What folding is used for
deployables is one function. Everything below came out of it during this
build, at arguments taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
At its defaults
which: [solar, airbag, stent, starshade, map]
which: [solar, airbag, stent, starshade]
which: [solar, starshade, map]
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- the chart carries 5 deployed applications ×4
- and it costs it as a power: the packing falls by 20.0 across that range, which is the cycle range to the power of the fatigue exponent ×3
- a deployment that needs every one of its hinges is p to the n, so the crease count is a reliability budget — at 0.9995 a hinge it runs 99.4%, 98.0%, 93.7%, 81.9% across 12, 40, 130, 400 hinges ×2
- and for every requirement short of the whole area a sheet of several freedoms beats one freedom — 90 per cent is best met with 100 modules, 100.0% against 73.3%; 75 per cent is best met with 100 modules, 100.0% against 73.3% ×2
- and the compaction given up to get there is 14.4 per cent, which buys 16.4 points of deployment probability ×2
- and the testing needed to demonstrate 95 per cent grows with the hinge count — 701, 2,337, 7,593, 23,362 consecutive successes ×2
- counting the chance that every hinge works moves the best fold count from 87.6 to 54.4 — a pattern optimised for compaction alone is folded too finely ×2
- splitting 300 hinges among more separately driven modules lowers the chance that the whole area opens — 0.733, 0.726, 0.704, 0.670, 0.606, 0.448, 0.271 — and raises the share of it expected to open — 0.733, 0.852, 0.932, 0.961, 0.975, 0.984, 0.987 ×2
- a deployment that needs every one of its hinges is p to the n, so the crease count is a reliability budget — at 0.999 a hinge it runs 99.2%, 97.6%, 94.2%, 88.7%, 74.1% across 8, 24, 60, 120, 300 hinges ×1
- and the testing needed to demonstrate 99 per cent grows with the hinge count — 2,385, 7,154, 17,885, 35,769, 89,422 consecutive successes ×1
- at every exponent drawn the packing falls by more than five times across three decades of cycle count — 16× at b = 0.4, 32× at b = 0.5, 63× at b = 0.6 ×1
- every application charted is one this figure has a packed fraction for ×1
- every extra cycle a hinge has to survive costs the structure compaction — 43.8, 13.9, 4.380, 1.385 across 1 to 1,000 cycles ×1
- every extra cycle a hinge has to survive costs the structure compaction — 43.8, 21.2, 10.3, 4.987, 2.417 across 1 to 625 cycles ×1
- every extra cycle a hinge has to survive costs the structure compaction — 43.8, 9.794, 2.190 across 1 to 400 cycles ×1
- every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
- ranking the printed patterns by hinges per layer and by crease length per layer puts 6 of the 8 in different places, so choosing a pattern for fewest tests and choosing it for least folding are different choices ×1
- the Miura's hinges per layer of compaction rise with its fineness, 1.32, 2.20, 2.95, 4.33, 5.69 from 2 cells a side to 8, while the waterbomb's and the Yoshimura's settle — waterbomb 2.27, 2.36, 2.41, 2.47, 2.47; Yoshimura 1.37, 1.39, 1.44, 1.46, 1.47 ×1
- the smallest hinge that survives its whole life grows by more than eight times across three decades of cycle count at every exponent drawn — 16× at b = 0.4, 32× at b = 0.5, 63× at b = 0.6 ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
A property you can dial
Steel has one Poisson's ratio. A Miura-folded sheet has a surface of them, and where on that surface it sits is set by the panel shape and by how far it happens to be folded — which is why it is a mechanism rather than a material.
A wing that folds into nothing
A beetle stows a wing longer than its body under a case a fraction of that length, and the ratio is the whole engineering problem. What a fold achieves is computable from the pattern alone, and the four geometries available are not close to each other.
Folding that gets built
Solar arrays, airbags, stents and starshades. The requirement is always the same — large in use, small in transit, along a path nobody has to trust to chance — and folding is what answers it.
Found before it was designed
Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.
From a shell to a solar array
The Miura fold was published in 1970 and flew on a satellite in 1995. The gap is not ignorance — the pattern was known, understood and available the whole time — and the same twenty-five year lag appears between every folding result and the hardware that uses it.
Getting thickness round a corner
There are half a dozen ways to build a fold in a panel that has depth, and the useful way to arrange them is not by what the cross-section looks like. It is by what each one gives away.
Panels with somewhere to go
Every way of giving a folded panel real thickness costs something. Tachi's offset-panel technique costs the least interesting thing there is — it stops the panels being a surface, and leaves the hinges exactly where the zero-thickness pattern put them.
Paper through paper
Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.
Splitting a sheet buys area, not certainty
A folded deployable with one freedom needs every hinge and its one actuator, and three hundred hinges at 0.999 each open all the way 73 per cent of the time. Split the same hinges among ten separately driven modules and a stuck hinge costs only its own module: the share of the area expected to open rises to 96 per cent, and the chance of at least nine tenths of it rises to 94. The chance of all of it falls, to 67 per cent, because every freedom added is an actuator added. So freedoms, actuators and reliability trade in a definite way: one freedom is the best design only for a mission that is worthless without its whole area, and for any mission that can live with less, several freedoms win by a margin that no improvement in the hinges matches.
The crease count is a reliability budget
A deployment that needs every hinge to work is the hinge reliability raised to the crease count, so the fineness that buys compaction spends the probability of getting it. At a thousandth of a chance of a hinge failing, sixty hinges give a 94 per cent deployment and three hundred give 74. The fold count that maximises expected compaction is well below the one that maximises compaction — and demonstrating the result takes tens of thousands of successful tests on an article that itself deploys once.
The pattern cheapest to trust
Demonstrating that a one-shot deployment will open takes a number of successful tests proportional to its hinge count, so the pattern that needs fewest tests for what it delivers is the one with fewest hinges per layer of compaction. That criterion is a count nobody computes, and computed on the printed shelf it ranks the patterns differently from crease length per layer: the preliminary base is first, at exactly one hinge per layer, and the square twist rises from seventh to fourth. As patterns are refined the difference sharpens. The waterbomb settles at 2.47 hinges a layer and the Yoshimura at 1.47, but the Miura climbs without levelling — 1.32 at two cells a side, 5.69 at eight — so every finer Miura costs more tests for each layer it adds, and the pattern that gets built is the only one of the three that gets dearer to trust as it gets finer.
The tube that gets built
Every folded structure that leaves a laboratory is a sheet joined to itself — a boom, a stent, a bellows, an airbag, a packed antenna. The mathematics has been done on flat rectangles for the whole history of the subject, and the object is a cylinder, which is a different sheet with different counts and a condition the rectangle does not have.
What a second deployment costs
Every folded structure this field builds deploys once. The reason is a power law: a hinge asked to survive more cycles cannot be as sharp, a blunter hinge takes more surface out of the sheet, and the fold count that packs best falls as the cycle count to a fatigue exponent. A structure required to work a thousand times packs thirty times worse than one required to work once, and the exponent decides how fast rather than whether.
Every generator · The rigid folding field · The patterns a reader can fold