The excess does not choose its waves

A lettuce leaf’s margin waves a few times across its width. A kale leaf’s waves many more times, and each wave is smaller. Both are edges that have grown longer than the span they have to fit into, and the obvious question is what sets the count.

The answer this essay can give is a negative one, and it is unusually clean. The excess of length does not set the count at all. It sets a relationship between the count and the height of the waves, and every pair satisfying that relationship has exactly the same metric — the same arc length, the same excess, the same everything a measurement along the surface could detect.

What “the same metric” means for a curve #

For a one-dimensional edge the metric is arc length and nothing else. Two curves have the same metric exactly when they have the same length, and a creature confined to the curve — able to walk along it and measure distance, unable to look at it from outside — could not tell one from the other by any measurement whatsoever.

That is a low bar for a curve and it is exactly the bar the growth argument sets. What growth changes is length. So what growth determines is length, and it determines nothing else at all about how the length is arranged in the plane.

It is worth dwelling on how much information that discards, because the discarding is the result. A curve in the plane has a shape, a wave count, an amplitude, a symmetry and an orientation. Arc length is one number. The metric of a curve is therefore an almost comically weak description of it, and any argument that reasons from a growth field to a length and then expects a shape to fall out has thrown away nearly everything on the way.

Surfaces are better off, which is why the two-dimensional version of this problem is not hopeless. A surface’s metric determines its Gauss curvature at every point, which is a whole function rather than a single number, and that is enough to forbid flatness outright. It is still not enough to fix an embedding, and the residual freedom is what this essay is about — but the gap between what the metric knows and what the shape is narrows considerably in going from a curve to a surface.

The figure above makes that concrete. The generator solves each amplitude by bisection until the numerically integrated arc length matches the target, and then asserts that it did: the worst residual across the family is about five parts in a thousand trillion, and the spread of excesses across the four profiles is smaller still. They are the same length to the last digit a double can carry.

The one thing the excess does fix #

The family is not arbitrary. There is one relationship, and it is worth extracting because it is what makes the four profiles look like one picture at four scales.

For a sinusoidal profile across a fixed span, the excess of arc length over the span depends on the amplitude and the wave count only through their product. So a fixed excess means a fixed product: twice as many waves is exactly half as tall.

The generator checks this rather than asserting it. Multiplying each solved amplitude by its own wave count gives 0.07969 at every one of the four counts, constant to better than a hundredth of a percent, and the figure refuses to draw if that constancy fails by more than two percent.

Where the closed form stops being right, measured #

There is a textbook shortcut for this: assume the slopes are small, expand the square root in the arc-length integral, and the excess comes out as a simple expression whose consequence is that the product of amplitude and wave count equals the square root of the excess divided by π.

It is a good approximation and this site prefers to know how good. At an excess of two percent the shortcut predicts 0.04502 against a solved 0.04535 — low by 0.73%. At six percent it predicts 0.07797 against 0.07969, low by 2.21%. At fourteen percent it predicts 0.11910 against 0.12508, low by 5.02%.

The shortcut is low at all three, and by a growing amount rather than a wandering one. So the small-slope form is excellent for a mild ruffle and visibly wrong for a strong one, and its error has a direction: it always says a ruffle needs less amplitude than it does.

So the small-slope form is excellent for a mild ruffle and visibly wrong for a strong one, and the crossover is somewhere a reader can now locate. That is the kind of number this site exists to produce: not “the approximation is good”, but the size of the error at three stated conditions, computed by a solver that never uses the approximation.

The next term, which recovers all three #

The three errors are 0.73, 2.21 and 5.02 per cent at excesses of two, six and fourteen. Divide each by its own excess and the quotients come out at 0.365, 0.368 and 0.359 — the same number three times, which says the correction is proportional to the excess and can be written down.

Carrying the arc-length expansion one term further does exactly that. Writing $X$ for $\pi^2 n^2 A^2$, the integral gives an excess of $X - \tfrac{3}{4}X^2$ rather than $X$, so inverting to first order in the excess $e$ gives $X \approx e(1 + \tfrac{3}{4}e)$ and

$$nA \approx \frac{\sqrt{e}}{\pi}\left(1 + \tfrac{3}{8}e\right).$$

Three eighths is 0.375, which is the number the three quotients are circling. Evaluate it: 0.04536 at two per cent against a solved 0.04535, 0.07972 at six against 0.07969, and 0.12534 at fourteen against 0.12508.

Four significant figures at every excess measured, from one extra term, against a bisection that integrates the arc length numerically and knows nothing about either expansion.

What that buys and what it does not #

Two things, and the second is the more useful.

It makes the shortcut usable in the regime a strongly ruffled margin is actually in. The bare square-root form is five per cent out at fourteen per cent excess, which is enough to matter and enough to be quoted anyway; the corrected form is a fiftieth of that, and it costs one multiplication.

And it says the error is a bias rather than a scatter, which changes what a reader should do with a published figure. Anyone who has used the small-slope form has an amplitude-times-count that is too small, by three eighths of the excess, every time — so a set of measurements reported through the shortcut is systematically shifted rather than noisy, and the shift is correctable after the fact from the excess alone.

None of that touches the essay’s finding. The relationship between amplitude and count is now fixed a little more precisely and it is still one relationship between two numbers, so the family is still one-parameter and the excess still declines to choose a member of it. What the correction improves is the accuracy with which the indifference can be stated.

The only ranking geometry can offer #

If the excess cannot pick a member, something has to, and it is fair to ask whether geometry has anything left to say. It has exactly one thing, and the honest move is to name it and then name its limits.

The one functional available with no material in it is the total squared curvature of the profile — the integral of the square of the curvature along the arc. It is a purely geometric quantity, and computing it across the family gives a clean result: dividing each profile’s value by the square of its wave count gives 4.311 at every one of the four counts, constant to four significant figures.

Total squared curvature goes as the square of the wave count, at fixed excess. So the geometry does have an ordering: the fewest waves is the smoothest solution by this measure, and eight waves costs sixteen times what two does.

That ordering is a fact and it is not an answer. Nothing establishes that a physical sheet minimises this functional, and a great many physical sheets demonstrably do not — a kale leaf has many waves and is not choosing the two-wave solution. The functional is offered here as the boundary marker of what geometry can supply, not as a prediction.

Where this argument stops, stated precisely #

Two boundaries, and both are boundaries with other sites in this fleet rather than gaps in the reasoning.

The selection of a wavelength by a physical mechanism is phyllotaxis.xyz’s ground, under its claim on reaction-diffusion and pattern formation, and this essay is deliberately clear of it. The argument above needs no chemical species, no diffusion, no dispersion relation and no instability; it needs an excess of arc length and nothing else. That is the separating test, and it is worth stating because the two arguments arrive at superficially similar pictures.

What a real sheet does with the excess is elasticity, which is illustrated-physics.com’s. The physical selection balances the cost of bending against the cost of stretching and against whatever the boundary conditions are, and it involves a thickness, a modulus and a length scale. None of those appears anywhere in this repository and none of them should.

So the honest statement of the result is a conditional with a large antecedent: given that the edge has a certain excess, the admissible profiles form this one-parameter family, and picking among them is a question this site poses and does not answer.

The thing that does break the tie, on a real sheet #

There is one tiebreaker this site is entitled to talk about, because it is about folding rather than about elasticity, and it is worth putting on the page.

A real sheet cannot bend to an arbitrarily small radius. Every crease on this site has a radius rather than a line, and the same is true of a smooth bend: below some radius the material yields, cracks, or delaminates. That puts a floor under the tightest curvature in the profile, and since the maximum curvature of the family grows with the wave count, it puts a ceiling on the count.

That is a geometric bound with a material input, which is the right shape for this site: the constraint is stated as a radius, the radius is somebody else’s measurement, and what is computed here is what a given radius forbids.

It is also a bound rather than a selection. It removes the top of the family and leaves everything below it, so a sheet with a small minimum radius still has a great many admissible profiles and still needs something else to choose.

Underdetermination is a familiar shape here #

It is worth noticing that this site has met this situation before, in a setting with no biology in it at all, and that recognising the shape is most of the skill.

Two local theorems decide whether a vertex folds flat, and they decide it completely. What they do not decide is the order the layers end up in, and the layer ordering is not a detail — it is what a folder is actually doing with their hands. The local conditions constrain the orderings to a set and are indifferent within it, which is precisely the relationship between the excess and the wave count here.

The same thing happens one level up, where every vertex of a pattern can satisfy both theorems and the sheet still fail to fold, and again at the checker, where the honest account of what a local test establishes is a list of things it does not.

The common structure is that a necessary condition carves the possibility space and does not point inside it. That is not a weakness of any of these results; it is what a necessary condition is. The mistake worth avoiding is reading a constraint that happens to be tight in a familiar case as though it were a formula.

What would settle it, and who would have to do the measuring #

Since geometry declines to pick, it is worth being explicit about what evidence would.

The wave count on a particular organ is settled by a mechanical calculation with the tissue’s own properties in it, run against the actual boundary the organ has. That calculation needs a thickness, a modulus, the growth field as a function of position, and the constraint the interior of the sheet imposes on its own edge — four inputs, none of which this repository has or could obtain.

What the geometry supplies to that calculation is the space it must search, and that is not nothing: knowing that the admissible profiles form a one-parameter family with a fixed product of amplitude and count reduces a search over all curves to a search over one number. A model that did not know the constraint would spend its effort rediscovering it.

This is also the point at which the field’s rule about organisms does real work rather than being a disclaimer. There is a version of this essay that reports a wave count for a named species, cites a paper for it, and lets the reader assume the count came out of the computation. The computation cannot produce a count and saying so is the whole result.

Where the excess came from #

It is worth closing the loop back to growth, because the excess in these figures arrived as an argument rather than as a measurement.

An edge acquires excess length because the rim grew more than the interior, which is the rim-heavy profile and the negative sign of curvature. The excess is then not a number somebody chose but a consequence of a growth field, and the growth field is what a biologist would measure.

The two-dimensional version is more subtle than the one-dimensional one, and this essay’s model is deliberately the simpler case. A real ruffled margin is a surface with negative curvature, not a curve with excess length, and the waves are a cross-section rather than the object. What survives from the simpler case is the underdetermination, which gets worse rather than better in two dimensions.

The idealisation, named #

The profile is a sinusoid, and nothing says it should be. The family is drawn as sinusoids because they are the one-parameter family a reader can hold in mind, and the real space of curves with a given length across a given span is infinite-dimensional.

That makes the argument stronger rather than weaker. If restricting to a single shape family already leaves a one-parameter indifference, allowing every curve of the right length leaves vastly more. The figure understates the underdetermination it is drawn to establish.

The second idealisation is that the span is held fixed while the arc grows, which treats the interior of the sheet as rigid. A real sheet can relieve some of the excess by pulling its own interior about, and where it does the effective excess at the edge is smaller than the growth field implies.

Where this ladder goes next #

The next rung changes what the curvature is allowed to do rather than how much of it there is.

Growth spreads curvature over an area. A crease concentrates it onto a point. Matching the two — building a growth field carrying exactly the deficit of a cone — turns out to be possible and to produce two surfaces that share one number and nothing else, which is the sharpest statement this field has about what folding and growing have in common.

Sideways from here, the underdetermination is not a defect peculiar to growth. A crease pattern’s layer ordering is the same shape of problem in a completely different setting: the local conditions leave a set of possibilities and the physical answer needs something the local conditions do not contain. Recognising when a model has stopped determining its answer is a skill this site keeps having to use, and it is easiest to learn on a case where the leftover freedom can be drawn.