Trade-off — where it appears
Named by 31 essays across 7 fields — each of them below, with the objects they name alongside 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.
Spelling a tree on a grid
Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.
Spending the cheap paper
A hole makes a sheet cheaper by the square inch, because a flap against its rim claims only half a disc. Whether a design can spend that was left open, because no packing search here could express a region that is not paper. It can now, and the mechanism turns out not to be the discount at all: across a hole, two flaps may overlap, and two tips three fifths of a sheet apart carry flaps of 0.500 rather than 0.300.
How much line is on the paper
A crease pattern is described by its creases: how many, at what angles, in what arrangement. What a folder spends is length. The Yoshimura this site prints has eighty-six creases and 2,380 millimetres of folding on a sheet seventeen centimetres across; the Miura has thirty-eight creases and 1,049, and the fold-and-cut triangle has six and 258 — and the two counts do not rank the eight printed patterns the same way.
Fourth of eight, and still not chosen for it
A deployable is sold on compaction: large in use, small in transit. Measured, the pattern that actually gets built converts folding into compaction at 0.67 sheet-widths of crease per layer, which is fourth of the eight printed patterns — nearly three times worse than the Yoshimura, which nobody deploys, and nearly three times better than the hexagon twist, which nobody deploys either. The ranking does not pick out the pattern that flew from anywhere on the shelf, and that is the finding.
How deep is a crossing
A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.
A notch is not a hole
Remove the same rectangle of paper from the middle of a square and from its edge, and the two sheets are not worth the same. The hole is cheaper paper at every flap length measured — 4.71% cheaper than a plain square of equal area against the notch's 2.58% — because what a cut is worth is rim with paper on both sides of it, and a notch spends one of its four sides on an edge the sheet already had.
Stopping is cheaper than finishing
A search whose cost varies by a factor of two hundred with nothing but the order of its guesses should not be waited out. Give up after a hundred steps, reseed and start again, and the whole job costs five hundred and twelve steps in expectation; run each attempt to twenty thousand and it costs sixteen thousand two hundred and ninety-one. Patience is thirty-two times more expensive than impatience.
Four materials, four optima
The convergence argument gets its pattern and stops there. A hinge has a radius, the radius takes a fixed length of surface out of every fold, and the fold count that gets the most packing out of a sheet is inversely proportional to it — so a leaf, a wing, a gut lining and a metal array agreeing on a corrugation still disagree by an order of magnitude about how many creases to put in one.
The surface has to be supplied
The curve that turns over does so because the sheet's own thickness fills the box it is folding into. A surface in a body has to be reached as well as fitted, and the channel that reaches it takes depth out of the same box on exactly the same terms — so the best fold count and the surface it delivers both fall by the ratio of the sheet's thickness to the sheet and its supply together.
Two surfaces in one box
A body folds several surfaces into one volume and each of them does a different job. Dividing the depth between them looks like a fair split costing nothing overall, and it is not: the area a single surface can reach goes as the square of the depth it has, so m surfaces sharing a depth reach a total of exactly one m-th of what one of them would have reached alone.
A nest pays four a level
A corrugation folded inside the panels of another looks like the arrangement that multiplies surface rather than dividing it. It multiplies the factors and divides the depths, and the depths cancel: a packed level can hold at most its depth over four times what it folds, the thing it folds is as thick as the depth the level below was given, and so a nest of L levels reaches at most the box over 4ᴸ sheet thicknesses. One level with the whole box beats any nest of two by exactly four.
The fold count sets the spring
A corrugation sweeps the same span at every fold count, and the count decides only how much room the zigzag needs while it does it — which was counted as a gain with nothing pushing back. Something does push back. Every hinge is a spring, a finer corrugation has proportionally more of them, and the force to hold a given span rises exactly as the clearance falls: the product of the two is the same number whatever the count, and at each material's own best count the spring goes as one over the square of the hinge radius.
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.
Standing up beats lying down by eight
A level that fills its clearance with plies lying flat makes every ply share the clearance, and the sharing costs a factor of four. A level that fills the same clearance with walls standing on the base gives every member the whole height and charges them only for footing. The ratio is exactly eight, at every clearance and every thickness — and it is the difference between a cost charged against depth and a cost charged against the space beside it.
The channel grows with what it feeds
A comb of standing walls beats a stack of plies by eight because its members do not share the clearance that pays them. Supply takes that back, and asymmetrically: a wall's channel has to be sized for the surface the wall carries, so it grows with the wall's height and is charged against the pitch, while a ply's channel is a constant charged against the clearance. The comb then saturates at twice the reciprocal of the channel's share, the stack does not saturate at all, and the two cross at a clearance the model gives in closed form.
The price of a limb is not its length
Lengthening an edge of a subject's tree costs the design some of its scale, and the amount can be measured by re-solving the arrangement. It is not proportional to the edge, and it is not the body that is dearest. On a bird whose wings are twice its legs, the head — nine tenths of a unit against the wings' one and six — costs three times what a wing costs, and two edges of a lizard cost nothing at all.
Crowding outward costs almost nothing
Placing the tuck starts for equal error crowds them toward the rim, where the gathered paper is already at its thickest, and the obvious worry is that the accuracy is bought with depth. Measured, it is not: on a hemisphere the crowded placement's deepest start sits in 3.37 sheets against the even placement's 3.32, because a start is three sheets of its own and the gathering beneath it is only one and a half.
Three answers, one count
Seams, curved creases and a few per cent of stretch are the three ways round the sphere, and a tuck is a fourth. All four dispose of one quantity — the excess circumference a flat disc has over the sphere's circle — and all four dispose of it by dividing the circle. So the number of divisions needed is the same whichever answer is chosen: nineteen for a hemisphere in a material that gives two per cent, eight at five, four at ten. What differs is what a division costs, and one of the four runs out.
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.
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.
Gauss's polygon is the expensive one
Which regular polygons a fold reaches is a condition on the factorisation of Euler's totient, and every polygon that passes it also has a height — the number of extension steps the shortest tower to it takes. Read that column instead of the verdict and the field inverts: the heptagon, which no compass reaches, costs one step; the seventeen-sided polygon that made Gauss famous costs three, the most on the list; and the polygons a compass finds easy are the ones a folder pays most for.
Eighty layers and the sheet decides the rest
Five of these essays each bound one thing and none of them bounds a design. Put together they close. The crease floor fixes the thinnest usable paper at about a fibre and a half; that paper's stack runs out at eighty layers; the largest sheet two arms can make falls away as the square of the finished size. The region under both is every model anybody can fold, and it has a ceiling at eighty layers and a corner at about a hand's width — above which the paper is no longer the limit and the vat is.
A price holds until the arrangement moves
Every edge of a subject's tree has a price — the scale lost per unit of extra length — and the obvious use of a price list is to spend a fixed total of limb where it is cheapest. Done a tenth of a unit at a time, re-pricing at every step, it works and then stops: the bird's scale rises 6.3 per cent in four steps and no further. But the prices do not hold while it happens. The bird's free tail stops being free after the first tenth, and its legs nearly treble in price without being touched. The lizard's prices hold for four steps, because its arrangement keeps the same three pairs at their limit for four steps. A price is a statement about which pairs are at or near their limit, and it lasts as long as they stay there.
Rounding in the cheap direction
A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.
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 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 angle the eight does not know
A comb's members are always drawn standing square to the base, and nothing has asked why. Lean one to an angle and it must be longer to reach the same clearance, which is more surface; it also takes more of the base to stand on, which is fewer members. The two are reciprocal and cancel exactly — the surface a comb holds is the same number from a right angle down to one degree, where each member is fifty-seven times the clearance long and there are two of them where there were a hundred.
In a tube the standing members lose
Members standing across a clearance beat layers lying along it by eight, and every drawing of that argument has a flat base under it. Curve the base into a tube and the ranking inverts: radial fins converge, so the room they need is the room at their tips, and their best arrangement fills exactly half the cross-section. Concentric layers fill all of it. The eight becomes a half, and the half is exact.
The wedge belongs to one length
Radial fins inside a tube reach at best half of what any lining of sheet could hold, because converging fins leave empty wedges behind their tips. Tapering the fins cannot help: the tip already sets the count, and a fin cannot be thinner there than the sheet it is made of. Fins of several lengths can. Counted along the radius they are a staircase under a straight line, and the staircase with m steps is best with its steps equally spaced, where it holds exactly m ⁄ (m + 1) of the ceiling. The factor of two belonged to fins of one length, not to fins.
The deepest point pays for the paper
A folded design uses its sheet according to its mean layer count and its paper according to its deepest point, and the ratio of the two is a property of the crease pattern. Measured on every printed pattern it runs from exactly one to nearly six — and it does not split tessellations from bases, as expected. It splits patterns whose every panel lies over every point from patterns that keep a footprint with structure in it. The ratio moves the corner of the substrate map by its square root, so the fold-and-cut triangle can reach the paper's eighty layers at 326 millimetres where the preliminary base must stop at 134.
Named alongside it
The objects these essays reach for when they reach for this one.
Packing ratioScalingSurface in a volumeCrease radiusLayer countThicknessConstraintDeploymentTree methodCorrugationReliabilitySurface area