Origami & Folding
what one sheet can do
Concept

Deployment — where it appears

The motion from a packed state to an open one. It needs a continuous path between the two and something able to traverse it, which is a stronger requirement than the two end states existing.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

00.20.40.60.811.21.402468fold angle (radians)exposed spanθ = 0.06θ = 0.75θ = 1.449 panels · one parameter · the span rises at every step, so nothing has to reverse

Opening with nothing to pull

A leaf is not opened by a hand, a hinge or a motor. It opens because it keeps growing — which puts a condition on the pattern that no folder ever has to satisfy, because a person can always push.

biology · Leaf folding geometrypacks tocorrugation8 panels at 0.42 rad40.8% — 2.5× smallerMiura6 × 4, 15 interior vertices16.6% — 6.0× smallerfan8 sectors about one point25.0% — 4.0× smallerroll8 turns12.5% — 8.0× smallerpacked area as a fraction of deployed, computed from each geometry — not measured from any animal

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.

biology · Insect wings geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each11Miura, 5 × 412 interior vertices, still one freedom11parallel corrugationno interior vertex to couple1112 vertices, uncoupledwhat a pattern costs when nothing constrains it1212or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all

No motor in the fold

An insect's wing has muscles at its base and nothing out along its length, so the pattern has to carry the deployment by itself. The condition that makes that possible is a count: one degree of freedom means one number determines every panel, which means one thing has to pull.

biology · Insect wings patternhow much smaller it foldscreasingper sheet-widthpreliminary8.0 layers, 8 at the deepest8.0×4.81.65×footprint × depth = 1.004 of the sheetmiura8.1 layers, 16 at the deepest8.1×6.21.31×footprint × depth = 1.001 of the sheetyoshimura32.0 layers, 36 at the deepest32.0×11.82.71×footprint × depth = 1.000 of the sheetwaterbomb31.6 layers, 32 at the deepest31.8×14.32.22×footprint × depth = 0.992 of the sheettwist3.0 layers, 9 at the deepest3.0×4.70.64×footprint × depth = 0.995 of the sheetthe shrinkage is the pattern's, not the paper's — nothing here knows what the sheet is made of

What a corrugation costs

Every tessellation this repository can fold, measured the same way: how much smaller it gets, how deep the stack becomes, and how much creasing was needed to buy it. The last column is the one nobody quotes and the one a folder feels.

tessellation · Shrinkage 24682468times shorter across the sheettimes shorter down itaccordionMiuraleaf corrugationYoshimurasquare twistwhat each one doesaccordion8.00× across, 1.00× downMiura2.73× across, 1.75× downleaf corrugation3.57× across, 1.22× downYoshimura4.00× across, 4.00× downsquare twist1.52× across, 1.52× downthe dashed line is a directionthe fold never touchesone direction untouched: 1 of 5 · equal both ways: 2 · neither: 2the single number a corrugation is usually quoted by is these two multiplied together

A shrink is two numbers

How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.

tessellation · Shrinkage parallelstraight creases spread 0.0°flat, they spread 0.0°fan 5.2°straight creases spread 25.5°flat, they spread 30.9°fan 9.2°straight creases spread 46.6°flat, they spread 55.0°fan 13.8°straight creases spread 72.4°flat, they spread 82.5°

The corrugation that curves

A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.

tessellation · Miura Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubepackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable

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.

rigid · Deployables 00.20.40.60.81050100150span held, as a share of the sheetforce needed to hold it there4 folds8 folds16 foldsthe pale line under each: what it takes at flat, 4kc ⁄ Ssheet 10, hinge radius 0.05 · the corrugation springs open, and this is what has to stop it at each state

How far open is a question about the grip

A corrugation of hinges that rest flat is loaded when it is shut, so it opens by itself and the force in the held-state calculation is a restraint rather than a drive. Followed from shut to flat that restraint only ever falls, and by exactly π over two — so every partly open state a structure can occupy is squeezed into a band a third wide, and a grip that weakens by a third leaves the sheet nine tenths open.

biology · Convergence folding, priced by how often it has to happena crease of radius ρ strains its outer fibre by t ⁄ 2ρ, and a material takes less strain the more often it is askedcycles it must survivesmallest hinge radiusbest fold countpacking it reachesagainst once10.05087.643.8—100.15827.713.93× worse1000.5008.7604.38010× worse1,0001.5812.7701.38532× worsesheet 10, thickness 0.1, fatigue exponent 0.5 · the radius goes as N^0.5 and the packing as N^−0.5

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.

rigid · Deployables what a finer pattern costs in confidenceeach hinge working 0.999 of the time, against a target of 99 per cent for the whole deploymenthingesthe system openseach hinge needssuccesses to show itand the article itself899.2%0.9987442,385cannot be tested2497.6%0.9995817,154cannot be tested6094.2%0.99983317,885cannot be tested12088.7%0.99991635,769cannot be tested30074.1%0.99996689,422cannot be testedr consecutive successes put a 95 per cent lower bound of 0.05^(1⁄r) on a hinge, and a flight article deploys once

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.

rigid · Deployables one freedom, or severala stuck hinge or a failed actuator loses its own module and nothing elsemodulesall of it opensshare expectedat least 90%at least 75%173.3%73.3%73.3%73.3%272.6%85.2%72.6%72.6%570.4%93.2%70.4%96.0%1067.0%96.1%94.4%99.4%2060.6%97.5%98.7%100.0%5044.8%98.4%100.0%100.0%10027.1%98.7%100.0%100.0%300 hinges at 0.999 each, split evenly · each module's actuator works 0.99 of the time

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.

rigid · Deployables tests needed for each layer of compaction, pattern by patterna test campaign grows with the hinge count, so the pattern with fewest hinges per layer is cheapest to trustpatternhingeslayersper layertests per layerby testsby lengthThe preliminary base88.01.002981st3rdThe Yoshimura pattern8660.01.434272nd1stThe waterbomb tessellation7631.52.417203rd2ndThe square twist123.03.981,1854th7thThe Miura fold389.24.121,2285th4thFold and cut — the triangle61.25.051,5056th6thThe tapered corrugation458.35.401,6117th5thThe hexagon twist183.35.541,6508th8thtests are consecutive successes demonstrating 99 per cent for the whole pattern at 95 per cent confidence

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.

rigid · Deployables four Miuras at a slant of 0.35, folded flat and drawn at one scalethe first two differ only in rows, the second two only in columns4 columns by 2 rows0.939 by 2.4364.07 layers deep4 columns by 6 rows0.939 by 2.43612.21 layers deep2 columns by 4 rows0.939 by 1.7506.04 layers deep6 columns by 4 rows0.939 by 3.1229.23 layers deep

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.

rigid · Deployables a 16 by 4 Miura cut along its zigzags into modules, folded and stackedslant 0.35; each module folds to its own box and the stacked modules share itmodulescolumns eachbox lengthlayers, stackedhingeshinges a layeractuators1166.55111.01089.791283.80819.71045.292442.43632.6962.954821.75048.3801.6681611.40764.0480.7516a cut removes one zigzag's hinges and adds one module to drive; the box and the hinges a layer both fall

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.

rigid · Deployables 00.511.520.60.70.80.91failures crowded into one deployment in 10^xchance10 modules, share expected10 modules, at least 90%one freedom10 modules, all of it300 hinges at 0.999 each on average, actuators 0.99 · to the right, the same failures crowd into fewer deployments

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.

rigid · Deployables a 10 by 10 Miura at a slant of 20.1°, with two of its places markedzigzag line 3: 10 hinges, one a rowthey all fold onto one segmentcolumn 3, odd lines: 5 straight hingesthey fold onto one segment toocut between columns into 5 modules

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.

rigid · Deployables

Named alongside it

The objects these essays reach for when they reach for this one.

Trade-offCorrugationPacking ratioReliabilityDegrees of freedomMiuraFootprintHingeActuationCrease radiusExpected-valueLayer count

All concepts