A leaf ends its pattern

A leaf in a bud packs by corrugating, and the corrugation has to end — at the margin, at the tip, at the midrib. How it ends is a question this collection has asked of its own drawings a great deal lately and never asked of the organism, which is the wrong way round: the leaf solved it first.

It does not end by being cut off. The pleats narrow along their length until there is nothing left of them, and the leaf’s edge is the place where the pattern has reached zero rather than the place where somebody stopped drawing it.

Why the taper is free #

The reason a corrugation can taper at all is a fact about which parameters its conditions read, and it was derived here before it was drawn.

At an interior vertex of a corrugation, the four sectors are decided by the zigzag angle and by the heights of the row above and the row below. Kawasaki’s alternating sums come to a half turn exactly when those two heights are equal — and the column widths never enter the calculation. Every width is therefore free, and free independently: no relation binds one column’s width to its neighbour’s.

So a corrugation may be tapered along its fold lines as sharply as anybody likes. Widths of 0.005, 0.04, 0.25, 0.41, 0.25, 0.04 and 0.005 of the sheet — a ratio of eighty-two to one between the middle and the outermost column — give a pattern with eighteen interior vertices and a worst Kawasaki residual of 4 × 10⁻¹⁶, which is the same residual an untapered one gives. No crossing, no crease stopping in the middle of the paper, no vertex out of condition.

And it may not be tapered across them. Varying the row heights instead — the obvious way to draw a leaf that narrows toward its tip — puts the two alternating sums at 186.4° and 173.6°, and the sheet does not close. Twelve of the eighteen vertices fail.

That asymmetry is the whole of the leaf’s boundary strategy. One direction of taper is free and the other is impossible, and the free one is the direction that lets a corrugation come to nothing at a margin.

What “reaching zero” means on paper #

A column of width zero is not a column, so the taper approaches the margin rather than arriving at it — which is what a leaf does too. Between two veins the pleat gets shallower and shallower, and at the margin itself the lamina is flat.

The drawn version has the same structure with the same limit. The narrowest column in the extreme taper above is a two-hundredth of the sheet, and nothing stops it being a two-thousandth; what stops it is the paper, at the point where the column is narrower than the crease is wide. A crease has a radius, so a pleat narrower than that radius is not a pleat.

The consequence for a folder is a limit and not a boundary condition: the pattern remains valid all the way down, and the material gives out first. That is a much more comfortable place to be than the alternative, where the pattern itself fails at some width and the failure has to be found.

The taper sweep, in full #

Five corrugations, differing only in their column widths, each put through every condition at every one of its eighteen interior vertices.

Even, seven columns all 0.16 of the sheet: folds, worst residual 4 × 10⁻¹⁶. Gentle, a ratio of 1.7 between widest and narrowest: folds, same residual. As drawn, the pattern this collection prints, a ratio of 2.4: folds, same residual. Sharp, a ratio of 10: folds, same residual. Extreme, a ratio of 82: folds, same residual.

The residual does not move because the widths are not in the calculation. That is worth saying flatly: this is not a case where a parameter has a wide but finite tolerance. It is a case where a parameter is absent, so the pattern’s validity is constant along that whole axis and the sweep is confirming an algebraic fact rather than probing a limit.

The value of running it anyway is the value of every refusal in this collection: an assertion that has never been exercised proves nothing, and a sweep that comes back constant is evidence the derivation is about the right quantity.

The residual is the arithmetic’s floor #

The number that does not move across the sweep is worth identifying, because knowing what it is turns the sweep from a tolerance measurement into a confirmation of an identity.

Four times $10^{-16}$ is about twice the smallest gap double-precision arithmetic can represent near one. It is not a small error; it is rounding, and a computation that returned exactly zero would be reporting the same thing.

So the sweep is not saying the widths affect Kawasaki very little. It is saying they do not appear in it, and the residual is what the machine produces when adding four numbers that sum exactly to a straight angle.

That also bounds any dependence there might have been. The taper ratio spans a factor of eighty-two across the sweep and the residual does not move at all, so any sensitivity to the widths is below $10^{-16}$ divided by eighty-two — which is below what the arithmetic can express and therefore below what any measurement could find.

Fourteen orders of magnitude apart #

Set that against the taper the pattern refuses and the asymmetry stops being a comparison at all.

Tapering across the folds puts the two alternating sums at 186.4° and 173.6°, which is a residual of 6.4 degrees, or 0.112 radians. Against the column taper’s $4 \times 10^{-16}$ radians, that is a factor of $10^{14}$.

The two tapers are not two points on a scale of tolerance. One is an algebraic identity and the other is a violation fourteen orders of magnitude outside it, with nothing in between that any pattern reaches.

And the six that survive are the symmetric ones #

One detail of the failure is worth reading, because it confirms the mechanism rather than merely recording the count.

Twelve of eighteen vertices fail, not eighteen. The pattern has four rows and therefore three interior row boundaries, six vertices along each. A taper that narrows toward both ends symmetrically makes the first and fourth rows equal and the second and third equal — so at the middle boundary the row above and the row below have the same height, which is exactly Kawasaki’s requirement.

Six vertices there pass; the twelve at the two outer boundaries do not.

So the survivors are not survivors at all; they are the vertices where the taper happens to be locally flat, and a one-sided taper would fail at all eighteen. That is the condition doing precisely what the derivation says it does — reading the two row heights and nothing else — and it is a sharper confirmation than the sweep, because it predicts which vertices fail rather than how many.

Against a patch that was cut #

Set that beside what a tessellation patch does at its edge and the contrast is the point of this rung.

A patch has no strategy. Its units are the same size everywhere, so the sheet’s edge falls where it falls: some units whole, some cut through the middle, some pleats reaching the rim and stopping there because the paper does. At the sizes anybody draws, between a quarter and nine tenths of the units on the paper are cut ones, and every one of them is a unit doing something the construction never specified.

The leaf’s pattern reaches its own boundary. The patch’s pattern is interrupted by one. Both are finite folded objects with a rim, and only one of them decided what happens there.

And the failure modes follow. Interrupting a pattern is what produced the crossings and the creases stopping mid-sheet that this collection spent a long time finding; a pattern that tapers to nothing has neither, because nothing is ever cut. The leaf’s boundary is a limit of the construction and the patch’s is an accident of the paper.

The measurement makes the contrast sharper than the description does. Count the units of a twist patch against the sheet it was clipped from and the interrupted ones are a large fraction of the total at every size anybody prints at, rising toward all of them as the units grow. A tapered corrugation has no such fraction, because there is no unit for the sheet to interrupt: the outermost column is a column, narrow but entire, and the count of interrupted units is nought at every taper in the sweep above and at every taper that could be drawn. One pattern spends most of itself on its boundary and the other spends none of itself there.

What the margin looks like folded #

The taper’s effect on the folded packet is worth one paragraph, because it is what the bud actually experiences.

An untapered corrugation folds into a packet whose depth is the same everywhere: every pleat contributes the same two layers, so the stack is a slab. A tapered one folds into a packet that is deep in the middle and thin at the ends, because a narrow column contributes a narrow pleat and the layers it adds occupy less width.

So the taper does two things at once, and they are not the same thing. It lets the pattern reach zero at the margin, which is this rung’s subject; and it makes the folded packet the shape of the container, which is the rung below it. One parameter, two consequences, and the reason a leaf’s corrugation is tapered is almost certainly the second — the first comes free with it.

That is a pleasant arrangement and it is not general. A pattern family whose free parameter tapered the pattern to nothing without shaping the folded object would still solve the boundary problem, and would give a designer nothing else.

What the organism is actually doing #

The comparison should not be pushed past what it supports, and two things about the biology are worth stating carefully.

A leaf does not fold a sheet. It grows, and the corrugation is a consequence of differential growth rather than of anything folding: the lamina between the veins grows more than the veins do, the excess has nowhere to go in the plane, and it buckles. The pattern is an outcome and not a plan.

Which is why the taper is free there too, for a different reason. Growth is local. Each patch of lamina grows at whatever rate it grows at, and a rate that falls off toward the margin produces a corrugation that falls off toward the margin, with nothing coordinating the two ends. The drawn pattern’s freedom comes from Kawasaki not reading the widths; the leaf’s comes from there being no global condition at all.

Two independent reasons for the same shape, and the coincidence is not a coincidence: a growth field that produces a flat-foldable pattern is producing one that satisfies the same local conditions, so what is free in one description is free in the other.

The bud is the reason for the taper #

A leaf tapers because of what it is packed into, and the shape of the container decides the shape of the pattern.

A bud is a volume with a shape, and a leaf inside it has to fit in the direction across its own width as well as along it. A corrugation of constant depth fits a container of constant depth; a bud is not one. So the taper is not decoration or a limit of growth but the pattern’s answer to a container that narrows.

That also explains why the taper is where it is. The deepest point of a folded leaf’s packet is 1.926 times its average thickness, at every patch size measured, and the deepest point is what has to clear the bud. A pattern that tapered across its folds instead would move that peak rather than lowering it — and would not fold at all, which settles the matter.

The other way a pattern can end #

Tapering is not the only honest ending, and it is worth naming the second because this collection uses it too.

A pattern can end at a boundary the construction itself produced. The fold-and-cut patterns do this: the outline being cut is part of the construction, the skeleton lives inside it, and the pattern’s edge is the shape’s edge rather than the sheet’s. Nothing about that pattern is interrupted by the paper — the paper is simply larger than the pattern.

So there are three endings, and only two of them are decisions:

Taper to nothing, which is the leaf’s and needs a parameter the conditions do not read. Stop at a boundary the pattern brought with it, which is the fold-and-cut construction’s and needs the pattern to have a natural extent. Be cut off by the sheet, which is a tessellation patch’s and is not a decision at all.

The third is what most drawn tessellations do and what this collection did for years. It is not always avoidable — a tessellation genuinely has no natural extent — but it should be recognised as the absence of an answer rather than as an answer.

What a drawn pattern should take from it #

Three things, and the first is the one the drawings here keep running into.

A pattern should end at zero, not at the rim. A construction that reaches the edge of the paper still going is a construction that has left its boundary to whoever chose the sheet. Tapering to nothing is one way to avoid that; cutting the pattern out of a larger drawing is another and a lesser one.

Ask which parameters the conditions read. The taper is free because the widths are not in Kawasaki’s calculation, and that fact was available from the beginning of this collection and used only much later. Any pattern family has such parameters, and they are exactly the ones a designer may spend on a boundary.

And check the plausible drawing. Tapering across the folds is what anybody would try first, it is what a leaf looks like it does, and it fails. The refusal is asserted on every build here for that reason: a plausible drawing that fails a theorem is the failure this collection exists to catch.

The same freedom, three patterns over #

This is the third time in this collection that a condition holding identically for one family of parameters has made a second family look free when it is not — or, read the other way, has made a genuinely free family look constrained.

The Yoshimura’s row height looked free because Kawasaki holds at every half-angle, and the big-little-big lemma caps it at sixty degrees.

A twist’s sector angles looked free because the construction’s polygon is built from half-planes, and the pleat’s matching condition fixes one twist’s size from its neighbour’s.

And a corrugation’s column widths look constrained — a drawn leaf narrows, so surely the pattern is doing something delicate — and are completely free.

Two of the three went one way and one went the other, which is the reason the question has to be asked rather than guessed each time: which parameters does the condition actually read? On this pattern the answer is the row heights and the zigzag angle, and the widths are not in it at all.

That is the transferable part of this rung. The leaf’s boundary strategy is available to any pattern family with a parameter its conditions ignore, and finding such a parameter is a matter of reading the derivation rather than of experimenting with the drawing.

Where the analogy stops #

A leaf is not one sheet. It has veins with their own mechanical properties, a midrib, a thickness that varies, and two surfaces that are not the same tissue. Everything above treats it as a corrugated sheet, which is the idealisation these essays are careful to name.

The taper measured is in a drawn pattern. The eighty-two-to-one ratio is a fact about what the conditions permit, not a measurement of any leaf. Real laminae taper by rather less and for reasons involving vascular supply, not foldability.

The margin is not the only end. A leaf’s corrugation also has to stop at the midrib and at the petiole, and those are different problems: the midrib is a line of symmetry where the two halves meet, and the base is where the whole lamina narrows to a stalk. Only the margin is addressed here, and it is the easiest of the three.

And the pattern is one of several. Different plants use different corrugations — some fan, some roll, some pleat in two directions — and only the pleated ones are described by this pattern at all. What generalises is the question rather than the answer: how does this pattern reach its own edge, and did anything decide that it should?