What folding is used for
deployables is one function. Everything below was drawn by it when the site was
last published, at settings 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.
With nothing specified
view: "walk", show: "folded"
which: [solar, airbag, stent, starshade, map]
view: "places", show: "pattern"
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- folded flat, the 4 by 2 Miura at a slant of 0.35 fits a box 0.9394 by 2.4361, which is cos a by sec a plus 4 sin a ×34
- 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
- a tether that passes a failure on erodes nearly-everything long before it erodes the average: the split sheet's chance of at least 90 per cent falls to the one freedom's 73.3 at a contagion of 0.57, its expected share only at 0.93 ×1
- across 8 slants, a place reaches two column modules exactly when 2 ⁄ sin 2a is an even whole number — 45°, 15°, 9.7°, 7.2° — and never at an odd one or between ×1
- and at a slant of 45°, where 2 ⁄ sin 2a is 2, column 3's straight hinges on the odd lines share their place with column 5's on the even lines ×1
- and column 3's straight hinges on the odd lines share a place with each other and with nothing in any other column ×1
- and for every requirement short of the whole area a sheet of several freedoms beats one freedom — 90 per cent is best met with 20 modules, 98.7% against 73.3% ×1
- and the chance that all 10 modules open stays exactly 0.914 times the one freedom's at every clustering, because both need every hinge ×1
- and the chance that all 10 open is 67.0 per cent at every contagion, because contagion needs a failure to start from ×1
- and the rows' losses are nearly all total: a jam anywhere stops all 5, which happens 1.88 per cent of the time ×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
- as failures move into the places, the cut between columns keeps ahead of the cut between rows at every share, and with every failure a place's its lead over one mechanism is 2.34 points against the rows' 0.97 ×1
- as hinge failures crowd into fewer deployments, the lead of 10 separately driven modules over one freedom in the share expected to open falls — 22.7, 16.0, 6.8, 1.1, 0.0 points, from failures arriving independently to failures arriving in one deployment in 100 ×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
- clustering lifts a large module far more than a small one: with failures crowded into one deployment in 100, 300 hinges all work 34 per cent more often than independent hinges would, and 30 hinges 2.1 per cent more often ×1
- cut between columns, no place reaches two modules except at the slants where 2 ⁄ sin 2a is even, and there none reaches more than two ×1
- cut between rows, every one of the 19 places that keeps a hinge reaches all 5 modules ×1
- every application charted is one this figure has a packed fraction for ×1
- every bar fits under 0.75 at this contagion ×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
- folded flat, the 10 hinges of zigzag line 3 — one in every row — lie on one segment and nothing else does ×1
- of 5 Miuras with the same 64 cells, the one 2 columns by 32 rows needs 1.94 hinges a layer and the one 32 by 2 needs 16.14 ×1
- one mechanism at 20.1°: 29 places, 0 reaching two or more modules; 5 modules of rows at 20.1°: 19 places, 19 reaching two or more modules; 5 modules of columns at 20.1°: 25 places, 0 reaching two or more modules; 5 modules of columns at 15°: 19 places, 6 reaching two or more modules; 5 modules of columns at 45°: 17 places, 8 reaching two or more modules ×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
- so hinges per layer settle along the rows, at 3.19, 3.24, 3.28 on the last three ×1
- 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 — and raises the share of it expected to open — 0.733, 0.852, 0.932, 0.961, 0.975 ×1
- the extra hinges a layer each column costs is greatest near a slant of 45 degrees, where the walk along the strip adds most footprint against its width ×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 slant moves none of the column cut's expected share — every module touches 5 places at every slant — only how its losses arrive ×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
- with 4 columns the folded box is the same at every row count from 2 to 12: every row lands on the one before ×1
- with 4 rows the hinges per layer rise at every added column, 1.66, 2.33, 2.95, 4.12, 5.29, 6.43, 7.58 ×1
- with every failure a place's, the cut between rows loses two or more of its 5 modules 1.98 per cent of the time and the cut between columns 0.22 — the same hinges, the same modules' worth of paper, 9.1 times as often ×1
- with every failure the hinge's own, the two cuts are the same design: 96.27 per cent of the area expected to open either way ×1
- with no contagion the row is 10 independent modules: the share expected to open is 96.1 per cent, each module's own chance ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
A cut along the zigzag needs a tether
A Miura's folded stack walks along itself by the sine of its slant for every column, so a sheet cut along its zigzags into modules folds into boxes a fraction as long and stacks far deeper for the hinges. But the pieces cannot stay joined by a crease: a zigzag crossed by row creases is a line of four-crease vertices, and joined there the modules are the one Miura again. The join has to be a tether and every module its own actuator — and a cut, which removes a zigzag's worth of hinges, pays for itself in complete success only when an actuator fails less often than those hinges together.
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 stack is a shared fate
Hinges that fold onto one segment of a folded stack share whatever happens to that segment. On a Miura every zigzag line puts one hinge from every row in the same place, so a sheet cut between its rows into five modules leaves every place in every module, and one jam anywhere stops all five. Cut between its columns, each place stays inside one module — except at the slants where two over the sine of twice the slant is an even whole number, the very slants that pile the stack evenly, where columns that far apart share places across the cut.
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.
Splitting insures only the failures that come alone
Three hundred hinges split among ten separately driven modules were found to open 96 per cent of their area on average against 73 for the same hinges as one mechanism. That gain rests on every hinge failing on its own. Let the same average failure rate arrive in a few bad deployments instead — a cold soak, a crushed stowage — and the one mechanism gains far more than its modules do, because a shared cause makes three hundred hinges behave like one. At one bad deployment in a hundred the lead is gone. A tether that passes a jam to its neighbour erodes it from the other side, and takes the chance of nearly all the area long before it takes the average. Neither touches the chance that everything opens.
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 rows are free
A Miura's hinges per layer of compaction were found to climb without a ceiling as the pattern is refined, and the footprint was blamed for growing with the sheet. Measured one direction at a time it grows in only one. Every band between two straight creases folds exactly onto the band before it, as an accordion's pleats do, and the folded box does not change at all as they are added. Every band between two zigzags walks the stack a step along itself, the sine of the slant each. So the hinges a layer costs settle as rows are added and climb as columns are, and the same sixty-four cells cost 1.94 hinges a layer laid out long in their rows and 16.14 laid out long in their columns.
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.
All the figures · The rigid folding field · The patterns a reader can fold