Only four creases decide a Miura

Which crease to push measures how far an error in a driven crease travels and notes, in passing, that the sheet repeating one vertex sometimes gives several answers to what shape it takes. The hardest instant meets the same thing at the far end: on the Miura the amplification measurement has to refuse to answer.

Both were reporting a symptom. The Miura genuinely has more than one folded state at a given driven angle, and which crease is driven decides how many.

Counting properly #

The count is an enumeration rather than an estimate. Fix the driven crease’s angle, walk the interior vertices in an order where each new one shares a crease with one already placed, and take every branch consistent with everything already known. A degree-four vertex has two configurations at a given fold angle, so a vertex arriving with only one crease known contributes a choice, and a vertex arriving with two contributes none.

Every complete assignment reached that way is a folded state: every vertex closes, every shared crease agrees, and no crease is left unfolded. There are four such states for eight of the Miura’s creases and one for four of them.

The counts are all powers of two — 1, 2, 4, 8 — which is what the argument above predicts: the choices are independent, so the total is two to the number of vertices that arrived knowing only one crease. That number depends on where the walk starts, which is to say on which crease is driven.

The four that decide it #

The four creases leaving a single state are c:3:2, c:3:3, r:3:2 and r:3:3 — two row creases and two column creases, all at the same corner of the mesh.

That is not a coincidence and the reason is the walk. A propagation starting at a corner expands along a diagonal front, and every vertex on that front touches two vertices already solved, so it arrives with two creases known and has no choice to make. A propagation starting in the middle expands outward in all directions, and the vertices on the leading edge arrive knowing one crease each.

So the deciding creases are the ones whose propagation never has a free vertex, and on a grid those are the corner ones. An actuator belongs at a corner, which is a placement rule nobody would derive from the mechanics and which falls out of the counting immediately.

It is the symmetry, not the mesh #

The finding that matters is the comparison, and it goes the opposite way from what the earlier essays assumed.

On a mesh whose vertices all differ, every one of its twenty-four creases leaves exactly one folded state. There is no ambiguity anywhere and no corner rule to find, because there is nothing to choose: at each vertex the two branches give different angles and only one of them agrees with what is already known.

On the Miura they agree. A Miura is one vertex repeated, so the two branches at a vertex are related by the pattern’s own symmetry, and a vertex arriving with one crease known cannot tell them apart. The ambiguity is the symmetry, and the pattern the whole deployable field is built on is the one pattern in this family that has it.

One crease decides the sheet established the determinacy on a generic mesh, and it was right. What it did not say is that the sheet it was describing is not the sheet anybody builds.

What the count is a count of #

One more clarification before the sizes, because the phrase “eight folded states” is doing a lot of work.

It counts assignments of fold angles to creases that satisfy every interior vertex’s closure and agree on every shared crease, with the driven crease held at its commanded value. It does not count shapes a reader would call different — two assignments may differ only in a region that looks much the same — and it does not count shapes reachable by a continuous motion from flat.

So the honest statement is that a driven Miura’s conditions have eight solutions where a driven generic mesh’s have one, and that a mechanism relying on the conditions to determine the sheet is relying on something that does not hold. Whether all eight are physically available is a further question with a further model in it.

What makes the count worth having anyway is the comparison. One mesh gives one and the other gives eight under identical treatment, so whatever the count is a count of, the two patterns differ in it by a factor of eight.

How it grows, and how it moves #

The deciding creases stay few as the sheet grows — four, four, six — while the worst ambiguity doubles with each row added, reaching sixteen states on a five-by-five. The certainty does not scale and the choice does, which is the wrong way round for anything that has to be built large.

And the count does not depend on the angle. Driven to 0.8 radians the four-by-four sheet is decided by the same four creases as at 0.6 and 1.2, and every crease leaves the same number of states. An earlier version of this essay reported fourteen deciding creases at 0.8 radians, and concluded that the deciding set was a property of the pattern at a state rather than of the pattern. That was the vertex solver losing one of a vertex’s two configurations at particular propagated angles, which halved the count of states at the creases whose walks passed through them; with the solver repaired, the deciding set does not move maps the census along the whole motion and finds it identical everywhere.

So the guarantee is a property of the mechanism after all. An actuator placed on a crease that decides the sheet at one angle decides it at every angle, which is the better news for a designer and the one the measurement now supports.

What the earlier measurements were seeing #

The refusals are the same phenomenon. Measuring an amplification means driving a crease a hair either side and dividing, and that only means something if both propagations land on the same branch. On a mesh with one state per crease they always do. On the Miura they need not, and seventy-six steps over eight creases were refused for exactly that reason.

So the earlier measurement’s caution was not conservatism. It was the method correctly declining to differentiate across a branch change, in a pattern where branch changes are available because the states are not unique.

The sweep shows the other face of the same symmetry: every crease of a Miura produces the same amplification, because every crease is the same crease. The pattern’s uniformity makes the amplification question trivial and the state question hard, and the two are the same fact.

That pairing is worth holding on to, because it reverses which pattern looks well-behaved. Asked where to put an actuator for accuracy, a Miura answers “anywhere” and a generic mesh answers with a spread of more than an order of magnitude — so the Miura looks like the forgiving one. Asked where to put an actuator for certainty, the generic mesh answers “anywhere” and the Miura answers “four of twenty-four”. Each pattern is easy on the question the other is hard on, and a designer who has only asked one of them has a false impression of the other.

What a second state actually is #

It is worth saying what the other seven configurations are, because “eight folded states” sounds more exotic than it is.

They are the same panels with the same fold angles at the driven crease and different mountain-valley choices elsewhere — a region of the sheet popped through, in the way a folded strip can be pushed inside out without unfolding anything. Every vertex still closes; every shared crease still agrees; the sheet is a different shape.

That is the thing a self-folding sheet cannot be trusted to avoid. Paper that folds itself puts the difficulty precisely: supplying the torque is easy and the two outcomes at a vertex are equally downhill, so nothing in the energy chooses. Scaled up, the eight states of a driven Miura are eight equally downhill outcomes, and a sheet released from flat will find whichever one the noise points it at.

The corner rule is therefore not a convenience. It is the difference between a mechanism with one outcome and a mechanism with eight, and it costs nothing but the position of a motor.

Why a generic mesh cannot do this #

The control deserves its own explanation, because “the vertices all differ” is doing more work than it looks.

At a degree-four vertex, the two branches give different fold angles on the three creases other than the driven one — unless the vertex’s sector angles have a symmetry that makes them coincide. A Miura’s vertex has exactly that symmetry: its opposite sectors are equal, and the two branches are mirror images that agree on the angles a neighbour can see.

So a neighbouring vertex, told one crease’s angle, cannot tell which branch produced it, and the choice propagates. On a mesh whose sectors are all different the two branches disagree on every crease, the neighbour’s constraint picks one out immediately, and the walk has no freedom at any point.

The Miura is not ambiguous because it is regular; it is ambiguous because its regularity is of exactly the kind that hides a branch. The hardest instant met the same property at the flat sheet, where every mesh’s branches coincide; the Miura has a version of that at every angle.

What this means for an actuator #

Paper that folds itself identifies the hard half of self-folding as choosing what the sheet becomes rather than supplying the torque, because the two outcomes are equally downhill. This is that problem at the scale of a sheet: a Miura with an actuator on the wrong crease has up to eight outcomes and no preference among them.

The remedy the counting suggests is cheap. Put the actuator at a corner, where the propagation has no free vertex, and the sheet has one state to fall into. That is a placement rule, it costs nothing, and it is stronger than any amount of torque.

What it does not give is a guarantee across the motion, since the deciding set moves with the angle. A sheet driven from a corner at 0.6 radians is determined; the same sheet at some other angle may not be, and nothing here says which angles are safe.

The corner, and what is special about it #

The placement rule is worth pushing on, because “put it at a corner” is the sort of answer that turns out to be about the walk rather than about the sheet.

It is about the walk, and that is fine. The propagation from a corner reaches each new vertex across a diagonal front, so each arrives sharing creases with two solved neighbours; the propagation from the middle reaches the leading edge across a single crease. The sheet does not know where the walk started — the states enumerated are the states — so the corner rule is a statement about which driven crease leaves the sheet with the fewest consistent alternatives, and that really is a property of the crease.

What is not settled is whether a corner is special or whether an edge is. Two of the four deciding creases are row creases and two are column creases, all adjacent to one vertex; at five-by-five there are six, spread along one edge rather than clustered at a point. So the rule is closer to “drive from the boundary, at the end of a run” than to “drive at a corner”, and the difference would matter on a sheet that is not square.

A larger census would settle it, and it is not expensive: every crease of a Miura at several sizes and several angles, with the deciding set drawn on the mesh rather than listed. Nothing here does that, and the six creases at five-by-five are the only evidence that the set is not always four.

What the counting cannot show #

The enumeration is exact about a model and the model leaves out most of a sheet.

It counts states that satisfy every vertex, and says nothing about whether they are reachable. Two of eight consistent assignments may be separated by the flat sheet or by a self-intersection, so a physical sheet driven continuously may never see seven of them. The count is of configurations, not of outcomes.

It has no energy in it. A real sheet with several available states falls into whichever is lowest, and if they differ in energy the choice is made for it. The counting treats all states as equal, which is the same idealisation paper that folds itself identifies as the source of the whole difficulty.

And it counts at a few angles. The census is run at 0.6, 0.8 and 1.2 radians and gives the same answer at each; the map along the whole motion is made separately, where the angle’s irrelevance is shown rather than sampled.

What the model assumes #

Every crease is genuinely folded. An assignment leaving a crease at zero is a folded state of a different pattern — the one with that crease rubbed out — and the enumeration discards it.

The panels are rigid and the vertices are degree four. That is the whole of the rigid-folding model in use here, and it is what makes each vertex a two-branch choice.

The enumeration is complete up to its limit, which is raised well past what any count here reaches. The default limit is eight solutions, which for a five-by-five sheet is a cap rather than an answer.

And the mesh folds. Both meshes compared are ones that solve at the angle drawn; a mesh that does not fold has no states and is not in the comparison.

How the numbers were checked #

The counts are required to be powers of two on the four-by-four Miura, which is what the independent-choice argument predicts and which would fail if the enumeration were missing or double-counting assignments.

The generic mesh is required to decide on every crease. That is the control: a comparison in which both meshes were ambiguous would be measuring the enumeration rather than the patterns.

The Miura is required to decide on fewer than half its creases at the angle drawn, so the contrast is verified rather than described.

And the total is checked against the crease count at every size, so no crease is counted twice or left out of the census.

Still open: two actuators, and a sheet asked to be in two places #

The corner rule removes the ambiguity and raises the question left open since which crease to push ended on it.

A sheet with one freedom driven by two actuators is over-determined. Not redundant — over-determined: two commands for one number, and if they disagree by a hundredth of a radian the sheet cannot satisfy both. It settles somewhere, and where is decided by the stored energy, which is decided by the gearing between the two creases.

That has a shape worth computing. With a gearing $g$ between the driven creases and equal stiffnesses, a disagreement $\delta$ should divide as $g\delta/(1+g^2)$ at the first and $\delta/(1+g^2)$ at the second — so a strongly geared pair would absorb the disagreement and a weakly geared one would not, which is the opposite of what coupling usually means. Whether that is right, and which pairs of a real mesh are worst, is one measurement away.

Sideways from here, the question that looked most troubling turned out to have a reassuring answer. If the deciding set changed at particular angles, those angles would be configurations at which a driven sheet acquires a choice it did not have a moment before — a bifurcation, and exactly the event a deployment cannot afford. Mapped along the whole motion, it does not change at all; the apparent changes were the solver’s.

The habit worth carrying is about determinacy claims. Check them on the symmetric case, not the generic one. A result that holds for almost every member of a family can fail on the one member anybody builds, and symmetry is what makes a member both worth building and exceptional.