The hardest instant

One crease decides the sheet: fix a single fold angle on a quadrilateral mesh that folds rigidly, propagate the closure to every vertex, and the whole sheet follows with one consistent answer. That is what makes a self-folding sheet buildable with one actuator rather than with one per crease.

Which crease to push asked the question that leaves behind, and answered it: drive each of a mesh’s twenty-four creases in turn and the amplification — how much an error in the driven crease is multiplied by the time it reaches the far side — comes out differently for each. Twenty-four creases, twenty-four answers, and a designer has a choice worth making.

Every one of those answers was taken at one fold angle. A self-folding sheet is not delivered to a fold angle. It is delivered flat, and it has to get to the other end under its own power, and an actuator is sized for the worst instant of that trip rather than for the middle of it.

The measurement, and what it is not #

Drive a crease to ρ − h and propagate; drive it to ρ + h and propagate; divide the change in every other crease’s fold angle by 2h. That is a directional derivative of the whole propagation with respect to the driven crease, taken numerically because the propagation is a search rather than a formula.

What it is not is a force. Nothing here has a stiffness in it, no torque is computed, and no material appears anywhere. The number is a ratio of angles: if the driven crease moves by a thousandth of a radian and some other crease moves by five thousandths, the amplification is five. A designer who wants a torque multiplies by a stiffness, which nothing in this measurement supplies.

The reason to measure it anyway is that the ratio is what decides whether the mechanism can be controlled. An actuator that has to place a crease to a tenth of a degree, on a sheet where that crease’s error arrives somewhere else multiplied by six, is placing that somewhere else to six tenths — and the mechanism’s usable precision is set by the worst such factor over the whole motion, not by the actuator’s own.

Every crease reports one somewhere #

The first thing the sweep produces is an anchor, and it is exact.

Every crease of the solved mesh reports an amplification of exactly 1.000000 at some point of its motion. Not approximately: the driven crease reports itself, and the largest ratio anywhere on the sheet is that one, at whichever fold angle the rest of the sheet is momentarily moving no faster than the crease being pushed.

That is the measurement checking itself. A derivative computed by finite differences on a search can go wrong in a dozen ways, and a routine that never returns a clean 1 anywhere is a routine to distrust. It returns one on every crease.

And it goes with a second fact: none of the curves is flat. Every crease’s worst amplification is at least 2.9 times its best, and on the worst of them 6.7 times. So the choice of when to look is a real choice on every crease of the mesh, not merely on the awkward ones.

Twenty of twenty-four are worst at the start #

The result is a direction and it is the unhelpful one.

On the mesh with no two vertices alike, twenty of the twenty-four creases have their worst amplification at 0.15 radians — the first angle sampled, about nine degrees from flat. The four exceptions are worst at the closed end. The peaks at the flat end run from 3.4 to 6.7; the peaks at the closed end run to 2.9.

A self-folding sheet starts flat. So the instant a designer would most like to ignore — the first few degrees, when nothing appears to be happening — is the instant when an error in the driven crease is amplified most across the rest of the sheet.

That is not a statement about difficulty in the sense of force. It is a statement about authority: near the flat state, small errors in where the actuator has put its crease correspond to large errors in where the rest of the sheet is, so the sheet’s shape is least determined by the thing controlling it exactly when the control has done least.

Which is a property of the mesh, not of the question #

The obvious worry about a result like that is that it is a property of quadrilateral meshes, or of this solver, or of the sampling. The control is the other family, and it answers all three at once by behaving differently.

Take a Miura — the same size, the same solver, the same sweep. Ask the same question and the measurement refuses to answer on twenty of its twenty-four creases.

Not fails: refuses. And the refusal is the finding.

What a difference quotient is not #

A derivative is a small change over a small step. The driven crease is moved by a ten-thousandth of a radian in each direction; if some other crease’s angle changes by a fifth of a degree over that, nothing has been amplified — the propagation has landed somewhere else entirely, and the quotient is a number about two configurations that are not on one motion.

So the measurement carries a threshold. A crease whose angle moves by more than a fiftieth of a radian across the step is counted and excluded, and the count is reported beside the result.

On the solved mesh that count is zero, across every crease and every fold angle. On the Miura it is 430.

The reason is a thing this site has already published from the other side. The Miura folds two ways: at one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both close exactly. The propagation, asked at ρ − h and at ρ + h independently, is free to find one at one and the other at the other — and the difference between two states half a sheet apart, divided by two ten-thousandths, is a number in the thousands.

A measurement without the threshold would have reported those thousands as gearing. They are the branch structure of the Miura, arriving in a routine built for something else.

The refusal is the same discipline twice #

This is the second time recently that a measurement on this site has had to learn to say that a number it can produce is not the quantity it is measuring.

Closing is not building followed a solved mesh’s motion along one continuously chosen branch and refused to report a collision between two states that were not on one motion, because solving each angle independently hops between configurations and a hop produces an intersection that no physical panel ever performs.

This is the same hazard in the same family of meshes, met by a routine that was measuring something else. And the diagnostic is the same: the two meshes are put through one threshold, and the one with a single folded state per angle produces no refusals at all. That is what makes the Miura’s 430 evidence rather than a threshold artefact.

How large the numbers are, in a unit somebody can use #

The amplifications above are pure ratios, and a ratio is easy to shrug at. Put a length on it.

A quadrilateral mesh 150 mm across with four panels to a side has panels about 37 mm wide. Suppose the actuator can place its crease to a tenth of a degree, which is a good hinge. At an amplification of 6.7 the far side of the sheet is then placed to two-thirds of a degree — and a panel 37 mm long swung by two-thirds of a degree has its far edge 0.43 mm out of position.

Half a millimetre on a 150 mm sheet does not sound like much until it is compared with the other number this anchor has: the cutting allowance at the packed end is 0.033 mm. The positional error the actuator’s own precision produces near the flat state is an order of magnitude larger than the dimensional error the mesh will tolerate near the closed one.

The two are errors in different things and they do not add. What the comparison says is that a design conversation about precision has two separate budgets in it, they are worst at opposite ends of the motion, and the larger of the two is the one nobody measures.

Where the two ends come from #

The measured shape — a peak at one end, a floor of one in the middle, a second rise at the other — is not an accident of this mesh, and a single vertex says why.

At a degree-four rigidly folding vertex the two fold angles either side are tied by a fixed ratio of half-angle tangents: $\tan(\rho_2/2) = \mu \tan(\rho_1/2)$, with $\mu$ set once and for all by the sector angles. Differentiating that relation gives the amplification directly. Writing $t = \tan(\rho_1/2)$,

$$\frac{d\rho_2}{d\rho_1} = \mu\,\frac{1+t^2}{1+\mu^2 t^2}$$

which is $\mu$ when $t \to 0$ and $1/\mu$ when $t \to \infty$.

So a crease’s amplification at the flat end and at the closed end are reciprocals. A pair that gears up by 6.7 near flat gears down by 0.15 near closed, and the partner driven the other way does the opposite. That is the two-ended structure the sweep found, arriving from one line of algebra.

It also says the quantity is monotone between the ends, so at a single vertex the worst instant is always one end or the other and never the middle. Twenty creases worst at the start and four worst at the close is exactly the split that law predicts: the four are the ones whose ratio sits below one.

What the whole sheet adds #

The whole-sheet number is not one vertex’s ratio, and the difference is where the floor comes from.

An error in the driven crease reaches a distant crease through a chain of vertices, and the amplification along the chain is the product of the ratios on it. Some are above one and some below, so the product can pass through one on its way from the flat end to the closed end — which is the floor of exactly 1.000000 the sweep reports on every crease, and the reason it sits inside the motion rather than at an edge.

The single-vertex law even places it. The ratio equals one when $t^2 = 1/\mu$, so the crossing is at $\rho = 2\arctan(1/\sqrt{\mu})$: about 0.74 radians for the sharpest crease on this mesh and about 1.00 for the mildest. Those land inside the low window the curves show, which is the check worth making — the algebra was derived for one vertex and is being asked about twenty-four.

The practical reading is that the window is not a tuning parameter. Its position is fixed by the sector angles through $\mu$, so a designer who wants a wider controllable band changes the pattern’s geometry, not the actuator.

What it costs a designer #

Three readings, and the second is the one that changes a test plan.

Size the actuator for the first ten degrees. The worst amplification on this mesh is 6.7 and it occurs at 0.15 radians. A sizing taken at half a radian would see 2.0 and be wrong by a factor of three.

Test near flat, not near closed. That is the opposite of where a cutting tolerance binds, which is at the packed end — so the two qualifications are at opposite ends of the same motion and neither substitutes for the other. A sheet qualified only in the middle has been qualified against neither.

And on a Miura, ask a different question. A sheet with two folded states at every angle does not have a gearing in the sense measured here; it has two, and which one it is in is decided by something outside the closure. Choosing which crease to push on such a sheet is choosing which of several answers to get, and the useful design question is how to bias the sheet into one of them rather than how much the chosen one amplifies.

Reading the curves rather than the peaks #

The peaks are what an actuator has to survive, and the shapes are what a designer would use to place one.

Every curve starts high, falls to its floor of exactly one somewhere in the first third of the motion, and then climbs again — gently on most creases and sharply on four of them. So the mesh has a window, roughly between a third of a radian and a radian and a half, in which every crease on it is amplifying by less than two.

That window is where the sheet is most controllable, and it is neither end. A deployable that spends its working life inside it is a well-behaved object; one that has to hold a position at either extreme is being asked to hold it with the least authority it ever has.

The four creases that climb sharply at the closed end are worth separating out. They are the ones whose propagation path is longest — the error has the furthest to travel — and on them the amplification at 2.8 radians is comparable to the amplification at 0.15. Those are the creases a designer would not choose to drive, and the survey says so at both ends rather than at one.

What the sweep does not decide #

Two things, and both are the kind that would be easy to imply.

It does not say the sheet is hard to fold near flat. A mechanism’s mechanical advantage and its kinematic amplification are different quantities, and only the second is computed. A sheet whose fold angles are badly determined near flat may still be trivial to push.

And it does not say which crease is best overall. The ranking of creases changes along the motion — a crease that is best at one angle is not best at another — and the survey here reports the peak of each rather than an average, because a peak is what an actuator has to survive and an average is not a thing a mechanism ever experiences.

The same sweep on the sheet that is actually built #

The Miura is the pattern everything in this field is actually made of, so its refusals are not an inconvenient control — they are the case a designer meets.

What can be said about it is this. Of the twenty-four creases, four are followed cleanly all the way and twenty are not. On the four, the amplification behaves like the solved mesh’s: a floor of one, a rise at one end. On the twenty, the propagation changes state somewhere in the motion, and where it changes is not a property of the crease being driven — the same crease changes state at different angles depending on which side of the step the propagation started from.

The design consequence is not that the Miura is uncontrollable. It is that a Miura’s controllability is not a number, and the question a designer has to answer first is a different one: which of the two states the sheet is to be in, and what biases it there. The sheet has to be told, and no amount of authority in the actuator tells it.

Why the anchor keeps coming back to the ends #

Three rungs of this anchor have now found their answer at one end of the motion or the other, and the pattern is worth naming.

Paper that folds itself found the difficulty at the very start: a flat sheet with a crease made ambiguous has two outcomes that are equally downhill, and no amount of torque chooses between them. One crease decides the sheet found that the ambiguity vanishes once anything is folded at all. This rung finds that the precision has the same shape — worst where the ambiguity was, better afterwards.

So the flat state is not merely the starting configuration. It is a degeneracy, and every difficulty this anchor has measured lives within a few degrees of it.