Nothing grown was cut out of anything

A plicate leaf packs into its bud by corrugating: a fan of folds running out from the midrib, opening as the leaf expands. The pattern is drawn here at four widths, and searching each of them for a consistent mountain-and-valley assignment costs twelve steps on twelve panels, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four.

Exactly one step per panel, at every geometry. The plant’s pattern is not a hard case of anything, which has been recorded here before and is worth saying again — nine, twelve, fifteen, eighteen, twenty and twenty-four nodes at nine to twenty-four panels across six geometries, with no backtracking anywhere.

What has not been said is that one step per panel is not the middle of the collection’s range. It is the top of it.

The range #

Every pattern here costs one step per panel or less, and most cost less.

The box-pleating grid, the Miura, the crumples and the leaf sit at exactly 1.00. A tessellation patch costs between 0.47 and 0.67, on all five tilings at every size measured. A corrugation drawn at a proportion where the big-little-big lemma bites costs 0.23.

So the leaf is at the ceiling, together with the three other families that fill their own sheet — and the objects below it are, in one case, patterns whose vertices are more constrained, and in the other, patterns that have been cut out of something larger.

What a cut takes away #

A twist tessellation fills the plane. Cut a square out of it and the square’s edge falls through the middle of the pattern’s own structure — through a pleat, across a twist polygon, wherever the paper happened to stop.

Every crease that edge divides was one crease of the tessellation, and its two ends had to agree. On the square they are two creases and they need not. A cut converts constraints that spanned the pattern into local ones, and a pattern with fewer long-range constraints is a pattern with more freedom in its letters and a cheaper search.

That is why a tessellation patch costs half what a corrugation costs, and it is the only reason the two differ in this respect: joining a patch’s opposite edges so that nothing is divided takes the same drawing from forty-eight steps to fifty-six thousand seven hundred and seventy-two.

The size of the effect is worth quoting because it is not marginal. On a four-period square rectangle of the twist tessellation the two readings — cut out of the plane, and joined so that nothing is divided — differ by a factor of more than a thousand, on the same drawing at the same vertices under the same conditions. Whatever a rim is doing, it is doing most of what there is to do.

A leaf has nothing to take away #

The leaf’s pattern ends at the leaf’s margin, and its creases end there because the plant’s growth stopped there.

There is no larger leaf pattern of which this one is a square. The fan runs from the midrib to the margin because that is where the lamina is; the corrugation has as many folds as the leaf has, and the outermost fold is the outermost fold rather than the place a cut fell. Nothing has been severed, because nothing extended past it in the first place.

So a leaf sits at the ceiling for the same reason the grid does: it is a complete pattern. One step per panel is what a pattern costs when nothing has been taken from it, and everything below the line has had something taken.

The reading this reverses #

The natural way to describe the two objects is that a leaf’s boundary is natural and a patch’s is arbitrary, and that the natural one is somehow better behaved.

The measurement says the arbitrary one is better behaved, if better means cheaper — and that the description had the causation backwards. A boundary is not a frame around the interesting part of a pattern. It is a place where constraints were either honoured or destroyed, and which of those happened depends entirely on whether the pattern continued past it.

A leaf’s margin honours them: every condition the pattern has is present inside the leaf, because the pattern is the leaf. A patch’s rim destroys them, four per period of edge, and the destruction is what makes it easy.

Three complete patterns and two specimens #

The collection’s patterns sort into the two kinds cleanly, and the sort is worth having written down.

Complete. A box-pleating grid, whose creases end at the paper’s edge because the grid was drawn to fill it. A Miura and a leaf, whose corrugations run out to the edge of the sheet they are designed for. A crumple, whose folds are wherever the crumpling put them and which extends nowhere. All four at one step per panel.

Specimens. A twist tessellation patch, cut out of a plane-filling construction. That is the whole of the second list, and it is at half the line on every tiling.

There is a third category worth naming for completeness, since it is what a folder actually usually holds: a complete pattern on a larger sheet, which is a corrugation drawn in the middle of a bigger square with plain paper around it. That has no severed constraints either, so it belongs with the first list — and it makes the point that the distinction is not about whether the paper ends but about whether the pattern was interrupted.

What the plant is not doing #

It is worth being careful here, because there is a tempting and wrong reading.

A leaf is not paying a cost. Twenty-four steps on twenty-four panels is a trivial search on a trivial object, and the sense in which the leaf is at the top of the range is a sense that concerns a machine reading the pattern rather than anything about the plant.

Nor is the leaf’s completeness a design achievement. A leaf’s pattern is complete because the leaf is its own boundary; there is no alternative arrangement in which a plant grows a fragment of a larger corrugation. The observation is about what a pattern is, not about what growth optimises.

What the plant does have is the property this collection keeps needing and rarely gets: a pattern whose every condition is present and whose measurements are therefore about the object rather than about a specimen. A leaf answers honestly to every question a patch answers approximately.

Where the leaf sits among the other grown patterns #

Plants fold in more than one way here and it is worth checking that the completeness argument is about growth rather than about corrugations in particular.

The hornbeam and beech pattern the leaf ladder draws is a fan of pleats: a corrugation, complete, at the ceiling. An insect’s wing folds by a different mechanism, and a flower bud by another again, but every one of them shares the property this essay turns on — the pattern and the object’s outline are produced together, so there is no larger pattern to be a piece of.

The one biological case that might not share it is a pattern that continues past what has been drawn: a leaf whose fan is modelled for four columns when the plant has thirty. That is a specimen of a leaf rather than of a tessellation, and its outermost creases were cut by the modeller. Every leaf pattern drawn here is complete in the sense of running from midrib to margin, which is why the four widths give exactly one arithmetic and not four.

So the claim is about completeness rather than about biology, and biology supplies complete patterns because growth does not produce fragments.

What a leaf can be measured for that a patch cannot #

Three quantities, and they are exactly the ones a cut inflates or invents.

Creases per unit area. A patch’s rim divides creases and each half is counted, so a patch reports ten to fifty per cent too many. A leaf’s count is a count of creases.

Panels. The same: a patch’s rim slices panels and counts the pieces.

The bottom of the layer stack. A patch has one and it sits at the rim; a periodic pattern has none at all. A leaf’s folded state has a bottom layer that is a genuine piece of the leaf, in a position that means something.

So the leaf, being complete, is a better object to measure than a patch of a tessellation is — and that is a slightly odd thing for a collection built on generated patterns to have to concede to a plant.

Where the leaf’s own boundary does something #

The leaf is not a perfectly uniform object either, and its own margin has an effect worth separating.

Its outermost vertices have creases that end at the margin, so they carry fewer conditions and their letters are freer — exactly as at any rim. But those creases were never anything else. They are the pattern’s ends rather than a cut’s, so no constraint has been broken and the freedom is the pattern’s own.

The consequence is that the leaf’s cost is exactly one step per panel rather than slightly under it. A pattern whose boundary creases answer to one vertex instead of two has one free choice each there, and the arithmetic comes out at exactly the panel count on all four geometries — which is the same arithmetic the box-pleating grid gives and for the same reason.

The bud, which is where the pattern comes from #

There is a reason a leaf’s pattern is complete that goes past the tautology, and it belongs here because it is the biological half of the argument.

A plicate leaf’s corrugation is laid down in the bud. The lamina is folded before it is grown to size, and the fan of creases is a consequence of how the tissue is packed into the space available — the bud chooses the pattern, and the number of folds and their spacing follow from the bud’s diameter and the leaf’s eventual span.

So the pattern is not laid over the leaf. It is produced by the same process that produces the leaf’s outline, at the same time, and the two cannot come apart. A leaf with the corrugation of a larger leaf would be a leaf that had grown to one size and folded for another.

That is the strongest form of complete available: not merely that nothing was cut, but that a cut is not a thing that could have happened. A tessellation patch is a pattern somebody chose a piece of; a leaf is a pattern that has no pieces.

What the taper does not do #

The leaf’s most interesting geometric property is its taper, and it has nothing to do with any of this.

A corrugation may be tapered along its fold lines and not across them: the column widths are free and the row heights are not, because Kawasaki’s alternating sum at an interior vertex involves the row heights and not the widths. The taper decides nothing about foldability, and it decides nothing about cost either — four geometries, four different tapers, one arithmetic.

That is a satisfying null result to have beside this essay’s main one. What matters for the cost is whether the pattern is complete; what a plant varies is the shape, and the shape is free.

What this says about how the collection measures #

A collection built on generated patterns has an incentive worth being aware of, and this essay is where it shows.

Generated patterns are convenient. A tiling construction produces a pattern at any size, at any pitch, on any lattice, and a sweep over its parameters is a population. A leaf is one shape at four widths and nothing more can be asked of it. So most of the measurements here are on generated patterns, and most of the generated patterns are cut out of plane-filling constructions.

Which means most of the measurements here are on specimens, and specimens systematically report a pattern’s search cost too low, its crease count too high and its panel count too high. Not by a little: ten to fifty per cent on the counts, and a factor of a thousand on the cost.

The leaf, the grid, the Miura and the crumple are the objects that do not have that problem, and three of the four are less interesting to sweep. So the collection’s easiest measurements and its most honest ones are almost disjoint sets — which is worth knowing when reading any number here that came off a patch.

Which theorem was checked, and how #

Every one of the four leaf patterns is put through the four vertex conditions before it is searched, and a pattern that failed would be refused rather than measured. The taper is the one this collection has checked in both directions: varying the columns folds and varying the rows does not, and the refusal is asserted rather than assumed.

The costs are measured under a fixed letter order, so each is the number that pattern gives every time. And the claim that no decision is withdrawn is asserted directly: every pattern in the ladder is required to cost no more than one step per panel, and the requirement fails on the first that does.

The comparison against the tessellation patches uses the same search, the same order and the same test on both, so the leaf’s 1.00 and the patch’s 0.51 are two readings of one instrument.

The sentence worth keeping #

A boundary is not a frame. It is the place where a pattern’s constraints either run out honestly or are cut through, and which of those happened is not visible in the drawing.

A leaf’s margin is the first. A tessellation patch’s rim is the second. They look identical — creases meeting an edge and stopping — and they mean opposite things, and the difference is worth a factor of a thousand in what the pattern costs to reason about and ten to fifty per cent in what it appears to contain.

The only way to tell them apart is to know whether the pattern continued past the edge, which is a fact about where the pattern came from rather than about the paper. That is an unusual kind of thing for a crease pattern to depend on, and it is the reason a specimen and a pattern have to be kept apart even when they are drawn identically.

What the picture cannot show #

A ladder with every point on the line is a picture of an equality, and equalities are dull to look at and hard to disbelieve. What it cannot show is that the line is a ceiling — that nothing can be above it — which is an argument about how many labellings a degree-four vertex admits rather than a measurement of anything.

Nor does anything here show a leaf. The pattern is a corrugation drawn from the geometry a plicate leaf uses, and a real leaf has veins, a curved margin, a thickness and a growth history, none of which is in the drawing and each of which would matter to a botanist more than the arithmetic does.

And the essay’s central comparison is between a plant’s pattern and a pattern nobody grows, which is a comparison of two mathematical objects rather than of two organisms. Whether anything in nature ever has a crease pattern interrupted — a corrugation whose folds were laid down and then cut through by something else — is a question about plants that this collection has no way to ask. The nearest thing here is a leaf that ends its own pattern, which is the opposite case and the one the measurements are about.