Fourth of eight, and still not chosen for it

Folding that gets built lists what deployables are for: solar arrays, airbags, stents, starshades, and always the same requirement — large in use, small in transit, along a path nobody has to trust to chance. The first two thirds of that sentence is a ratio, and a ratio invites a comparison.

Here is the comparison, and the pattern that flew comes fourth.

The two numbers #

Compaction is the sheet’s area divided by the footprint it packs into, which is the same thing as the average number of layers over that footprint. It is measured off the folded state by sampling, and the sampling is checked against a quantity known exactly — the layer count integrated over the footprint has to come back as the sheet’s area.

Folding length is the total length of crease in the pattern, in sheet widths, summed from the pattern’s own coordinates with the boundary excluded and divided by the pattern’s own width. That last divisor is not a detail: a length needs a scale, and without it these essays reported the Miura at six times its folding length for four earlier essays and this essay drew the wrong conclusion from it.

Divide the second by the first and the result is sheet-widths of crease per layer of packing.

The Miura is fourth. It is 2.9 times worse than the Yoshimura, 2.8 times better than the hexagon twist, and within twenty per cent of the two patterns immediately above and below it.

What a middling rank means #

A ranking is useful when it separates. This one does not.

Three patterns beat the Miura and none of the three is a deployable. The Yoshimura is a buckling mode a crushed cylinder finds for itself. The waterbomb is a traditional tessellation. The preliminary base is the base under half the classical repertoire, folded by children, and it beats the pattern that flew on a satellite.

Four patterns lose to it and none of the four is a deployable either. Two twists, a fold-and-cut construction and a tapered corrugation.

So the ranking has the built pattern in the middle with non-deployables on both sides, which is the strongest form the refusal can take: a figure of merit that puts its one successful instance fourth of eight has not measured what selected it. A rule that put the Miura first would at least be consistent with packing being the criterion. A rule that puts it last would be a paradox worth an essay. A rule that puts it fourth is a rule about something else.

That is a weaker claim than these essays made before the arithmetic was corrected, and it is a better-founded one. The earlier reading — that the Miura is the worst converter on the shelf by a factor of eighteen — was a striking number arrived at by multiplying by a sheet size without dividing by the pattern’s own width, and the striking number was doing the work.

What it is buying instead #

The answer is in a sheet with one freedom, and it has nothing to do with how small the packet is.

The Miura folds and unfolds as a mechanism. Its whole patch moves through a continuous family of rigid states with one parameter, so pulling two opposite corners opens the entire array at once. That is the property a solar panel needs and it is the property nothing else on this shelf has: the Yoshimura is a buckling pattern that a crushed cylinder finds for itself, and it does not deploy — it collapses.

It folds into a flat rectangle rather than into a point or a line. Fifty-nine per cent of its panel pairs share ground, against the Yoshimura’s ninety-nine and the preliminary base’s hundred, and a shape that keeps two fifths of its pairs apart is a shape that stays rectangular. A packet is a thing that has to travel in a bay.

And neither of those is in the ratio at all, which is why the ratio has the Miura in the middle. A pattern is not penalised in this table for lacking a motion and not rewarded for having one; the table sees crease and layers and nothing else.

Where the length actually goes #

The Miura’s six sheet-widths is worth taking apart, because the corrected figure overturns a reading these essays had of the pattern’s geometry.

A Miura of six columns and four rows has thirty-eight creases in the census, and they are segments between vertices rather than lines from edge to edge. Each horizontal segment spans one cell and each vertical one spans one row, so at six cells across, a segment is about a sixth of a sheet width. Thirty-eight of those is 6.2 sheet-widths, and the arithmetic has no slack in it.

The interesting part is the comparison. The Yoshimura’s eighty-six creases average 0.163 sheet-widths each; the Miura’s thirty-eight average 0.162. To three figures, the two patterns lay in creases of the same length, and they differ only in how many.

That is the opposite of what these essays said before the correction — which had the Miura’s average crease at 1.03 sheet-widths, longer than the paper is wide, and built an argument about grids whose creases run past their vertices. The grids do run past their vertices; the census does not count them that way. A crease between two vertices is one crease here, on every pattern, and the two constructions come out identical on the mean.

So the ratio is set by the number of creases and not by their length, which makes it a count in disguise — and the first of these essays of the crease-length sequence is about how counts and lengths disagree.

Compaction is the wrong figure of merit #

Stated as a design rule: a deployable is not chosen for its packing ratio, and the shelf makes that measurable rather than arguable.

If packing were the criterion, the Yoshimura would be the deployable of choice everywhere — sixty layers for a third of the folding. It is not used, and the reason is not obscurity: it is the pattern a thin cylinder finds for itself under compression, it has been known since the 1950s, and it reached this site’s own shelf as a found pattern rather than a designed one.

What it lacks is the motion. A Yoshimura has no useful continuous rigid folding: it goes from cylinder to collapsed and there is no controlled path between the two that a mechanism could drive. A packet that cannot be opened on command is not a deployable, however small it is.

So the ratio in the table is a measurement of one axis of a two-axis problem, and the patterns at the top of it are at the top because they are not solving the other one.

What a correction of this size does to a sequence #

It is worth being explicit about which of these essays’ essays the arithmetic touches, because the answer is one of them and that is not obvious.

Folding that gets built lists what deployables do and quotes no length. From a shell to a solar array is chronology. The tube that gets built is about cylinders and their counts. What a second deployment costs and the crease count as a reliability budget both price the number of creases rather than their length, and a count is exact whatever the coordinates.

So one essay of six quoted the affected quantity, and it is this one. That is a property of the essays here worth noticing rather than luck: the essays that survive are the ones that price a count or a probability, and the one that did not is the one that priced a physical length. A length has a scale in it and a count does not, which is the whole of why the error had somewhere to hide.

The lesson for the essays above is not that they are safe but that they were never exposed. Every one of them would have been, had it multiplied a crease count by a length per crease — which is a thing the reliability budget very nearly does and stops short of.

What the exchange rate is good for #

It is not useless — it is a cost, and a cost is worth knowing even when it is not the criterion.

It prices manufacture. Six sheet-widths of crease is six sheet-widths of scoring, creasing, perforating or embossing, and on hardware that is a process step with a length. A pattern with the same compaction and half the crease is half that step.

It prices error. Every crease is laid in with a tolerance and the errors are folded too, so a pattern with twice the crease has twice the length over which a placement error can accumulate. The rate is not the whole of that argument but it is its first term.

And it prices the choice between corrugations. The tapered corrugation at 0.79 and the Miura at 0.67 are both quadrilateral corrugations that fold rigidly, at similar compactions of 8.3 and 9.2 layers, and the difference between them is under twenty per cent. A designer choosing between them on this number is choosing on noise, which the table now says plainly and previously concealed behind a factor of four and a half.

Reading the rest of the table #

The other seven are worth a line each, because the ratio sorts them in a way that has a reason even though it has no predictive power.

The Yoshimura, 0.23. A triangulation: its creases end at vertices, it has many of them, and sixty layers come cheap. It folds to something with no motion in it.

The waterbomb tessellation, 0.46. Second best for the same reason, and it folds to something with structure rather than to a line — which makes it the entry closest to being a counterexample to the whole argument.

The preliminary base, 0.60. Eight creases, eight layers, and the shortest folding of anything that compacts at all. A traditional base outranking the pattern that flew is the clearest single sign that this ranking is not about deployability.

The tapered corrugation, 0.79. A quadrilateral corrugation like the Miura and within twenty per cent of it on every column.

The fold-and-cut triangle, 1.45. Barely compacts — 1.19 layers on average — because it is not trying to. It brings one line of the paper together and leaves the rest almost flat.

The two twists, 1.56 and 1.88. Three layers for five and six sheet-widths of crease. A twist is a rotation mechanism rather than a packing, and the ratio says so.

Nothing in the ordering is a surprise once each row is read, which is the useful property of a figure of merit: it should reward what it measures and be honest about what it does not. What this one does not measure is everything the field is about.

And the spread is the point rather than the order. From 0.23 to 1.88 is a factor of eight across eight patterns of five different kinds, which is a narrow range for a quantity compared across a base, two twists, three corrugations and a fold-and-cut construction. Patterns that do entirely different things convert crease into layers within a factor of eight of one another, so the quantity is close to being a constant of folding rather than a discriminator between designs — and a near-constant is the last thing anybody should be selecting on.

The twenty-five year lag, and what it is not made of #

From a shell to a solar array records that the Miura was published in 1970 and flew in 1995, and that the same quarter-century gap sits between every folding result and the hardware using it. The gap is not ignorance: the pattern was known, understood and available the whole time.

An earlier reading of this table offered one small part of an explanation — that a pattern with nearly seven metres of crease is not a thing anybody makes at engineering tolerances, so the length was the obstacle. That explanation is now unavailable. The Miura’s printed pattern carries a little over a metre of crease, which is a third of what the Yoshimura carries and no more than any of several patterns a hobbyist folds by hand.

So the length is not what took twenty-five years. Whatever the gap is made of — tooling, materials, qualification, somebody deciding a satellite needed it — this measurement does not reach it, and the honest result is that a plausible contribution has been removed rather than added. A correction that takes away an explanation is worth as much as one that supplies one, and these essays are now short of an account it thought it had.

The proxy for the second axis, and what it rests on #

Compaction is the axis this essay prices. The other one — whether the pattern moves — is not a number here, but it can be given a proxy that is.

How much of the folded object lies on the rest of it. The Miura’s panels share ground in eighty-three per cent of their possible pairs; the Yoshimura’s in ninety-nine, the preliminary base’s and the square twist’s in a hundred. A pattern whose every panel lies over every other has collapsed to a point, and a pattern that has collapsed to a point has nowhere left to move.

Read the column and the evidence for it is thin. The Miura is at eighty-three per cent and everything else is at ninety-nine or a hundred, so the proxy takes essentially two values on this shelf — the Miura’s, and everybody else’s — and the pattern that deploys is the one with the low value.

A proxy that separates one case from seven has been tested on one case. It could be measuring the thing it claims to, and it could equally be measuring that the Miura is the only pattern here that folds to a rectangle, or the only one drawn as a grid, or the only one anybody built. With a single discriminating row there is no way to tell those apart, and a rule fitted to one observation is a description of that observation.

What can be said is that it separates where the packing ratio does not, and that is a comparison between two proxies rather than a defence of either.

Still open: a figure of merit with the motion in it #

The table prices one axis and the proxy shadows the other, and a design would want them combined.

The quantity nobody has written down is compaction times the probability that it opens, which the reliability budget computes for a pattern’s crease count and which this table’s numerator is the input to. A pattern with more creases packs better and opens less reliably, so the product has a maximum, and the maximum is at a fold count this measurement could supply. That is a one-line combination of two things these essays already has and nobody has taken it.

And the second axis wants a real number rather than a shadow. The honest measure of whether a pattern deploys is the dimension of its rigid-folding configuration space at the folded state — one for a mechanism, zero for a pattern that only flexes. Computing it for the eight printed patterns would replace the shared-ground proxy with the thing it stands for, and it would test the proxy at the same time, which is what a shelf of one discriminating row most needs.

The habit worth carrying is about rankings with one known good answer in them. Find where the already-trusted instance lands before reading the ranking. If it lands in the middle, the ranking is measuring something orthogonal to what selected it — which is more useful than a ranking that agrees, because it says the selection happened somewhere the numbers are not looking.