A wing that folds into nothing

A beetle’s hindwing is longer than the beetle. It lives folded under a rigid case that covers only part of the animal’s back, it comes out in a fraction of a second, and it goes back in afterwards — which is the harder half and the one nothing engineered does routinely.

The number that makes this remarkable is a ratio: packed area against deployed area. This essay computes that ratio for four folding geometries, from each one’s own parameters, and the useful result is not any single number but how far apart they are.

What is being computed, and what is not #

The bar chart above is not a measurement of any animal. This needs saying at the start because a chart of packing fractions beside a discussion of beetles reads as data unless it is prevented from doing so.

Each row is a geometry with stated parameters, and its packed fraction is arithmetic on those parameters. A corrugation at a given zigzag angle collapses its span by the sine of that angle. A Miura collapses in both directions from the same single parameter, so its fraction is that sine squared. A fan of a given number of sectors and a roll of a given number of turns each have their own one-line expression.

The generator asserts that the geometries it was asked to compare differ by more than half again, and refuses to draw if they do not — because a comparison whose entries are all within a few percent of each other is a figure arguing that the choice does not matter, dressed as a figure arguing that it does.

Why squaring is the whole story #

The single most useful fact in the chart is that the Miura beats the corrugation by a factor of two and a half, and the reason is not that it is a better fold.

A corrugation collapses across its creases and does nothing along them. Its packed footprint is one shrinking length times one unchanged length. A Miura collapses in both directions at once, from the same one parameter, so its footprint is the product of two shrinking lengths — the same factor, squared.

Squaring a fraction below one always helps, and it helps most when the fraction is already small. That single observation explains why two-directional folds dominate every application where volume is the binding constraint, and why one-directional folds survive wherever making the pattern accurately is harder than making it compact.

The roll, which wins and is not used #

The roll packs best of the four and is the least useful of them, which is worth understanding because the reason generalises.

A roll has no creases at all. It has a continuously curved sheet, and its packed radius falls with the number of turns for as long as the material tolerates the curvature. It is beaten only by its own bend radius.

What a roll does not have is a flat state it returns to cleanly, or a mechanism. Unrolling requires either something to pull the free end or a stored curvature that drives it, and the sheet passes through every intermediate radius on the way, which means every part of it is doing something different at any moment.

That is the general shape of the trap. A packing ratio ranks geometries on the one axis that is easiest to compute and is almost never the axis a design is decided on. The other axes — how it opens, what drives it, whether it returns, how accurately the pattern must be made — are the ones that eliminate candidates, and none of them appear in the chart.

What an insect actually needs, listed #

Setting the numbers aside, the requirements on a hindwing fold are unusually demanding, and the list explains why the answer is not simply “use the best packer”.

It has to pack under a rigid case with a fixed footprint. It has to open in a small fraction of a second, and close again afterwards. It has to survive doing that thousands of times without the creases failing. It has to be stiff when open, in bending, against aerodynamic loads. And it has to do all of that with actuation only at the base — there is no muscle out along the wing, which is the next rung and the sharpest constraint of the five.

That last one is what removes the roll. A roll’s motion has to be driven along its whole length or from a free end that is inside the packed state, and neither is available.

The list is also the answer to a question the chart would otherwise leave open, which is why an animal does not simply use the best number. It is not using a worse fold because biology is constrained to be clumsy; it is solving a problem with five requirements in which the packing fraction is one, and a candidate that wins on one and fails another is not a candidate at all. The geometries clearing all five are a much smaller set than the geometries that pack well, and the intersection is where the plain folds live.

That is the same conclusion the deployables anybody actually builds arrive at from the engineering side, and the two arriving there independently is worth noticing. The patterns that get flown are not the tightest packers either.

The corrugation’s compensation #

The plainest fold packs worst and has a property none of the others has, which is why it is everywhere in wings as well as in leaves.

A corrugated sheet is enormously stiffer in bending than a flat one of the same material — the corrugations act as a set of parallel beams — and the stiffening is in exactly the direction a wing needs it. So a fold that was made to pack the wing also stiffens it when open, which is one component doing two jobs.

This site does not compute bending stiffness and will not: what a section does with a material’s properties is another site’s subject entirely. What is worth noting here is only that the geometric and structural functions coincide, which is a fact about the fold rather than about the material.

The number a packing ratio hides #

There is a specific way a packing fraction misleads, and it is worth making explicit because the chart invites it.

The fraction is an area ratio in a model with no thickness. Real packed material has a thickness, the packed stack is a solid rather than a footprint, and the quantity that matters to a beetle is a volume. A geometry that halves the footprint by doubling the number of layers has not halved the volume — it has kept it the same, and made the stack twice as deep.

That correction is not small at the fold counts these geometries reach, and it has a name in this field’s arithmetic: it is the same term that gives the bud problem its interior optimum and it turns an unbounded improvement into a bounded one.

Two of the four do not improve with folding #

The four expressions are worth reading for what their parameters are, because two of them are not the parameter the chart’s row labels suggest.

A corrugation’s packed span is its length times the sine of the fold angle, and the fold count does not appear. Eight folds or eighty, at the same angle, give the same packed fraction: each panel contributes its own length times the sine, and the total is the whole sheet times the sine. The Miura is the same statement squared, and the count is absent there too.

A fan’s fraction is two over the number of sectors, and a roll’s is one over the number of turns. Both improve as the count rises.

So two of the four are set by an angle and two by a count, and comparing all four “at eight” is comparing two geometries at a parameter they do not have. That is not a defect in the arithmetic — every number is right — but it does mean the chart’s four bars are not four points on one axis.

Which makes the spread a depth, not an efficiency #

There is a fair common axis available, and putting the four on it collapses the comparison entirely.

Every point of the sheet lands somewhere in the packed footprint, so the packed area times the mean number of layers over it equals the deployed area. The mean depth is therefore the reciprocal of the packing fraction, exactly.

Run the four through it. The corrugation at 0.408 packs to 2.45 layers. The fan at 0.250 to four. The Miura at 0.166 to six. The roll at 0.125 to eight.

The packing fraction and the layer count are the same number written twice. So the chart’s factor of three between best and worst is a factor of three in stack depth, and there is no sense in which the roll is more efficient than the corrugation — it is simply folded more.

That reframes what the comparison is for, and improves it. The question is not which geometry packs best, since any of them packs as well as its depth allows. It is which reaches a given depth for the least cost in creases, mechanism and reversibility, and on that question the chart is silent while the five requirements above are not.

It also explains why the thickness correction bites so hard. Depth is exactly what a real stack pays for, and a packing fraction is a depth in disguise — so a figure that ignores thickness is ignoring the one quantity its own headline number is measuring.

Where the four numbers come from #

It is worth opening the arithmetic, because a number in a chart with no derivation beside it is the thing this site exists not to publish.

The corrugation is a set of parallel panels hinged along their edges. Folded to a half-angle, each panel’s projection onto the packed direction is its length times the sine of that angle, and the packed span is the sum of those. So the packed fraction is the sine, and at the angle used it is 0.408.

The Miura is the same collapse happening in two directions from one parameter, so the packed area is the product of two such factors and the fraction is the sine squared: 0.166. The generator builds an actual Miura to get there, checks its interior vertex count, and reports the number of vertices in the row’s note — which is how a reader can tell the figure computed a pattern rather than evaluated a formula.

The fan is a set of sectors about a single point. Packed, the sectors stack, and the packed footprint is essentially two sectors’ worth of the disc: two divided by the number of sectors, or 0.250 at eight.

The roll is a sheet wound into a cylinder. Its packed footprint is the cylinder’s cross-section, and with the number of turns as the parameter the fraction goes as one over the count: 0.125 at eight turns.

All four are stated as functions of a parameter, and every one of them improves without bound as its parameter is pushed. That is the tell that the model is missing something, and what it is missing is the thickness.

The fold that has to close as well as open #

Every deployable this site has looked at opens once. A wing does the round trip, and the asymmetry is worth naming because it eliminates designs that pass every other test.

Opening can be driven by stored energy: something wound up, something elastic, something that releases. Closing cannot use the same trick, because the energy has been spent. So a mechanism that opens by release needs a second, separate mechanism to reset it — or it needs the motion to be reversible under the same actuator, which is a much stronger condition on the pattern.

That comparison is the honest one to end the packing discussion on. The engineered deployables reach packing fractions comparable to the geometries above and solve a strictly easier problem, because they get to be one-shot. An insect’s fold has to be a mechanism in both directions, and that requirement is not visible in any packing number anywhere.

Three or four geometries is not the space #

The comparison is over four named geometries, which is a convenience rather than a claim about what exists.

The real space of folding geometries is enormous, and the ones an insect uses are not any of these four. Published descriptions of beetle hindwing folds show combinations — a corrugation along the wing, with transverse folds, with regions that bend rather than crease, and with a distinct pattern near the base where the actuation happens. Nothing in this repository models such a pattern and nothing here should be read as though it did.

What the four do supply is the range: a factor of three between plain and clever, on geometry alone, before any material or mechanism is considered. That range is the useful output, because it says how much is available to be won by choosing the pattern well.

It also sets a scale for what other choices are worth. A factor of three is a lot, and it means the pattern is the first thing to get right — but it is also a bounded amount, which puts an upper limit on what a cleverer geometry can rescue. A design that is a factor of ten away from fitting is not going to be saved by choosing a better fold, and knowing that before starting is worth the arithmetic.

The same reasoning runs the other way and is how this kind of number is best used. The design field of this site spends its effort on exactly this question in a different setting — how much of a square a set of flaps can be made to use — and the answers there have the same character: a computable bound, a range between naive and good, and a gap between the best known arrangement and any proof that it is best.

What the picture cannot show #

The chart shows a packed fraction and nothing about the motion between the two states.

Two geometries with the same fraction can have completely different paths, and the path is where the requirements live. Whether a state is reachable at all, whether the motion is monotone, whether it needs a sequence — none of that is visible in a bar. A reader who takes only the numbers away has taken the least decision-relevant part of the figure.

The chart also treats each geometry at one set of parameters. Every one of them has a parameter that can be pushed — the zigzag angle, the number of sectors, the number of turns — and pushing it improves the fraction without limit in the idealised model. The figure fixes the parameters and says so; the bound that stops the pushing is a thickness and it is not in this figure.

The idealisation, named #

Zero thickness, zero hinge width, panels that are perfectly flat, and creases that are lines.

The most consequential of those here is the hinge width, because a wing’s folds are not creases in the paper sense at all. They are compliant regions — patches of thinner or differently structured cuticle that bend rather than crease — and a compliant region has a length along the wing that is not available for packing. That is the third rung of this ladder and it is the correction that makes these numbers optimistic rather than merely approximate.

The second is that a wing is not a uniform sheet. It has veins, which are stiff, and membrane, which is not, and the fold pattern is laid out with respect to the veins rather than freely. A geometry chosen without that constraint is solving an easier problem than the animal is.

Where this ladder goes next #

The packing number says what is achievable and says nothing about how it is achieved.

The next rung is about the achieving, and its answer is a count rather than a ratio: one degree of freedom means one actuator, which is what makes a deployment possible with muscles only at the base. It is the same property that makes the Miura useful to spacecraft and the same property that lets a leaf be opened by growth, arriving for the third time in this field from a third direction.

After that, the correction that makes all of these numbers optimistic: nothing in a body folds on a line, and what a hinge with a radius costs is computable and larger than most estimates allow for.