The deciding set does not move

Only four creases decide a Miura counted, for every crease of a Miura driven to a fixed angle, how many consistent folded states the rest of the sheet can take. On a mesh with no two vertices alike every crease leaves one. On the Miura four creases of twenty-four leave one and the others leave two, four or eight, so an actuator placed on one of the four determines the sheet and an actuator anywhere else leaves it a choice.

It also reported that the four were not always four. Driven to 0.8 radians instead of 0.6 or 1.2, fourteen creases appeared to decide the sheet, and the conclusion drawn was the uncomfortable one: the deciding set belongs to the pattern at a state rather than to the pattern, and a guarantee made at one angle may not hold at another. A later census found a five-by-five crease leaving fifteen states where every other count was a power of two, and recorded it as unexplained.

Both were left as questions about the sheet. Both were answers from the solver, and the way to tell was to ask the question at every angle rather than at three.

The census along the whole motion #

A census at one angle drives each crease to that angle in turn, propagates the constraint to every vertex, and counts the complete assignments of fold angles that satisfy every vertex. Doing it at many angles is the same work many times.

At ten angles along the four-by-four Miura’s motion, the first figure is ten copies of one row: four creases leave one state, eight leave two, four leave four and eight leave eight, and the four that leave one are the same four — c:3:2, c:3:3, r:3:2 and r:3:3, at one corner — at 0.2 radians and at 2.8. Extended to twenty angles from 0.1 to 3.0, on three sizes of sheet, the answer is the same kind:

• three by three: four deciding creases of twelve at every angle, and the other eight leave two or four;
• four by four: four of twenty-four at every angle, the rest two, four or eight;
• five by five: six of forty at every angle, the rest two, four, eight or sixteen.

Every count at every angle on every sheet is a power of two, which is what the counting argument predicts: each vertex the propagation reaches with too little known to pin it contributes an independent factor of two. The fifteen that broke that pattern is gone.

What the solver was doing #

A census rests on one small computation repeated many times. When the propagation reaches a vertex knowing one of its fold angles, it has to find every configuration of the other three that closes — every way the four panels round the vertex can sit with that crease held. A degree-four vertex is a spherical four-bar linkage, and at any fold angle away from flat it has exactly two configurations.

The solver found them by Newton’s method from a fixed lattice of 216 starting guesses and kept the distinct results. That was chosen deliberately over random starts, because random starts sometimes found only one, and a walk that loses a configuration loses every assignment built on it. The lattice was better. It was not enough.

Solving every interior vertex of the four-by-four Miura with each of its creases held at 240 angles — 8,640 solves — the lattice found one configuration instead of two on 138 of them, all with the held angle between 1.0 and 2.5 radians. At those angles every one of the 216 starts converges to the configuration already found, and the second has a basin of attraction that no start lands in.

A driven angle of 0.8 radians is below that range, but a propagation does not hand each vertex the driven angle. It hands on whatever the neighbouring vertex produced, and on the Miura a crease driven to 0.8 propagates to angles of 1.8151 at several vertices — squarely in the range where the lattice misses. Each miss removes one configuration from a vertex and halves the count at every crease whose walk passes through it, and halving a count of two gives one: a crease that leaves two states was reported as deciding the sheet. At 0.8 radians ten creases were misreported that way.

The same solver, run at other angles nobody had published, was wrong in other ways. At 0.2 and 1.5 radians its census of the four-by-four sheet included counts of six and seven — not powers of two, and so not counts any walk of independent binary choices can produce — and at 0.76 it reported twelve deciding creases. At 0.6 and 1.2, the two angles that happened to be drawn, the propagated values stayed clear of the solver’s blind range and the census came out right. The published angles were the lucky ones, and the one unlucky angle that was published was read as a finding about the sheet because it was a single number with nothing beside it to contradict it. Twenty angles side by side would have looked wrong at once; three did not.

The five-by-five crease with fifteen states is consistent with the same failure one level less tidy. A walk that loses a configuration on some branches of its tree and not others counts only the surviving assignments, and a single lost configuration deep in one branch of a sixteen-state walk leaves fifteen. With both configurations found everywhere, that crease leaves sixteen.

Carrying a configuration from where it is easy to find #

The repair uses a fact about the vertices rather than a better search.

A Miura vertex is flat-foldable: its alternate sector angles sum to a half turn. For such a vertex each of the two configurations keeps simple relations along the whole motion. In each configuration, some creases turn through exactly the same angle as the held one, and every other crease keeps the tangent of its half-angle in a fixed ratio to the held crease’s. Those relations are the standard kinematics of a flat-foldable degree-four vertex; they are what make its motion a one-parameter family.

So when the lattice finds fewer than two configurations at an awkward angle, the solver finds both at a well-conditioned one — half a radian — reads off each configuration’s relations, and carries them to the angle it was asked about: a crease equal in size to the held one is set equal, and any other is set by the fixed ratio of half-angle tangents. On a flat-foldable vertex the carried configuration closes exactly, and Newton’s method only confirms it. On a vertex that is not flat-foldable, like the generic meshes, the relations are not exact, and the carried configuration is a start close to the right basin.

On all 138 misses the carried configuration is the second one, and the census built on the repaired solver is the one mapped above.

Why the angle cannot matter #

The map is a measurement, and there is a reason behind it that makes it unsurprising once seen.

When the propagation reaches a vertex knowing one crease, the vertex has two configurations and both extend the assignment, so the count doubles. When it reaches a vertex knowing two or more creases, each configuration either agrees with every known value or does not, and only those that agree go on. Whether a configuration agrees is decided by the same relations the repair used — this crease equal to that one, that crease’s half-angle tangent a fixed multiple of this one’s — and those relations hold at every angle of the motion. So a vertex that pins its configuration at one angle pins it at every angle, and a vertex that leaves a choice leaves it everywhere.

The count at each crease is therefore two to the number of vertices that leave a choice on that crease’s walk, and the number is fixed by the pattern and the crease, not by the angle. Exceptions would need a coincidence — an angle at which two configurations happen to agree on a crease they normally disagree on — and none of the forty sampled angles on three sizes of sheet hits one.

What this gives back to a designer #

Only four creases decide a Miura drew a corner rule — drive one of the four creases at the end of a run, and the sheet has no choice to make — and then withdrew some of its value by making the rule local to an angle. The rule is not local. An actuator on a deciding crease determines the sheet from nearly flat to nearly closed, and an actuator elsewhere leaves the same choice at every stage of the fold.

That matters most for the self-folding sheets paper that folds itself describes, whose actuators are materials that swell or contract and cannot be re-placed partway through a motion. A placement chosen once has to work throughout, and a deciding set that moved would have made that impossible for any single placement. It does not move, so one placement suffices.

It also sharpens what the census says about the pattern. One crease decides the sheet found that on a generic rigid mesh any single fold angle settles every other; the Miura’s exception is now a clean statement. The Miura’s symmetry offers a fixed set of binary choices, independent of how far it is folded, and which crease an actuator sits on decides how many of those choices it removes. That is a property of the pattern as a mechanism, which is the kind of property a designer can use.

Where the two actuators come back in #

A gearing reflects stiffness squared placed two actuators by the gearing between their creases, and noted that on a Miura the analysis needs the sheet to stay on one branch. The fixed deciding set supplies that condition: drive one deciding crease and the branch is chosen for the whole motion, so a second actuator on any other crease acts within a branch that does not change underneath it. The stiffness rule and the placement rule compose.

What else rested on the same computation #

A vertex solver is shared ground, and a blind spot in it is not confined to the measurement that found it.

Which crease to push measured how far an error in a driven crease travels by driving the crease a hair either side of an angle and dividing, and the hardest instant followed that measurement along the motion. Both refuse a step when the two drives land on different branches. A lost configuration could imitate a branch change, so the refusals were checked against the repaired solver: the Miura’s count of refused steps over eight creases is seventy-six before the repair and seventy-six after, and the mesh with no two vertices alike refuses none either way. The refusals were real branch ambiguities, not solver misses, which is what those essays said they were.

The census is more exposed than the amplification because it enumerates rather than follows. An amplification measurement stays on whichever branch it starts on and needs only that branch to be found; a census needs every branch at every vertex, and a single miss anywhere in a walk changes the count. A method that follows one solution tolerates a root-finder that sometimes misses; a method that counts solutions does not, and the census was the first measurement here that counted.

The condition that is not flat-foldability walks round the four vertices of a face and asks whether the fold angles close, which also takes both configurations at each vertex; its meshes are generic, where the carried configuration is a start rather than an exact answer, and the lattice’s misses were measured here on the Miura rather than on those. Whether they occurred there too is a check the repaired solver makes on the next drawing of every such figure.

What the measurement assumes #

The vertices are exact degree-four vertices of a flat-foldable pattern, and the panels are rigid. A Miura cut with a small error is not flat-foldable, its configurations are not related exactly along the motion, and whether its deciding set holds still is not tested here.

A crease left at zero is not a fold. Assignments that leave any crease flat are discarded, as in every census, because they are folded states of a pattern with that crease removed.

And the count is of consistent assignments, not of reachable ones. Two assignments may be separated by passing through the flat sheet, so a sheet driven continuously may never see some of them; the census counts what the vertex conditions allow.

What the map cannot show #

It samples the angles. Twenty angles on each sheet and 240 in the solver check are many, and a coincidence at a single angle between them would not appear; the argument above says why none is expected, and it is an argument rather than a proof.

It maps the Miura at one lean. Every sheet here is drawn with the same zigzag, so its vertices share one pair of sector angles; a Miura with a steeper or shallower lean has different ratios in its relations and the same structure, and the census should come out the same at every angle for the same reason, but it has been drawn for one lean only.

It stops at five by five. Larger sheets take longer to enumerate and have not been mapped; the doubling of the worst count with each row, and the slow growth of the deciding set from four to six, are three sizes’ worth of evidence.

And it does not say which of a crease’s states a real sheet takes. That depends on energy, imperfection and history, which paper that folds itself identifies as the whole practical difficulty. The census says how many choices there are; the physics of choosing is elsewhere.

Still open: whether an imperfect Miura keeps its choices #

The fixed set rests on the relations a flat-foldable vertex keeps. Cut a Miura a little wrong and its vertices stop being flat-foldable, its two configurations at a vertex stop being related by fixed ratios, and a pair of configurations that agreed on a known crease at one angle may disagree at another. The census of a slightly imperfect Miura could therefore move with the angle after all — not because the pattern’s symmetry changes, but because the symmetry that made the choices clean has been broken by an amount that matters at some angles and not others.

That is the physical version of the question the solver’s misses imitated, and it is the more important one, because every manufactured Miura is imperfect. A tolerance is a direction measured which directions of error a solved mesh forgives; the same errors, applied to a Miura and censused along the motion, would say whether an actuator placement survives manufacture.

Sideways from here, the repair itself is worth generalising. Any enumeration built on a root-finder inherits the root-finder’s blind spots, and a count that comes out lower than its structure predicts — fifteen instead of sixteen, one instead of two — is the signature. The other measurements in these essays that walk a mesh vertex by vertex rest on the same solver, and they now rest on the repaired one.

The habit worth carrying is about surprising variation. When a quantity that should be structural changes with a continuous parameter, check the instrument at the parameter values where it changed before believing the change. A deciding set that moved at 0.8 radians and nowhere else was a result about one value of an angle, and a result that holds at exactly one value is more often a property of how it was measured.