Twice as thick where it is thickest

A leaf packs by corrugating: the lamina is pleated along the veins, the pleats nest, and a great deal of blade goes into a small bud. The usual way of pricing that is an area calculation — take the lamina’s area, take the packed footprint, divide, and the quotient is how many layers deep the packet is.

The quotient is the average depth over the footprint. Nothing has to fit an average.

The measurement #

A leaf pattern here is a corrugation: columns of pleats along the midrib and rows of them across it, drawn so that every vertex satisfies the conditions in the subject. Folding it is a composition of reflections, one per crease, and every panel lands somewhere in the plane.

The layer count is then sampled on a grid over the folded footprint: at each sample, how many panels lie over it. That is a sampled quantity and the sampling has to be trusted, so it is checked against a number known exactly — the layer count integrated over the footprint has to come back as the area of the sheet, because that is where all the lamina went. On every patch here the two agree to four decimal places.

The last column does not move. Not approximately: 1.926 at every size, to three figures, across patches with fourteen to thirty-five panels.

Why the ratio is fixed #

The mechanism is in the second column and the fourth.

The deepest point is four layers per row. Adding a row of pleats adds four panels to the deepest column of the packet, every time, because the pleat is a fixed piece of geometry repeated. Eight, twelve, sixteen, twenty.

The average is two and a bit per row, and the “bit” is what makes the ratio interesting. The average is the sheet’s area divided by the footprint, and both of those grow — but the footprint does not.

That is the fact worth stopping on. The footprint is 0.0900 of a sheet width squared at every size. Two rows, three rows, four rows, five rows: the same footprint. The paper grows from 0.374 to 0.935 and the packed outline does not move at all, which is exactly what a corrugation is for and is why the average depth grows linearly.

So the deepest point grows linearly, the average grows linearly, and the ratio between two linear things with the same intercept is a constant. The constant is 1.926 because the panels at the deepest column are the full-height ones and the panels away from it are the tapered ones, and the taper is fixed by the pattern.

What has to fit in the bud #

A bud is a volume with a wall, and what has to go inside it is the folded packet — all of it, including its thickest part. So the quantity a leaf’s development is constrained by is the deepest point and not the average, and the two differ by a factor of nearly two.

Read the other way round, that is a limit on how much lamina a bud of a given size can hold. If the bud’s clearance is d sheet thicknesses, the deepest column must be at most d, so the number of rows is at most d/4 — and the lamina that packs is 2.077 × d/4 footprints’ worth rather than d footprints’ worth. An area calculation overstates what fits by ninety-three per cent.

That is a large enough error to change what the numbers are for. A leaf reported as packing “eleven times its footprint” is packing eleven times on average and twenty-one times where it is thickest, and the second number is the one a bud has to be built around.

What the constant is made of #

The 1.926 can be taken apart, and taking it apart is what makes it a fact about the pattern rather than a coincidence in four numbers.

Write the deepest point as 4r for a patch of r rows — that is the second column, exactly. Write the paper as A and the footprint as F, both measured off the folded state; the average is A/F, and the table’s fourth column is 4rF/A.

The footprint is constant, so F comes out as a fixed 0.0900. The paper grows by 0.1870 per row, exactly — 0.3739, 0.5609, 0.7478, 0.9348 — because each row adds the same cell. So the average is 2.077r and the ratio is 4/2.077, which is 1.926 and has no r in it.

Nothing about that argument needs the folding. It needs the deepest column to gain a fixed number of layers per row and the footprint to stay put, and those are the two properties that make a corrugation a corrugation. Any pattern with them has a constant of its own; what the measurement supplies is the value.

The constant is a cell’s property, not a patch’s #

The decomposition gives the constant as four times the footprint over the paper added per row, and both of those quantities belong to one cell. So the constant can be had without folding a patch of any size at all.

Draw a single row of the pleat. Measure the paper it holds and the footprint it packs into. Four times the second over the first is the ratio, and it will be the ratio at two rows, at five, and at fifty. The four patches in the table are a check on that rather than the way to obtain it — which is worth knowing, because folding a fifty-row patch to find out what a one-row cell already says would be a great deal of work for a number that does not move.

And it is exactly what a uniform packet would beat it by #

The constant has a second reading that turns it from a caution into a target.

Suppose the bud’s clearance is $d$ sheet thicknesses. The deepest column may not exceed it, so the packet holds at most $d/4$ rows and therefore $2.077 \times d/4$ footprints of lamina, which is $0.519\,d$.

Now suppose a pattern of the same footprint whose packet were uniformly deep — every point of the footprint carrying the same number of layers, so that the maximum and the average coincide. Such a packet holds $d$ footprints of lamina in the same clearance.

The two differ by $1/0.519$, which is 1.926 — the constant, arriving as the factor by which a uniform packet outperforms this one. That is not a coincidence: the ratio of maximum to average is the factor by which the peak wastes clearance, and reading it that way makes it actionable rather than merely cautionary.

So the corrugation is leaving about half its bud unused. The deep band spends the whole clearance and the shallow ground either side of it spends about a quarter, and the difference is lamina that could have been there and is not.

Which identifies what a better pattern would have to do, and it is not a smaller cell or a different taper. It is a flat profile — a packet whose depth is the same everywhere, which is what the Yoshimura achieves at a ratio of exactly one and what nothing built out of tapered pleats can. Whether a pattern can be uniformly deep and hold its footprint and open the way a leaf has to is a question about three requirements at once, and the ratio is what says how much is at stake in answering it.

Against the other folded objects #

The same ratio measured on this site’s printed crease patterns comes out nothing like as tidy.

The preliminary base, the square twist and the fold-and-cut triangle each have a point where every panel of the sheet lies over it, so their deepest points are 8 of 8, 9 of 9 and 7 of 7 — and their averages are 7.99, 3.02 and 1.19. The ratios are 1.00, 2.98 and 5.88. The Yoshimura is sixty layers deep everywhere, so its ratio is exactly 1.

A leaf is the only family here whose ratio is a constant, and the reason is that a leaf is the only one built by repeating a cell. The others are single objects, and a single object’s deepest point is wherever its own geometry happens to put it.

The deep column is thin #

The other reason an average misleads is visible in where the deep place is.

The deepest point of these packets is not a broad plateau. On the four-row patch, the twenty-eight panels stack to sixteen layers over a narrow band and to far less over most of the footprint — which is what makes the average 8.31 rather than something close to 16. A bud enclosing it has to clear sixteen along that band and nothing like sixteen anywhere else.

That has a consequence for the shape of the packet rather than only for its size: the folded leaf is a wedge, thick along one line and thinner away from it, and a bud that fits a wedge is not a cylinder. The measurement here does not go on to the bud’s shape, and nothing in this file is a claim about a plant. What it establishes is that the packet has a profile, that the profile has a maximum, and that the maximum is where the constraint lives.

What the corrugation buys and what it costs #

The constant footprint is the whole trick and it is worth stating as a trade rather than as an achievement.

A corrugation converts area into depth at a fixed footprint. That is ideal for a bud, whose constraint is a diameter, and it is why the bud chooses the pattern rather than the other way round: the packing has to fit through an opening, and a pattern that spreads would not.

What it costs is that depth is the expensive direction. A sheet of zero thickness can be corrugated without limit — folding buys unlimited surface inside a fixed volume when the sheet has no thickness — and a real lamina has one. The deepest point is where that thickness is spent, and it is spent nearly twice as fast as an average would suggest.

Where the same reading applies elsewhere #

The distinction between a maximum and an average is not a fact about leaves, and this site has met it twice before under other names.

The pile, not the panel is the engineering version: every technique for building a fold out of panels with depth is drawn and priced at one crease between two panels, and a folded model has two layers nowhere except at its last fold. The printed patterns reach eight, sixteen, thirty-two and sixty. The quantity that decides whether a thick-panel mechanism can be built is the largest of those and not the mean.

Nothing to average over is the metamaterial version, and it is the same complaint about a different statistic: a folded corrugation is reported with a single Poisson’s ratio, which is an average over cells, and on a sheet with no repeating cell the cells run from −3.5 to +0.4.

The pattern across all three is worth naming. A folded sheet is an object whose interesting quantities are extremes, and the literature reports means, because a mean is what an area calculation produces and an area calculation is what a folded sheet invites. The reason a repeated cell is the honest case is that on it the mean and the extreme differ by a constant, so the mean can be converted; on anything else it cannot.

What a leaf gains by not being flat #

There is a question the numbers pose and do not answer, and it belongs here rather than in the caveats because it is the obvious next thing to ask.

A corrugation of r rows holds 2.077r footprints of lamina and is 4r layers deep at its worst point. Both are linear, so the trade between them is fixed: every extra footprint of blade costs 1.926 layers of clearance, whatever size the packet already is. There is no economy of scale and no penalty for growing — a five-row packet is exactly two and a half times a two-row packet in both quantities.

That is unusual and it is a property of the repeated cell rather than of folding. Circle packing has no such linearity: adding a flap to a design changes what every other flap can do, and the efficiency of a packing wanders as the count rises. A corrugation adds a row and nothing anywhere else changes, which is what makes it the pattern a growing organ can use — the packing does not have to be redesigned as the leaf gets bigger.

What the linearity does not buy is a way past the thickness. The clearance runs out at some number of rows, and when it does the pattern gives no warning: the packet has been growing at a constant rate the whole time and simply stops fitting.

Three things this does not say #

It does not say a leaf is one of these patterns. These are corrugations satisfying the conditions in this subject, folded exactly. A real leaf’s pleats are curved, unequal, and attached to a midrib that is not a line, and the organism is not the model is the standing caveat over all of it.

It does not measure a thickness. Every panel here is a zero-thickness plane. The layer count is a count of sheets over a point, and turning it into a millimetre needs a lamina thickness, which is somebody else’s measurement and appears nowhere in this file.

And 1.926 is a property of this pleat, not of corrugations. Change the taper and the constant changes. What is general is that the constant exists — that a repeated cell gives a fixed ratio between the deepest point and the average — and that it is greater than one, which is the same shape of statement a corrugation’s shrink turned out to be.

The check the sampling gets #

The layer count is a sampled quantity and everything above rests on it, so the check it gets is worth setting out rather than mentioning.

A folded state places every panel somewhere in the plane, and the layer count at a point is how many of those panels contain it. Counting that exactly would mean overlaying twenty-eight polygons and computing the arrangement, which is a different program; here it is sampled on a regular grid, and a sampled area is only as good as its grid.

What makes it trustworthy is that the answer is known in advance for one quantity. The layer count integrated over the footprint has to come back as the area of the sheet, because every square millimetre of lamina is somewhere in the pile and is counted once for each layer it is part of. On the four patches here the sheet areas are 0.3740, 0.5610, 0.7480 and 0.9350 and the integrated layer counts are 0.3739, 0.5609, 0.7478 and 0.9348 — agreement to four figures, on a quantity the sampling was not tuned against.

A grid that gets that right is a grid fine enough to report a maximum, which is the quantity the rest of the essay is about. It is not fine enough to report the shape of the deep band, and no claim about that shape is made.

What a folder should take from it #

Average depth is the wrong number for anything that has to fit. It is the right number for how much paper there is and the wrong one for how thick the packet is, and on a corrugation the two differ by nearly a factor of two.

A corrugation holds its footprint. Adding rows adds depth and nothing else, which is the property the whole strategy is built on and is measurable in one line: the footprint of these four patches is the same number four times.

And the constant is worth finding for any repeated pattern. It costs one layer map, it does not move with size, and it converts an area calculation into a thickness.