The pattern cheapest to trust

The crease count is a reliability budget found that a deployment which must open once, and cannot itself be tested, has to be qualified by testing its hinges, and that the number of consecutive successes needed to demonstrate a system reliability grows linearly with the hinge count. It ended on a design criterion that follows and that nobody uses: choose a pattern to minimise the tests needed for what it delivers, rather than to maximise what it delivers. Splitting a sheet buys area, not certainty carried the same question through a divided design and left the criterion itself unmeasured.

What a deployable delivers is compaction, and what testing is charged for is hinges. The criterion is therefore hinges per layer of compaction, a count that can be read off any crease pattern and its folded state. It turns out not to agree with the measure of efficiency already in use, and on the pattern that gets built it grows as the pattern is refined.

Tests per layer #

Demonstrating a system reliability $R$ with 95 per cent confidence by consecutive successes takes

$$r = \frac{n \ln 0.05}{\ln R}$$

tests for a pattern of $n$ hinges. At $R = 0.99$ that is about 298 tests a hinge. A pattern’s compaction is the mean number of layers of paper over its folded footprint — the sheet’s area divided by the area it packs into — and it is what the tests are bought for, so the tests needed for each layer of compaction are 298 times the hinges per layer.

A hinge here is an interior crease segment: a length of crease between two vertices, or between a vertex and the sheet’s edge, which is the unit that can stick. The mean layer count is measured on each pattern’s own flat folded state, exactly as every other compaction figure in these essays is.

The shelf, ranked by tests #

On the eight printed patterns the hinges per layer run from 1.00 to 5.54:

• the preliminary base, eight hinges and eight layers: one hinge a layer, 298 tests a layer;
• the Yoshimura, eighty-six hinges folding to sixty layers: 1.43, and 427 tests a layer;
• the waterbomb tessellation, seventy-six hinges and 31.5 layers: 2.41;
• the square twist, twelve hinges and three layers: 3.98;
• the Miura, thirty-eight hinges and 9.2 layers: 4.12;
• the fold-and-cut triangle, the tapered corrugation and the hexagon twist, from 5.05 to 5.54.

Set beside the ranking by crease length per layer, fourth of eight, and still not chosen for it, six of the eight patterns move. The preliminary base rises from third to first, because its eight creases are long but few. The square twist rises from seventh to fourth, for the same reason. The Yoshimura falls from first to second, the waterbomb from second to third, the Miura from fourth to fifth, and the tapered corrugation from fifth to seventh.

The two rankings measure different costs. Crease length per layer prices the folding — the work a hand or a machine does laying the creases in. Hinges per layer prices the trust — the tests a programme runs before it will launch the result. A pattern of a few long creases is cheaper to trust than to fold, and a pattern of many short ones is the reverse.

Why one hinge a layer is the floor #

The preliminary base sits at exactly one hinge per layer, and no pattern on the shelf does better. That is not a coincidence of the shelf.

A folded state’s mean layer count cannot exceed its number of panels, since each point of the footprint is covered by at most every panel once. A pattern whose creases meet at interior vertices has fewer panels than creases plus one, by Euler’s count of a planar subdivision: every interior vertex closes a loop and saves a panel. So hinges per layer is at least about one, and reaches it only when every crease adds a panel and every panel lies over every point — which is a single vertex whose flaps stack exactly, the preliminary base.

The Yoshimura comes close, at 1.43, for the same reason: its triangular panels collapse onto a narrow footprint in which nearly every panel covers every point. The patterns far from the floor are the ones whose folded footprint is large compared with a panel, so that most panels cover only part of it and a layer costs several of them.

As a pattern is refined #

A deployable is not folded from a printed patch; it is folded from the same pattern at whatever fineness the compaction requires. So the criterion that matters is how hinges per layer behave as the cell count rises, and the three tessellations behave in two completely different ways.

The waterbomb at two cells a side has 18 hinges and 7.9 layers — 2.27 hinges a layer — and at eight cells 312 hinges and 126.6 layers, 2.47. Between six and eight cells the ratio does not move in the second decimal. The Yoshimura runs 1.37, 1.39, 1.44, 1.46, 1.47. Both patterns fold their whole sheet onto a footprint about the size of a few cells, so their layer count grows with the number of cells, as their hinge count does, and the ratio settles.

The Miura runs 1.32, 2.20, 2.95, 4.33 and 5.69. At eight cells a side it has 112 hinges and only 19.7 layers. Its folded state is a stack of zigzag strips offset from one another, and its footprint grows with the number of rows, so its layer count rises roughly as the side of the sheet while its hinges rise as the area. The ratio grows in proportion to the cell count, without a ceiling.

That makes the Miura unique among the three in a way no earlier measurement had shown. Finer Miuras pack better, which is why they are refined; but every layer a finer Miura adds costs more hinges than the last, and so more tests. By eight cells a side it needs 3.9 times the Yoshimura’s tests per layer, and by sixteen, if the trend holds, nearly eight.

What this does to the choice of pattern #

Fourth of eight, and still not chosen for it concluded that the Miura’s place in the middle of the shelf’s conversion rates said nothing about why it was chosen. The test criterion adds a second measure on which it is middling at printed size and worst in the limit among the three families that tessellate.

The obvious objection is that the Miura is chosen for its motion, not its packing or its tests. It has one degree of freedom and opens by pulling two corners, and a sheet with one freedom is a precondition for most actuation schemes. The Yoshimura and the waterbomb are not rigidly foldable in the same way. So the comparison is not between interchangeable options, and nobody should swap a Miura for a Yoshimura on a test count.

What the criterion does establish is the price of that motion in qualification, and that the price rises with fineness. A programme choosing how finely to fold a Miura is choosing its test campaign too, and the campaign grows faster than the compaction. A pattern with the Miura’s motion and a settling hinges-per-layer ratio would be worth a great deal, and whether one exists is a question about crease patterns that no design practice currently asks.

Read against the families, the table says something the hinges-per-layer ratio hides. At eight cells a side the Miura has 112 hinges against the waterbomb’s 312, so its whole test campaign is the smaller of the two — 33,385 successes against 92,999 — and a programme that needed only a Miura’s compaction would pay less. It is per layer, at the compaction each pattern reaches, that the order reverses: the waterbomb’s 312 hinges buy 127 layers and the Miura’s 112 buy 20. The criterion is right to divide, because compaction is what the tests are for; but it is a ratio, and a mission that needs less compaction than a pattern’s maximum can take a coarser version of any pattern and pay less in total.

What a hinge count leaves out of a crease count #

The earlier budget counted creases and this counts hinges, and on a printed pattern they are close enough to use interchangeably. On a family they diverge in a useful way.

A long crease that runs straight across a Miura from edge to edge is several hinges, one between each pair of vertices on it, and each can stick independently. The hinge count grows with the number of vertices a crease passes through, which is why a finer pattern’s hinges grow as the square of its cell count even though its drawn lines grow only linearly. How much line is on the paper priced patterns by length and noted that length and count rank them differently; hinges are a third measure again, and they are the one reliability is charged on.

That distinction also bears on where a test campaign could save. A hinge in the middle of a long straight crease is geometrically identical to its neighbours, made in the same pass and folded by the same motion, so an argument that they fail together — and should be tested together — is stronger there than for hinges at different vertices. A pattern of long straight creases is a pattern whose hinges come in natural batches, and the Miura’s straight family is exactly that, which is a point in its favour the count does not credit.

Divided sheets, and the count again #

Splitting a sheet divided a deployable into modules so a stuck hinge costs only its own. Testing sees that division differently. Each module is a smaller article with fewer hinges, so demonstrating one module’s reliability takes fewer tests, and ten identical modules can share one qualification campaign if they are the same design.

On a Miura, division has a second effect the families figure makes visible. Ten Miura modules each four cells a side have hinges per layer of 2.95 each, where one Miura of the same total area — about thirteen cells a side — would be well above five. Dividing a Miura keeps each part near the flat end of its own curve, which is a reason to divide that has nothing to do with partial failure and everything to do with how its folded state grows.

What a designer would read off it #

The criterion is cheap enough to compute at the start of a design, and it answers three questions a programme usually answers late.

Which pattern, for a fixed compaction. Divide the hinges each candidate needs to reach the required layer count by that count, and the smallest is the pattern cheapest to qualify. On the printed shelf that is a base rather than a tessellation, which says the criterion alone would never have produced the structures folding that gets built lists — those were chosen for motion and area, and the criterion prices what the choice cost.

How fine, for a chosen pattern. On a settling family, finer is free in tests per layer and the total campaign grows only with the compaction bought. On the Miura, finer is dearer per layer at every step, so the fineness that minimises tests for a required compaction is the coarsest that meets it, and the campaign for any excess compaction is paid at an increasing rate.

Whether to divide. A pattern whose ratio climbs with fineness is a pattern that should be divided into coarser modules, since each module then sits lower on its own curve; a settling pattern gains nothing from division on this criterion and should be divided, if at all, for the partial-failure reasons alone.

None of the three is a statement about which pattern folds best. All three are statements about what folding well costs in confidence, and a programme that computes them before choosing a pattern knows the size of its test campaign before it has cut a sheet.

What the count assumes #

Every hinge must be tested as though it could fail. A programme that can argue some hinges are proven by heritage, or identical to hinges already qualified, needs fewer tests, and the argument would favour patterns with many identical hinges — which all three tessellations are.

Compaction is the mean layer count. A deployable is also constrained by its stowed thickness, its largest dimension and the shape of the package, none of which is the mean layers over the footprint; a pattern that is cheap per layer and awkward to stow is not cheap.

The tests are the simplest statistics. Consecutive successes with no prior, as in the reliability budget. A campaign with a prior shortens every row by the same factor and leaves the ranking alone.

And the flat folded state is the stowed state. Real deployables are stowed partly folded, with clearances for thickness, and their layer counts differ from the ideal flat fold’s.

What the table cannot show #

It does not include how hinges fail. A long crease is several hinges in the count and may fail as one unit or as several, and a short crease between two vertices may be stiffer or weaker than a long one. The count treats every segment as an equal, independent chance of failure.

It counts every hinge the same regardless of where it sits. A hinge at a sheet’s rim carries a free flap and one in the middle carries the sheet on both sides, and their loads, and so their chances of sticking, differ. It stops at eight cells a side. The Miura’s rise is measured at five sizes and looks linear; the claim that it has no ceiling is an extrapolation from how its folded footprint grows, which is itself measured only to eight.

And it compares only three families. The twist tessellations, the tapered corrugation and the many patterns with no printed version may behave like the Miura or like the Yoshimura, and which property of a pattern decides it is argued here from the footprint rather than established.

Still open: a Miura whose ratio settles #

The families figure suggests a design target. The Miura’s hinges per layer grow because its footprint grows with its rows, and its footprint grows because each row of the zigzag is offset from the last. A Miura variant whose rows stack onto one footprint — by alternating the offset, or by tapering the cells so the zigzag closes on itself — would keep its single degree of freedom and might have a settling ratio.

Whether such a variant exists among rigidly foldable quadrilateral meshes is a search the same mesh solver could make: generate meshes that fold with one freedom, fold them flat at several sizes, and keep the ones whose hinges per layer level off. The family the Miura belongs to describes that space of meshes from the inside, and it is large enough that the answer is not obvious either way.

Sideways from here, the criterion measures a cost that also falls on the folder. Every hinge is a crease that has to be laid in accurately, and what a second deployment costs found repeated use punishing every hinge equally. A pattern with fewer hinges per layer is cheaper to test, to fold and to wear out, and the three costs point the same way even though each has so far been priced on a different measure.

The habit worth carrying is about what a cost is charged on. Before ranking designs by efficiency, find the unit each cost is actually paid in. Folding is paid in length, testing in hinges and packing in layers; a ratio of any two ranks the same designs differently, and the ranking that matters is the one whose numerator is the bill that is largest.