The bud chooses the pattern

The previous rung left a pattern with a free parameter. A corrugation folds flat for any number of folds, and the geometry has nothing to say about which number to use.

Something has to say. In a leaf it is not the leaf: it is the bud, and the bud is a cylinder of a given radius that the packed leaf has to fit inside. That constraint comes from outside the sheet entirely, and it is sharp enough to leave a window of two or three workable counts out of a range that all fold equally well.

The two costs, and why they pull opposite ways #

Take a leaf of a given area and corrugate it into some number of folds. The packed object is a bundle, and the bundle has two dimensions that matter.

Its width is the width of a single strip, which is the leaf’s area divided by the number of folds and by the leaf’s length. More folds means narrower strips, so this cost falls as the count rises.

Its thickness is the number of layers times the thickness of one. More folds means more layers, so this cost rises as the count rises.

A bundle that has to fit into a cylinder is bounded by whichever of those is larger — really by the diagonal of the two — and a quantity that is the diagonal of one falling term and one rising term has a minimum strictly between the extremes.

That is the whole argument, and its shape is more interesting than its arithmetic: the optimum exists because the sheet has a thickness. For a sheet with no thickness the stack costs nothing, more folds is always better, and the answer is as many as possible. The idealisation this site spends most of its time in removes the trade entirely.

What the generator refuses to draw #

The figure has three conditions attached and each one is a way the picture could be a lie.

It refuses if no fold count fits the container, because a figure showing only failures argues nothing about selection. It refuses if every count fits, because then the container is not choosing anything and the caption would be describing a constraint that is not binding. And it refuses if the smallest bundle is at either end of the range of counts it was given, because a minimum at an endpoint is the signature of a range that was drawn too narrow to contain the phenomenon.

That third condition is the one that matters. A trade-off figure whose optimum sits at the edge of the plot is the standard way to draw a trade-off that has not actually been demonstrated, and it is very easy to produce by accident.

The cross-section has a fixed area #

The two costs are described as pulling opposite ways, and there is an identity underneath that says exactly how hard each pulls.

Write $W$ for the leaf’s width across the folds and $t$ for its thickness. Corrugated into $n$ folds, each strip is $W/n$ wide and the stack is $nt$ deep. Multiply those together and the $n$ cancels:

$$\text{width} \times \text{thickness} = W t,$$

whatever the fold count is. The bundle’s cross-section has a fixed area, set by the leaf and not by the pattern, and the fold count chooses only its shape — long and thin at few folds, short and deep at many.

That turns the whole trade into one line of geometry. A rectangle of fixed area fits inside the smallest circle when it is a square, so the tightest bundle is the one whose stack is as deep as its strips are wide. Setting $W/n = nt$ gives

$$n^{*} = \sqrt{W/t},$$

and at that count both dimensions are $\sqrt{Wt}$ and the circumscribing radius is $\sqrt{Wt/2}$.

Which sharpens the prediction to an exponent #

That closed form does the comparative work the essay does by moving one input and re-plotting, and it does it with an exponent rather than a direction.

The optimum goes as the square root of the leaf’s width. So a leaf twice as wide should use $\sqrt{2}$ times as many folds — about forty per cent more, not twice as many — which is exactly what the figures show: doubling the area moves the tightest bundle from twelve folds to sixteen, and $12\sqrt{2}$ is seventeen.

It also says what the fold count is a measurement of. The ratio $W/t$ is the leaf’s width in units of its own thickness — a pure number, the same for a large thin leaf and a small thick one in proportion — and the fold count is its square root. So two leaves of quite different sizes with the same width-to-thickness ratio should corrugate into the same number of folds, and the count carries no information about absolute size at all.

That is a considerably more testable claim than larger leaves use more folds, and it is falsifiable in the useful direction: a survey finding the fold count proportional to the width rather than to its square root would say the bundle radius is not the binding constraint, and would send the question back to the veins.

The constraint is not in the sheet #

This is the point worth carrying away from the essay, and it is a methodological one.

Everything else on this site is a property of a pattern. Flat-foldability, the degrees of freedom, the packing ratio, the crease length: given the pattern, they are determined, and no information from outside is needed. The fold count is not like that. The pattern is equally valid at four folds and at forty, and choosing needs a fact about the world the pattern will live in.

That is why this essay’s figure has a horizontal line across it that is not computed from anything. The bud radius is an input. It is the shape of an argument this site makes rarely, and being explicit about it is the honest alternative to quietly deriving a number that was assumed.

There is a temptation worth resisting here, and it is the one that makes optimisation stories about organisms so easy to write badly. Having found that the bundle radius has a minimum, the next sentence almost writes itself: the leaf uses the count that minimises it, because natural selection would have found the optimum. That sentence explains nothing and is unfalsifiable in the form it is usually offered — any observed count can be rationalised by adjusting which quantity was being optimised.

What the computation supports is weaker and more useful. It says which counts are available, because a bundle that does not fit does not fit and no amount of selection changes that. Availability is a hard constraint the geometry establishes; optimality is a hypothesis about a process the geometry knows nothing about. Keeping the two apart is most of what separates a model from a story.

The same discipline applies to the engineered cases, where it is easier because the record exists. A stowed array that did not fit the fairing was not flown; that is a constraint. Whether the pattern chosen was the best available is a question about a trade study, and the trade study is usually the thing nobody published.

The same structure appears wherever folding meets hardware. A stent’s diameter is set by the artery, not by the pattern; an airbag’s fold is set by the housing; a solar array’s stowed volume is set by the fairing. In every case the pattern is a family and the world picks the member, which is exactly the relationship the excess of length has with its wave count one ladder over.

Where thickness enters, and how much of it there is #

The trade depends on a thickness, so it is worth asking how big that thickness is relative to everything else.

For paper the answer is that it is small and still matters: the thickness problem is the reason an ambitious tessellation is folded from thin stock and the reason a large grid comes out short. For a leaf the answer is that it is not small at all. A leaf is a slab with structure in it — epidermis, mesophyll, veins — and its thickness is a substantial fraction of the spacing between its folds.

So the bud problem is not a small perturbation of the zero-thickness problem. It is a different problem, with an interior optimum where the idealised version has none, and the interior optimum is the whole phenomenon.

The same trade, one level up #

The bundle is a one-dimensional version of a question that has a two-dimensional answer, and the two-dimensional version is worth seeing because it turns over rather than merely bottoming out.

The bud figure asks whether a given amount of surface fits. This one asks how much surface can be made to fit, which is a rung of its own and the question an organ that exists to have surface area is actually solving.

Both curves have the same cause and different shapes, and the difference is instructive. The bundle radius has a minimum because two costs trade. The held surface has a maximum because one benefit and one cost trade. Neither has anything to do with the folding being clever.

A bigger leaf in the same bud #

The model has two inputs and it is worth moving the other one, because the direction the answer moves is not obvious.

Doubling the leaf’s area with the container unchanged does not simply shift the optimum; it moves it toward more folds. The reason is that the falling cost — the strip width — is proportional to the area and the rising cost is not, so a larger leaf pays more for having few folds and exactly the same for having many.

That is a prediction of a kind this model can honestly make, because it is a comparison rather than a value: among leaves packed in similar containers, the larger ones should use more folds. It does not depend on the bundle being a rectangle or the thickness being uniform, only on one cost scaling with area and the other not.

It is also, usefully, the kind of claim that could be wrong. If larger leaves turned out to use the same number of folds as small ones, the trade modelled here would not be the one operating, and the most likely alternative — that the fold count is set by the vein pattern, which is set by something else entirely — would be the better explanation.

The count that fits is not the only requirement #

A bundle that fits is not sufficient, and it is worth saying why before the next rung says it at length.

The packed leaf has to come out. That is a constraint on the motion rather than on the packed state, and a fold count chosen purely to minimise the bundle radius could easily produce a pattern that opens badly — jamming against the bud wall, or needing to widen before it can narrow.

So the real selection is a conjunction: fit the container, open monotonically, and do both with a pattern the leaf’s own structure can supply. This essay computes the first, the next rung computes the second, and the third is not a geometric question at all.

What a real bud does that this does not model #

Three departures, and the first is the largest.

A real leaf is not packed alone. A bud contains several leaves, nested, each folded and each occupying part of the volume the others need — and the packing of several folded objects into a cylinder is a much harder problem than the packing of one. Nothing here touches it.

It is worth saying how much harder, because the difference is not incremental. Packing one bundle into a cylinder is a comparison of two numbers. Packing several deformable bundles that must also be able to leave the cylinder in some order is a problem with the same flavour as the packing questions this site meets in design, where the arrangement rather than the pieces is the difficulty and the honest answers are computational rather than closed-form. A bud is a container problem with a scheduling problem inside it.

The nesting also changes what each leaf’s container is. The outermost leaf is bounded by the bud; every leaf inside it is bounded by the leaf outside, which is itself folded and is not a cylinder. So the constraint the model treats as an input is, for most of the leaves in a real bud, an output of the same problem one level out.

A real leaf’s fold pattern is not chosen at packing time. The leaf grows into its folded state; the fold and the packing develop together, and the fold count is a consequence of where the veins are, which is set long before the question of fitting arises. So the model describes a constraint the leaf must satisfy and not a decision anybody makes.

And the bundle here is treated as a rectangle in cross-section, circumscribed by a circle. A real bundle is neither, and the difference is a factor of order one that would move the optimum without changing its existence.

The idealisation, named #

The leaf is a rectangle of stated area with a stated uniform thickness. It is neither.

More consequentially, the packed bundle is assumed to have every layer flat against its neighbour, which is what “a stack of n layers is n thicknesses deep” means. Real folded material does not stack that way: the fold at the end of each pair has a radius, the layers separate near it, and the stack is deeper than the count suggests. That correction is the next rung on the wings ladder and it makes the thickness term worse rather than better, which pushes the optimum toward fewer folds.

The container is a circle. A bud is closer to a cylinder with a taper and a cap, and the cross-section a leaf actually has to fit is not the widest one.

The container is also treated as rigid, which is the assumption most obviously false. A bud grows, and it grows while the leaf inside it grows, so the constraint is a moving one and the packed state is a snapshot of a race rather than a solution to a fixed problem. Modelling that properly means a bud radius that is a function of time and a leaf area that is another, and the question becomes whether the two stay compatible throughout rather than whether they are compatible at one instant.

Where this ladder goes next #

The pattern is chosen and the leaf is packed. What remains is getting it open.

That is a harder requirement than it looks, because there is nothing in a leaf to pull on. The opening has to be driven by growth, which means it has to run one way in one parameter with nothing anywhere that has to reverse — a condition on the pattern that has no analogue in anything a person folds, because a person can always push.

Sideways, the same container logic is what governs everything anybody deploys from a canister. The engineering version has the advantage that the container is a specification rather than a fact, and the interesting cases are the ones where the specification and the pattern are chosen together.

The wider point is one this site keeps arriving at from different directions. A pattern is a family, and a family needs something outside itself to become a particular thing. Flat-foldability leaves the layer ordering open; an excess of arc length leaves the wave count open; a corrugation leaves the fold count open. In each case the temptation is to look harder at the geometry for an answer that is not in it, and in each case the useful move is to identify what kind of fact would settle the question and say plainly that this site does not have it.