A leaf packs by corrugating

The cheapest fold in the subject is a corrugation. Parallel creases, alternating mountain and valley, running from one edge of the sheet to the other. It has no interior vertices, so none of the local theorems have anything to say about it, and it folds flat for the least interesting possible reason: there is nothing there that could fail.

A leaf that packs by corrugating is therefore using the simplest tool available. What makes it interesting is that a leaf is not a rectangle, and the moment the corrugation has to taper, the theorems acquire something to say.

What a taper does to a corrugation #

A pure corrugation on a rectangle is a set of parallel lines. It has no vertices at all, because no crease meets another crease anywhere inside the sheet.

The moment the fold lines stop being parallel, or the moment a second family of creases is added to let the pattern change its width, interior vertices appear — and those are governed by the same four conditions as everything else on this site. The pattern used here is the second kind: a zigzag corrugation of the Miura family, where a set of transverse creases lets the sheet narrow.

At an interior vertex of that pattern, four creases meet. Two of them are halves of one straight transverse line, so they leave the vertex in exactly opposite directions. The other two are the column creases above and below, and they lean by an amount set by the zigzag.

Which parameter is free, derived rather than tried #

The consequence is short enough to do in the text, and it is the useful half of the essay.

The two transverse creases at a vertex are collinear, so the four sector angles are determined entirely by the directions of the two column creases. Those directions are the angles whose tangents are the zigzag offset divided by the row height — the row above and the row below, each with its own height.

Kawasaki’s condition is that the alternating sums of the four sectors both come to 180°. Working it through, they do exactly when the two angles are equal, which happens exactly when the row above and the row below have the same height.

The column widths never enter the calculation. They do not appear in any sector angle, and they are therefore free — every one of them, independently, with no constraint relating one to the next.

So the rule is: a corrugation of this family may taper along its fold lines and may not taper across them. That is derived before anything is drawn, and it is the opposite of what a person drawing a leaf would try.

The refusal, asserted rather than described #

The obvious way to draw a leaf that narrows toward its tip is to make the rows shallower as they go. It produces a plausible picture, it looks like a leaf, and it does not fold.

Building it and running the checker gives the numbers: the alternating sums come out at 186.4° and 173.6° instead of 180° and 180°, and the checker refuses the pattern by name and by vertex. The pattern library keeps that construction — the row-tapered corrugation, built and never printed — precisely so the refusal can be asserted in the gate rather than described in prose, in the same spirit as the preliminary base’s naive assignment and the fold-and-cut pattern’s obvious colouring.

This is the third time this shape of finding has come up here, and by now it is a pattern worth naming. The Yoshimura’s row height looked free because Kawasaki holds identically for any diagonal angle. Every regular polygon admits a twist because the sectors come out A, A, 180−A, 180−A whatever A is. Both cases have a condition that holds by identity over one family of parameters, which makes a second family look unconstrained when it is not.

The general lesson: a condition that is satisfied identically is not evidence that a parameter is free. It is evidence that the condition is not the one doing the work, and something else usually is.

There is a diagnostic that follows from it and costs nothing. When a condition comes out satisfied for every value of some parameter, the right response is not relief but suspicion: work out which quantities the condition actually contains, and check whether the parameter is one of them. Here Kawasaki contains the sector angles, the sector angles contain the row heights, and the column widths appear in neither — so the identity was never about the widths at all and says nothing about them.

Doing that check takes a minute and it is the minute that separates the two tapers. Both look like reasonable drawings. One of them satisfies a condition that has nothing to do with the choice being made, and the other violates a condition that has everything to do with it, and no amount of looking at the two pictures distinguishes them.

What the fold buys #

A corrugation’s job is to make a large surface occupy a small space, and the number is straightforward to compute from the pattern’s own parameters rather than estimated from a picture of the folded object.

Folded to a zigzag half-angle, a corrugation of any number of panels collapses its span by the sine of that angle. At the angle the default pattern uses, the packed footprint is 11.2% of the flat one — about a ninefold reduction, achieved with a fold anybody can make and no vertex anywhere doing anything clever.

The comparison is worth reading carefully because it does not say the corrugation is the best choice. It says it is the cheapest choice, and the geometries that pack better ask for more: a Miura needs its transverse creases to be right or it does not fold at all, and a roll needs the sheet to be flexible along its whole length rather than only at lines.

What the checker does to it, one vertex at a time #

It is worth being concrete about what “verified” means for this pattern, because the word is doing real work and is easy to wave.

Each of the eighteen interior vertices is taken in turn and put through four conditions. Developability: do the sector angles sum to 360°, which is the statement that the vertex was drawn on paper that exists and that folding created no curvature there. Kawasaki: do the two alternating sums both come to 180°, which is the condition the taper rule came out of. Maekawa: does the count of mountains differ from the count of valleys by exactly two. Big-little-big: where a sector is strictly smaller than both its neighbours, are its two bounding creases assigned different letters.

The important property of that procedure is that it is local and complete — every vertex is checked and no vertex is checked in the light of any other. It is also, for exactly that reason, not a decision procedure for the sheet. A pattern can pass at every vertex and still fail to fold, and the checker is not entitled to say otherwise.

For a corrugation the gap between local and global is unusually small, which is why this is a good pattern to fold. There is one degree of freedom, the panels do not have to pass each other, and the layer ordering is the obvious one. Nothing here is relying on the local conditions to establish more than they can.

What the free parameter buys the outline #

The taper rule says the column widths are free, and it is worth following that through to the folded object, because the freedom has a consequence a bud cares about.

Folding a corrugation collapses the sheet across its fold lines and leaves it untouched along them. The column widths lie along the fold lines. So every column keeps its width exactly, and the whole outline is compressed in one direction by a single factor and in the other by nothing at all.

The packed bundle therefore has the same taper as the flat sheet. A leaf drawn 2.44 to one from its broad middle to its narrow ends packs to a bundle 2.44 to one, and the profile survives the fold undistorted. Nothing in the pattern can change that ratio, because the only parameter that could — the column widths — is the parameter that sets it in the first place.

That is a genuinely useful property rather than an artefact. A bud is a taper, and a leaf that packs into a shape geometrically similar to its own outline is a leaf whose packed form matches its container without anything having to be arranged. The rule that forbade the obvious taper is the same rule that makes the permitted one behave.

What the fan’s apex actually costs #

The fan’s difficulty was stated above as a refusal — equal sectors with alternating letters have equal counts and Maekawa forbids it — and the size of the difficulty is worth having, because “forbidden” and “hard” are different and the fan is only the first.

At a vertex of $2k$ equal sectors, Kawasaki holds identically and the smallest-sector lemma is silent, so Maekawa alone decides. Of the $2^{2k}$ letterings, the admissible ones are those with $k+1$ of one letter and $k-1$ of the other, in either order: $\binom{2k}{k+1} + \binom{2k}{k-1}$.

At degree four that is eight of sixteen, a half. At degree six, thirty of sixty-four. At degree eight, a hundred and twelve of two hundred and fifty-six, which is seven sixteenths rather than the half a reader would guess. The share keeps falling, as roughly one over the square root of $k$.

So a fan does not become impossible as it gains creases; it becomes relatively harder while remaining absolutely easy. A radiating fold with twenty creases at its base still has about a third of its letterings admissible, which is hundreds of thousands of them.

What is never available is the one arrangement a drawing suggests. Strict alternation is the unique lettering with equal counts, and it is forbidden at every degree, which is why the base of a radiating fold has to do something that looks untidy and why it looks untidy in the same way every time.

Two directions instead of one #

The Miura in the comparison above packs better than the corrugation for one reason, and it is worth isolating.

A corrugation collapses across its folds and does nothing along them, so its packing is a single factor. A Miura collapses in both directions from the same single parameter, so its packing is that factor squared. Nothing about the fold is harder to make; what is harder is getting the transverse creases in the right places, which is what the derivation earlier in this essay is about.

That squaring is why the Miura shows up everywhere in things people build and why a plain corrugation shows up everywhere in things that grew. A manufacturer can afford to place creases exactly. A leaf gets its fold lines from where its veins are, and the veins have other jobs.

Why a corrugation rather than a crumple #

There is a competing option that packs better than any of these and is not on the chart, because it is not a fold: crumpling.

A crumpled sheet packs extremely well, needs no design, tolerates any shape of sheet, and is robust to being done badly — a crumple that goes wrong is another crumple. What it does not do is unpack. A crumple has no parameter, no motion and no reverse; the sheet arrives at a random configuration of ridges and stays there, and getting it flat again means working every ridge by hand.

A corrugation is worse at packing and has the property that matters: it is a mechanism. One number describes its state, running that number the other way opens it, and the opening is monotone, which is what lets a growing leaf drive it with nothing to pull on.

That trade — packing efficiency against having a motion at all — is the same one every deployable in this subject makes, and it is why the patterns that get built are almost never the ones that pack tightest.

Where the corrugation stops being simple #

Two things complicate the picture and both are real.

The first is that a leaf’s fold lines are not a rectangle’s. They radiate from a midrib, which means they are not parallel, which means the pattern is closer to a fan than to a corrugation — and a fan has a vertex where all the creases meet. That vertex is a genuine constraint: equal sectors with alternating letters gives equal counts of mountains and valleys, which Maekawa forbids outright. A fan that folds flat at its apex needs unequal sectors or an extra crease, which is why real radiating folds are messier at the base than in the middle.

The second is that the transverse creases in a real corrugated leaf are not all creases. Some are veins, some are regions of thinner tissue, and some are nothing at all — the sheet simply bends there. The pattern computed here is a model of the fold, not a tracing of a leaf, and that distinction is the rule the whole field runs under.

What is printable, and what folding it establishes #

The pattern at the top of this essay is on the pattern index with a sheet sized in millimetres, and folding it takes about ten minutes.

What folding it demonstrates is not that leaves do this. It is that the taper rule is real: the columns narrow and the sheet still collapses onto a single line, which is what a flat fold means. A reader who wants the negative half can redraw it with the rows tapering instead and discover that the sheet fights back — the creases will not all close, and the resistance is at the vertices rather than along the lines.

That is the site’s proposition in its cleanest form. The claim is geometric, the figure is generated from the claim, and there is a version of the experiment that runs in a reader’s hands and comes out the same way.

It is also the one place where this field’s rule about organisms is not a limitation. Nothing about a leaf can be tested by folding a sheet of paper, and nothing about a leaf is being tested. What can be tested is the geometric proposition the essay actually makes — that this family of patterns tolerates one kind of taper and refuses another — and that proposition is fully checkable by a reader with scissors, a printer and ten minutes, which is more than most claims about morphogenesis can offer.

The idealisation, named #

The sheet has no thickness. A corrugation of many folds is a stack, and a stack of any real material is bounded by its own thickness long before it is bounded by geometry — which is what the bud actually cares about and the subject of the next rung.

The creases are lines. A real leaf’s folds have a radius, and the radius is not small compared with the tissue: a leaf is thick relative to its fold spacing in a way paper is not. What that costs is a whole rung of the wings ladder and it applies here in full.

And the fold angle is uniform. A real corrugation is deeper in the middle of the leaf and shallower at the ends, which the model has no way to express and which changes the packing arithmetic in the direction of making it worse.

Where this ladder goes next #

The corrugation is a pattern with one free parameter — how many folds — and this essay has not said what sets it.

Nothing in the geometry does. The container does: a leaf packed in a cylindrical bud faces a trade between a strip too wide to fit and a stack too thick to fit, and the fold count that fits at all is a small window. That is the next rung, and it is the case where the answer comes from outside the sheet entirely.

Sideways, a corrugation repeated in two directions becomes a material rather than a folded sheet, with a Poisson’s ratio and a response to being pushed on. The leaf’s version stops one step short of that, and the step it does not take is what keeps it a leaf rather than a mechanism.