are not a smooth curve with a last point that happens to be large. Between a fifth of a millimetre and a half, the mesh stops having a folded state, and the difference between “a mismatch of 0.02 radians” and “no closure at all” is a difference in kind.

The reason is the branch structure. A degree-four vertex has two configurations at any fold angle, and finding a consistent set for the whole sheet means choosing one at each vertex so that every shared crease agrees. As the mesh is perturbed, the configurations move; at some point the two configurations at some vertex merge and vanish, and past that the vertex has no solution at the driven angle at all. There is nothing left to be nearly right.

A maker’s reading of that is unusually clean: a mesh cut slightly wrong is a mechanism with a small error, and a mesh cut badly wrong is not a mechanism, and there is no intermediate régime where it is a bad mechanism.

The other mesh, at fifty times the error #

A Miura whose panel width, height and slant are each wrong by five per cent — a hundred times the tolerance the solved mesh demands — closes to 4 × 10⁻¹⁵ radians. At one per cent it is 5.6 × 10⁻¹⁵ and at a tenth of a per cent 2.2 × 10⁻¹⁴, which is to say that the number is machine noise at every size and the errors are not doing anything at all.

The comparison is therefore not “which mesh is more accurate”. It is that the two meshes’ reasons for folding are of different kinds, and only one of the two kinds can be destroyed by a ruler.

What a third of a millimetre means #

A third of a millimetre on a 150 mm sheet is a relative accuracy of about one part in five hundred, and it is worth putting beside things a reader can hold.

A sharp pencil line is about half a millimetre wide, so the tolerance is narrower than the mark used to draw the pattern. A domestic printer places ink to about a tenth of a millimetre, which is comfortable; a laser cutter is better still. A person folding by hand places a crease to somewhere between two and five tenths of a millimetre, which an earlier rung measured from the other direction when it found that the reference points three folds reach are closer together than a crease is wide.

So the honest summary is that a solved general mesh is machine work. It can be cut, and it cannot be folded by hand from a printed sheet, and the reason is not that the pattern is difficult but that the property being relied on is an equality.

That last figure heads off an obvious explanation and is worth keeping. It would be natural to assume the fragility comes from errors being multiplied as they propagate across the sheet, in the way a gearing at a vertex multiplies a fold angle. It does not: the worst amplification anywhere on this mesh is a factor of 1.3. The sensitivity is arithmetic — twenty lengths, each contributing directly, none of them amplified.

Which theorem was checked, and how #

The derivative is taken one-sidedly and that is not a shortcut. A mismatch is a size — the largest of several absolute values — so as a function of a length it has a corner at zero, and a central difference across a solution reads the two arms of a V and returns no slope at all. Reporting that as insensitivity is the mistake this measurement is one line away from making, and it made it once.

The joint perturbation is seeded, so the numbers are reproducible, and it is a relative error applied to every length rather than an absolute one applied to a few. That is the model of a badly cut sheet rather than of a single mistake, and it is the harder case: twenty independent errors do not cancel.

The Miura comparison is run on the parameters that define a Miura — width, height, slant — rather than on its lengths, because those are the numbers a maker would get wrong. Perturbing a Miura’s coordinates at random would produce a mesh that is not a Miura and not flat-foldable either, and the resulting failure would be a failure of Kawasaki rather than of the closure, which is a different claim.

Where the model stops #

Everything here is geometry. There is no material in it — no stiffness, no yield, no assumption about what a panel is made of — so “the closure is gone” means the rigid folding is gone and not that the object falls apart. A real sheet with a mismatch of 0.02 radians does not refuse to fold; it folds with its panels bending slightly, which is a perfectly good way for a manufactured thing to work and is what most deployable structures actually do.

That is worth stating plainly because it changes what the number means. The tolerance computed here is the accuracy needed for a mesh to be a mechanism in the idealised sense. A maker who is content with panels that flex has a much larger budget, and how much larger is a question about materials that this site does not answer and says so about.

The second limit is the size. Nine interior vertices is a small sheet, and there is no reason to expect the tolerance to be size-independent — a larger mesh has more conditions and more lengths, and whether errors accumulate or average is not settled here.

The third is that the comparison is between one solved mesh and one Miura. The solution surface is sixteen-dimensional and its members need not be equally sensitive; what is established is that this solved mesh is fragile and that the Miura’s robustness is structural rather than lucky, and the second half of that generalises where the first does not.

What the picture cannot show #

A bar of length 0.0021 radians and a bar of length 4 × 10⁻¹⁵ cannot be drawn on one scale, and the hero figure does not try — the Miura’s rows are drawn as a mark rather than as a bar, with the number beside them. That is an honest way to draw a comparison between something and nothing, and it is not a picture of a ratio.

The allowance figure has a related difficulty. It draws four measured points and a line between them, and the interesting behaviour — the closure vanishing entirely — is the point that is not on the plot, because there is no value to draw. An honest curve here has a right-hand end that stops rather than a value that grows. Nothing here shows a badly cut sheet. Every figure is of a residual, and the object a reader would want to see — a mesh whose panels do not quite meet — is not drawable, because the whole content of the failure is that no folded position exists to draw.

The same result from the other side #

There is a version of this that needs no arithmetic at all, and it is worth having because it says why the numbers had to come out this way.

Ask what set of meshes folds rigidly. Inside the parameterisation used here it is the zero set of four smooth functions on a twenty-dimensional space, so it is a surface of dimension sixteen — which is what the previous rung measured by taking the rank of the derivative. A surface of dimension sixteen in a space of dimension twenty has volume zero: a point chosen at random is not on it, and a point on it, moved at random, leaves it.

The Miura family is a surface too, of much smaller dimension, and the difference is what the perturbation respects. Perturbing a Miura’s parameters moves along the Miura family and stays on the solution surface, because the Miura family is contained in it. Perturbing a solved mesh’s lengths moves in an arbitrary direction, and sixteen of the twenty directions stay on the surface while four leave it — so a random move leaves it, with probability one.

That is the same statement as almost every pattern fails, which made it about flat-foldability at a vertex, arriving one level up and about a different property. It is becoming the characteristic shape of this subject: the interesting sets are thin, and everything about making things is about whether the thinness is visible from inside the family being worked in.

The generalisation #

The distinction this rung is about has a name outside origami: the difference between a property that holds generically within a family and one that holds on a variety of measure zero. A Miura’s rigid foldability is the first kind — perturb it inside the Miura family and it survives, because the family is closed under the perturbation. A solved mesh’s is the second — the solutions are a sixteen-dimensional surface inside a twenty-dimensional space, and a random perturbation leaves the surface immediately.

Both are exactly true. Only one is robustly true, and manufacture is a process that tests robustness rather than truth.

That gives a design rule with an unusual shape. When a mechanism has to be made, prefer the family with a structural reason over the one with a solved reason, even when the solved one is better — because the structural reason is the only one that survives the workshop. It also explains why the deployable structures that get built are Miuras and their relatives, which had looked like a failure of imagination and is a fact about tolerances.

Who found it, and when #

Tolerance in rigid origami is a live engineering subject — the literature on deployable structures is largely about accommodating thickness and clearance rather than about the closure conditions themselves — and the observation that a Miura is forgiving is folklore among people who have built one.

What is measured here is the contrast, and the contrast needs both halves: a general mesh that folds, which is what the previous rung produced, and the same experiment run on both. Neither half is difficult once the first exists, which is a reasonable description of why nobody appears to have done it.

What it would take to make one anyway #

The rule “prefer the structural family” is a good rule and it is not the only option, so it is worth setting down what the alternatives cost.

Cut it more accurately. A third of a millimetre is within reach of a laser cutter and comfortably within reach of photochemical etching, so the mesh is manufacturable by machine at the sizes considered here. What that buys is a mechanism that works as designed, and what it costs is that the sheet cannot be made any other way.

Let the panels flex. A mismatch of a hundredth of a radian distributed over nine vertices is a small amount of bending, and a sheet of anything real will absorb it. This is what deployable structures do in practice and it works; the price is that the object is no longer a rigid mechanism and its behaviour is a materials question rather than a geometric one.

Adjust after cutting. The solution surface is sixteen-dimensional, so a mesh cut slightly wrong is near a solution rather than far from one, and four adjustable lengths would in principle be enough to return to the surface. That is a real engineering option and it converts a tolerance problem into a mechanism with four screws in it.

Where the ladder goes next #

The obvious continuation is to look for solved meshes that are less sensitive than this one. The solution surface is sixteen-dimensional and nothing says its members are equally fragile; the condition number of the closure varies over it, and a search for the best-conditioned member would be a search for the general mesh most worth building.

There is also a measurement worth making that would cost almost nothing: the same tolerance for the tuned three-by-three mesh the earlier rung produced, which satisfies one condition rather than four. If sensitivity scales with the number of conditions, a smaller general mesh is proportionally easier to build, and a maker choosing a size would want to know it.

The other direction is the one the limits above point at: what a mismatch of a hundredth of a radian actually costs a sheet whose panels can flex. That question needs a material and belongs to a site that has one, and the geometric half of it — how much bending a given mismatch demands — is a rung this site could write.