Each one moves, and together they do not

Here is an experiment that costs nothing and has no analogue anywhere else on this site.

Take a Miura pattern — a grid of parallelogram panels with the vertical crease lines zigzagging left and right, the fold everybody has seen on a map. Nine panels, twelve creases between two panels each, four interior vertices. It folds: one freedom, at every size, and the essay about that scaling takes it to a hundred and forty-four panels.

Now move each of its four interior vertices by about a tenth of a panel, in a direction that has nothing to do with anything.

Nothing structural has changed. The panels are still quadrilaterals, the graph is identical, every interior vertex still has four creases meeting at it, and the boundary is where it was.

The result does not fold. Not to half a radian, not to a fiftieth of one.

Asked to fold its first crease to 0.02 radians — a degree and a bit — the best any solve can do leaves a residual of $5.9 \times 10^{-6}$. At 0.1 radians it is $6.2 \times 10^{-5}$; at half a radian $1.6 \times 10^{-3}$. The Miura pattern, asked the same questions, closes at $10^{-16}$ every time, in seven Newton iterations. No seed and no number of iterations moves the moved grid, because there is nothing to move it to.

Every vertex is fine #

The first thing to check is whether the individual vertices have been broken, and they have not — not in any sense, and not in the sense that is usually reached for first.

Every interior vertex of a flat crease pattern is developable: its sector angles come to a full turn. That is not a condition anybody imposes on a drawing. A vertex drawn on a flat sheet has sectors that come to a full turn because the sheet is flat and there is no way for it not to. All four of the moved vertices report a full turn to nine decimal places, with sector angles of 62.561°, 115.539°, 114.793° and 67.107° at the first of them and similarly unremarkable numbers at the rest.

So developability distinguishes nothing here, and it is worth saying plainly because it is the property most often quoted as though it did.

More to the point, each of those vertices folds on its own. Four creases through a point, at fixed arcs from one another, is a spherical four-bar — the object the spatial field built — and a spherical four-bar has one freedom and two branches. Cut any one of these vertices out of the sheet with a pair of scissors and it will fold, through a large range, exactly as the Miura’s vertices do.

Four mechanisms that each move, joined at their shared creases into an assembly that does not. On a chain that cannot happen: joining two mechanisms at a common link gives an assembly whose mobility is at worst the sum less the joint’s constraints, and the arithmetic is local. On a network the arithmetic is not local and the assembly is the object.

The four vertices, one at a time #

The claim that each vertex folds is worth doing rather than asserting, because it is the half of the experiment that carries the weight.

Each interior vertex’s four sector angles are read off the flat drawing and handed to the field’s vertex routine as a pattern in its own right — a fan of four sectors around one point, with four creases and nothing else. The four of them come out as follows: 62.561°, 115.539°, 114.793° and 67.107°; then 121.692°, 116.345°, 61.742° and 60.222°; then 112.203°, 67.424°, 61.305° and 119.068°; and 116.042°, 60.438°, 63.556° and 119.964°. Every set sums to 360.000000000°.

Each of them folds. Driven to a third of a radian on any of its four creases, each solves to a residual at the arithmetic’s floor, and each has the two branches a spherical four-bar has.

None of them is degenerate, none is near a limit, and none of them is the one that has gone wrong — because none of them has.

The same failure in bars #

Nothing about this is peculiar to folding, and the bar-and-joint version takes one paragraph.

A four-bar has one freedom. Two four-bars sharing a link have two, and the count says so. Add a third link between one coupler and the other and the count falls to one — and whether the assembly actually has one freedom or none now depends on the length of that third link, which is a number the count does not contain. Chosen at random it has none; chosen to satisfy one condition it has one; and the condition is a single equation because there is a single dependency to arrange.

That is the same statement as the sheet’s, with $(n-2)^2 = 1$. What makes the folded sheet the better example is only that its unit count can be raised until the number of conditions is a hundred, at which point the difference between arranging them and not arranging them stops being a matter of care and becomes a matter of construction.

Where the obstruction actually is #

The reason is not hidden and does not need any new machinery to see.

Each vertex contributes three equations to the assembly’s constraint system, one loop closure per interior vertex. The unknowns are the fold angles, one per crease. On a three-by-three sheet that is twelve unknowns and twelve equations, and the counted mobility is nought.

A crease between two interior vertices appears in two of those closures. That is the whole of the difference from a chain: there is no order in which the four vertices can be solved one at a time, because whichever one is solved first has already decided part of the input to the next, and going round the four of them the decisions have to come back to where they started.

The Miura pattern is arranged so that they do. The moved grid is not, and there is no smaller statement of what has gone wrong than that — no particular vertex is at fault, no particular crease is the problem, and moving any one vertex back does not fix it.

That count has a clean answer, worked out in the essay on what a pattern has to satisfy: the number of conditions on the vertex positions is exactly the number of dependencies among the constraints, which on an $n$ by $n$ sheet is $(n-2)^2$. One condition at three by three, four at four by four, nine at five, a hundred at twelve. The three-by-three grid has eight coordinates free to move and one condition on them, so the foldable patterns form a seven-dimensional surface in an eight-dimensional space — a set with no thickness, which is why a displacement in a direction chosen for no reason lands off it.

The residual is the measurement #

It is worth being careful about what “does not fold” means, because a solver that fails is not by itself evidence of anything.

The residual reported above is the norm of the vertex closures at the best fold angles the solve can find, with one crease held at the angle asked for and the other eleven free. It is a least-squares problem in eleven unknowns, and the number is the distance from the closest attainable configuration to a closed one. It is not a convergence failure; it is a minimum.

Two things about it are the evidence rather than the number itself.

It grows with the fold, monotonically, from $5.9 \times 10^{-6}$ at 0.02 radians to $1.6 \times 10^{-3}$ at half a radian. A solve that had merely got stuck would not do that.

And it is many orders above the floor the same code reaches on the Miura pattern, which is run through the same routine with the same tolerances and the same starting seed. That comparison is the whole of the claim: one pattern reaches $10^{-16}$ and the other stops at $10^{-5}$, and neither number means anything on its own.

The same discipline is what makes an approximate straight-line linkage’s error a measurement rather than a complaint about a solver.

Why the flat state cannot tell them apart #

There is one measurement that gives the same answer for both patterns, and it is the one somebody would naturally reach for first.

At the flat state every crease lies in the plane. So the three rows each vertex contributes to the Jacobian are three components of a vector that has no third component: the rows span two dimensions rather than three, and the rank falls by exactly one per interior vertex. On a three-by-three sheet that is a rank of 8 rather than 12, and a nullity of four.

The moved grid gives 8 and 4 as well. Identical rank, identical nullity, identical everything the first-order analysis can produce — on a pattern that has one freedom and a pattern that has none.

That is not a numerical accident and it is not a shortcoming of the routine. The flat state is where every branch of every folding meets, so the tangent space there is the union of all of them and belongs to none; and whether any of those four directions is the start of a motion is a second-order question that no rank answers. It is exactly the situation the two-bar framework makes in two lines, arriving in a field where it cannot be avoided.

So the honest instrument for this question is the one used above: push the pattern off the flat state and see whether anything closes.

What this does not say #

It does not say the moved grid is a badly drawn Miura. It is a perfectly good crease pattern; it can be creased, folded by hand, and made into a shape. What it cannot do is fold rigidly — with every panel staying flat and every crease staying a straight hinge. Real paper bends, and the difference between a pattern that folds rigidly and one that does not is a difference between mechanisms rather than between sheets of paper. Everything in this field is about the mechanism.

It does not say that only the Miura folds. The foldable set is a surface of dimension seven in the eight-dimensional space of three-by-three patterns, and there is a great deal on it besides the Miura. What it says is that the set has no thickness, so a pattern drawn without regard for the conditions is not on it.

And it does not say the assembly’s failure is anywhere in particular. There is a temptation to look for the vertex that has been broken, and there is not one. The compatibility condition is a statement about the four of them together, in the same way that a coupler curve is not a property of any one link. Move all four vertices in a compensating pattern and the sheet folds again; move one of them back to where it was and it still does not.

Where the compatibility conditions live #

One last thing worth locating, because it is easy to imagine the conditions as living on the creases and they do not.

The unknowns of a folded pattern are its fold angles, and the conditions above are not conditions on those. They are conditions on the flat drawing: on where the vertices are, which is to say on the panels’ shapes. A pattern either satisfies them or does not, before anything is folded, and folding is then either possible or not.

That is a different arrangement from anything else on this site. A four-bar’s lengths decide whether its crank turns and how far it goes, but every set of four lengths that can be assembled at all is a mechanism of some sort. Here the shapes decide whether there is a mechanism, full stop, and almost all of them decide that there is not.

The measure of how special that is comes out of the same arithmetic. Eight free coordinates and one condition on a three-by-three sheet; thirty-two and nine on a five-by-five; two hundred and forty-two coordinates and a hundred conditions on a twelve-by-twelve. In each case the foldable patterns are a surface of codimension $(n-2)^2$, and in each case the surface has no thickness at all.

The obstruction has no address, so the repair has no target #

The essay declines to locate the failure at any particular vertex, and that refusal has a practical consequence worth drawing out, because it decides what a designer can and cannot do with a pattern that does not fold.

The obstruction is a property of a matrix, not of a part. The assembly’s constraint system has more independent rows than the fold angles can satisfy, and a rank is a global quantity — every row participates and none of them is at fault. There is no vertex to blame and no crease to move, in the sense that moving any of them changes every closure the crease appears in.

That rules out the repair a designer would naturally attempt. Given a pattern that will not fold, the instinct is to find the offending vertex and adjust it — the same instinct that works on a linkage whose one bad link length can be measured and corrected. Here there is nothing to find. Adjusting one vertex changes three closures, moves the rank condition in a way that depends on all the others, and generally produces a different pattern that also does not fold.

So the design method has to run the other way, and it is the method the field’s own examples all use. Start inside a family known to fold and search within it. A Miura pattern is a parameterised family: choose the panel angles and the panel sizes, and every member of the family folds because the compatibility conditions were solved once, symbolically, by whoever constructed the family. A designer moving inside it is moving along the foldable surface rather than off it.

That is why deployable structures are so overwhelmingly built from repeated identical units. Repetition is not an aesthetic choice and it is not a manufacturing economy first; it is the only cheap way of satisfying a number of conditions that grows with the pattern while the number of free parameters does not. A pattern of one repeated vertex satisfies its conditions once and inherits them everywhere.

The alternative — fitting a network of all-different units numerically — is a real method and it is a different activity entirely. It means solving the compatibility conditions as a system, on a pattern with as many unknowns as vertices, and accepting that the answer is a set of measure zero found by a solve rather than a family that can be drawn. That is how a freeform folded surface is designed, and it is why such surfaces arrived long after the periodic ones.

What it costs a designer #

The practical shape of this is worth stating, because it is the reason deployable structures are made of a few repeated units rather than many different ones.

A network of $N$ units drawn freehand has to satisfy a number of conditions that grows like $N$, on parameters that also grow like $N$. Satisfying them by search is not an option and satisfying them by luck is not either. What is available is symmetry: build the assembly out of copies of one unit, arranged so that the conditions are all the same condition, and the count of independent conditions collapses to one or two.

That is exactly what a Miura pattern is, and exactly what a deployable ring’s kink angle is: one number, chosen once, that makes every one of the assembly’s repeated conditions the same condition and therefore satisfiable. It is why this field’s mechanisms all look like tessellations, and it is a design constraint of a kind the rest of this site never meets, because a chain has no conditions of this sort at all.

The alternative — a network of units that are all slightly different, each fitted to its neighbours — is the general problem, and it is a system of $(n-2)^2$ equations in the vertex positions with no structure to exploit. Solving it is possible and is somebody’s subject. What this field measures is what happens when it has not been solved, which is: nothing moves, and the drawing gives no sign.