Every twist writes an equilibrium

One number where the corners wanted four ended with the distances solved and unread. The construction now places a twist polygon by giving each of its sides its own distance from the vertex, the corner conditions fix those distances up to almost nothing, and the twelve linear images that had refused to fold all fold. What the solution is was left standing: a list of numbers that satisfies a linear system, which is a description of how they were obtained rather than of what they are.

They have an obvious reading and it turns out to be the right one. Each side of a twist polygon faces one edge of the tiling. Divide the side’s length by the edge’s length. The quotient is dimensionless, it is a property of the pair rather than of either, and the two conditions the construction has been checked against since it was first written say two things about it that together name a familiar object.

Two conditions, read as one #

The first condition is that the twist polygon closes. It is so obviously true of a polygon that it is never stated: a closed convex figure’s side vectors sum to zero. Each side of the twist faces an edge and is perpendicular to it turned by the common angle, so writing the side as its own length times a unit vector and pulling the common rotation out, the sum being zero says

$$\sum_i \frac{\ell_i}{L_i}\,\mathbf{e}_i = 0$$

where $\mathbf{e}_i$ is the vector from the vertex to its $i$-th neighbour, $L_i$ its length and $\ell_i$ the length of the side facing it. The quotients, used as weights on the edges, balance at the vertex.

The second condition is the matching across a pleat: the two sides facing each other are the same length. Since they face the same edge, they divide by the same $L$, so the two ends of an edge assign it the same weight. That is what makes the weights a function of the edge rather than of the corner.

Put together, the quotients are a single positive number on every edge of the tiling, balancing at every vertex. That object has a name in a different subject: it is an equilibrium stress, the assignment of tensions to the members of a pin-jointed framework that leaves every joint in equilibrium, and a planar graph carrying a positive one is what is called a spider web. Nothing in the construction was aiming at it. The closure is a fact about polygons; the matching is a fact about pleats; and the two of them, divided through by the edge lengths, are the definition.

Measured on the twenty tilings and images, the weights the construction writes balance at every vertex to 3.1 parts in $10^{16}$ and the two ends of every edge agree to 1.4 parts in $10^{14}$. Those are the accuracy of the solve rather than tolerances anybody chose, and they are the sense in which this is a measurement and not a rearrangement of symbols: the weights are computed from the drawn polygons, and the balance is read off them afterwards.

The distinction matters because the two conditions are not equally obvious. Closure is a triviality once the polygon is drawn — no convex figure fails to close — so the balance at a vertex is guaranteed by construction and reading it as an equilibrium adds nothing to the geometry. The matching is not: it is the condition the older rule had to propagate across the tiling and could fail, and the whole of the propagation that never had to work is a study of what that propagation does. Making the weights a function of the edge is therefore the load-bearing half, and it is the half that stops being a condition at all once a vertex has a distance per side.

Whose object this is #

The weights are not a new invention and it is worth saying whose they are, because the borrowing runs in an unexpected direction.

An equilibrium stress on a planar graph is the nineteenth-century graphic-statics object: assign a tension to every member of a pin-jointed frame such that the forces at each joint cancel, and the frame stands without any joint doing work. Maxwell’s reciprocal figures are built from exactly that data — a graph with a balancing stress has a second graph, the reciprocal, whose edges are the first’s turned a quarter turn and stretched by their own stresses, and whose existence is equivalent to the balance. A graph carrying a positive balance is what is called a spider web, after the picture: a web of threads in tension, pinned at its boundary, every interior knot at rest.

Robert Lang and Alex Bateman showed that every spider web has a flat twist tessellation, and it is their theorem closing the loops is not folding quoted from outside to establish that the sheared lattices carry twists which the construction there could not draw. Their route is the reciprocal figure: the twist at a vertex is built from the reciprocal’s cell, which has one side per incident edge of exactly the right length, and the matching across a pleat is then automatic because both ends of an edge read the same reciprocal segment.

What the measurement above adds is the other direction, and it is the useful one for anybody standing inside this subject. The theorem says a balance is enough. The quotients say a balance is already there in every twist tessellation this construction has ever produced, including the ones drawn before any of this was noticed — any tiling makes a twist’s patterns, the rhombille’s mismatched pair, and every pattern the turn was dialled through in fenced at both ends. None of those was built from a reciprocal figure. Each of them writes one anyway, because writing one is what closing a polygon and matching a pleat amount to.

What that makes the existence question #

The construction’s way of asking whether a tiling carries a twist tessellation has always been to build one and check it. That is a fine method and it has one bad property: a failure is ambiguous. Closing the loops is not folding drew twelve patterns that failed at every vertex and concluded, wrongly, that something about those tilings was missing. The failure was about the construction.

Reading the sides as weights removes the ambiguity, because it turns the question into one about the tiling alone. A tiling carries a twist tessellation of this kind only if it carries a positive set of edge weights that balance at every vertex, and whether it does is a question with no drawing in it. It is a feasibility question about a linear system with a positivity constraint, which is exactly the shape a linear program answers: maximise the smallest weight subject to balance at every class of vertex and a fixed total, and read the answer’s status. An infeasible program is a proof that no such weights exist; an optimal one hands back the weights.

Asking it of the five tilings and their images returns the same answer everywhere: every one of the twenty carries a positive balancing set, and on four of the five the smallest weight can be made equal to the mean. The elongated triangular tiling is the exception and its number is 0.8453 — not a near miss but a genuine inequality, since its vertex has right angles beside sixty-degree ones and no balance can treat all four classes of edge alike.

The census is also a demonstration that the question was the right size. Five tilings and three maps is twenty drawings under the old method, each of them built at four turns and checked at every vertex — and twenty linear programs of at most four unknowns under this one, each returning a proof rather than an instance.

The test that could not have worked #

The four columns of that table are identical, and they are identical for a reason worth stating carefully, because it disposes of the whole of the earlier difficulty.

A linear map carries a balancing set of weights to itself. If $\sum_i \omega_i \mathbf{e}_i = 0$ at a vertex and every edge vector is replaced by $A\mathbf{e}_i$, then the new sum is $A\sum_i \omega_i \mathbf{e}_i = A\mathbf{0} = \mathbf{0}$: the same weights balance the image. The property is not merely preserved, it is preserved with the same numbers, which the table shows to the last digit reported. So no test of this kind can tell a tiling from any linear image of it.

The loop test can. It passes the rhombille as drawn and fails every image of it; it passes the square grid under every map and the rhombille under none. A test that separates a tiling from its images is therefore not a test of the property a twist tessellation needs — and that alone, computed in a line, would have said in advance that the loop condition could not be the right condition. What it tests is a construction with one number per vertex, and one number per vertex is not a property a linear map preserves.

This is a cheap check and it generalises past twists. Whenever a criterion is proposed for whether some construction runs on a tiling, apply a shear to the tiling and see whether the criterion moves. If the thing being constructed is insensitive to linear maps and the criterion is not, the criterion is about the method.

Where the weights stop being equal #

The weights are a single number per edge, so a tiling on which they come out all the same has a twist tessellation in which every side is proportional to the edge it faces. That is the familiar case and it has a clean characterisation. Measured across the twenty, the weights the construction writes are equal on every edge exactly when every sector at every vertex is the same size — the square grid’s four right angles, the triangular grid’s six sixties, the honeycomb’s three one-twenties, and the rhombille’s two kinds of vertex, each equiangular in itself. The elongated triangular tiling is the one that is not, and its spread is $\sqrt{3}$ before any map is applied.

Under a map the characterisation stops being about sectors. The honeycomb and the rhombille keep equal weights under every map tried; the square grid, the triangular grid and the elongated tiling do not, and the triangular grid’s spread reaches six under a stretch. The honeycomb’s case is the easy one to see — a triangle is determined by the directions of its three sides, so a degree-three vertex has no shape left to choose and the weights have nowhere to go. The rhombille’s is not, because its degree-six vertices have shape to spare and the weights stay equal anyway, and nothing here explains that.

The elongated triangular tiling deserves its own sentence, because its $\sqrt{3}$ is the one number here that was available from the beginning and was never computed. Its vertex is five tiles — two squares and three triangles — so its sectors are two right angles and three sixties, and the twist polygon at it is a pentagon with two kinds of corner. The weights being unequal as drawn means that tiling has never had a twist tessellation in which the sides are proportional to the edges, on any turn, under any rule. The dial and the tiling that is not alike singled out the rhombille as the awkward member of the set on the grounds that it was the only one whose distances had to be solved. By this reading the rhombille is the well-behaved one and the elongated tiling is the outlier, and the two readings disagree because they are measuring different things: one asks whether the vertices are alike and the other asks whether the edges are.

Which polygons twist found that a twist can be built around any regular polygon and that the restriction to three is the tiling’s rather than the paper’s — a fact about the plane. The weights sharpen that. The restriction is not about which polygons tile; it is about which tilings carry a balance, and the honeycomb keeping its balance through every shear is the strongest single instance of the difference.

What the twenty programs assume #

The weights are solved by class. One unknown per class of edge, one pair of equations per class of vertex, representatives supplied by a generated patch. That is a statement about the periodic tiling rather than about a finite piece of it, and a tiling with no repeating structure would need the unknowns back on the individual edges, where a patch’s boundary supplies freedom the plane does not have.

Positive means strictly positive. A weight of zero is a twist with a side of no length, which is a polygon with fewer corners than the vertex has edges and a pleat with nothing to join to. The program maximises the smallest weight precisely so that a tiling scraping the boundary reads as a small number rather than as a yes.

The balance is necessary and has not been shown sufficient here. The reasoning runs one way: a twist tessellation of this kind gives a positive balance. That a positive balance gives back a twist tessellation is the content of a theorem proved elsewhere, by a construction built from the reciprocal figure the balance defines, and it is not re-derived here.

And the weights reported are the construction’s, not the program’s. Where more than one balance exists, the program returns the one with the largest smallest weight and the construction returns whatever its own conditions land on. On the elongated tiling the two agree — the program’s 1.4641 against 0.8453 is a ratio of $\sqrt{3}$, which is the spread the pattern writes — and on the sheared square grid they do not.

What the balance does not settle #

It does not say how large the twists are. The weights are quotients, so they fix the sides relative to the edges and say nothing about the sheet. Two patterns with the same weights at different turns are different patterns and the balance cannot tell them apart.

It does not decide the layers. Every statement here is about the crease pattern’s geometry, and a pattern whose every vertex is satisfied can still have no consistent ordering of its layers, which is a global question of a different kind and a much dearer one.

It cannot see a tiling that is not a tiling. The balance is computed on the graph of edges and vertices, and a set of weights on that graph knows nothing about whether the faces are convex, whether edges cross, or whether the drawing covers the plane once.

The patch is a witness and not a proof. A generated piece three units across supplies one representative of every class, which is what the solve needs; it does not establish that the classes are all of them, and a tiling with a larger repeating unit than the patch would be silently under-described.

And a positive balance is not a drawing. A side distance can come out negative even when the weights are fine, which puts a side on the wrong side of its own vertex, and the construction’s window of pleat angles is where that shows up. Balance answers whether a pattern can exist; it never answers whether a particular one has.

Still open: what the equilibrium does when the sheet closes #

The weights were reached by asking what the sides are. There is a second question and it is about the paper rather than the drawing: what those weights do to the fold.

The reason to expect them to do something is arithmetic. A pleat is two parallel creases, so folding one translates the twist beyond it relative to the twist before it, and the size of that translation is set by the pleat’s width — which is the side’s length, which is the weight times the edge. So the displacement an edge contributes to the collapse carries its own weight, and a tiling whose weights are all equal contributes the same multiple of every edge while one whose weights differ does not.

Folding it flat is one similarity measured the collapse on the five tilings as drawn and found one answer for all five: a scale and a turn, agreeing to eight figures. Every one of those five has equal weights. The patterns that now exist on the images mostly do not, and if the arithmetic above is right their collapse cannot be a similarity — which would make them the first twist tessellations here whose folded form is a different shape from the flat one rather than a smaller copy of it. That is a measurement waiting to be taken and it has a prediction attached, which is the best position a measurement can be in.

There is a second thing the weights ought to predict and it is about substance rather than shape. One vertex, repeated is the essay that treats a tiling as a way of turning a sheet into a material — a thing with a stiffness and a packing behaviour the paper never had — and a material’s properties are directional. A shrink is two numbers already found that how much smaller a folded sheet gets is a pair rather than a single factor, and that a twist tessellation is the family member that draws in equally both ways while the Miura does not. Equal weights and equal drawing-in are suspiciously similar statements, and on the five tilings as drawn they hold together. Whether they are the same statement, or two consequences of a third, is not settled by anything here.

The habit worth carrying is about quantities a construction computes and never looks at. Divide the thing a construction produces by the thing it was given, and see whether the quotient is already known to somebody. The side lengths were in every drawing this subject has ever made; the edge lengths were in every tiling; and the quotient is an object a different field has theorems about, which is worth more than any amount of further drawing.