Which way the disc curves

Two of the most familiar shapes in a garden are opposites, and the opposition is exact. A ruffled lettuce leaf has too much edge for its span, so the margin waves. A doming petal or a cupped seed pod has too little, so the surface closes on itself.

Both are grown sheets. Neither was folded, bent or pressed by anything. The question this essay answers is what decides which of the two a given sheet becomes, and the answer is a good deal sharper than the phenomenon suggests.

The rule, stated first #

For an isotropically growing disc with linear growth factor Ω®, the curvature the metric forces is minus the flat Laplacian of ln Ω, divided by Ω². The division by Ω² is positive and cannot change any signs, so the whole question is the sign of the Laplacian of the logarithm.

That has a plain reading. The Laplacian of a function at a point compares its value there with the average of its values nearby. If the growth factor is larger further out — a rim that grew more than the middle — the Laplacian of its logarithm is positive and the curvature is negative. If the growth factor is larger in the middle, the curvature is positive.

So the rule is one line: growth concentrated at the rim opens a sheet, growth concentrated at the centre closes it. Nothing about the amount enters.

Running one parameter through zero #

The cleanest way to establish this is not to argue it but to build a family that passes through both cases and watch the sign flip.

The family used here is the simplest one that can: the growth factor is one plus a constant times the square of the radius. A positive constant means the rim grew more. A negative constant means the middle did. Zero means uniform growth, which is the control the first rung established as exactly flat.

The generator refuses to draw this figure unless both signs are present, which sounds like a technicality and is the assertion that makes the picture evidence. A version of the family that only reached one sign would still produce a handsome row of columns.

It also refuses to draw it if any column’s computed curvature has the same sign as its own growth parameter, which is the rule stated above turned into a condition the build has to satisfy. An indexing mistake in the differencing — the kind that produces a plausible curve pointing the wrong way — stops the site from building rather than appearing on a page under a confident caption.

The check that costs nothing and settles it #

There is a closed form available at the centre of the disc, and it is worth having because it is arrived at by a completely different route.

Expanding the logarithm of one plus a small quantity as a series and taking the flat Laplacian at the origin gives a curvature of exactly minus four times the growth parameter. No grid, no differencing, no boundary treatment — three lines of algebra. The computation on the radial grid has to reproduce it, and does, to about one part in a million.

That is the check that separates this figure from a diagram. A differencing scheme that had its sign convention backwards would produce the picture a reader expects and disagree with the series expansion in the first digit.

The closed form holds everywhere, not just at the centre #

The series expansion at the origin is a check on the differencing, and for this family the whole radial profile comes out in closed form as easily — which turns “the curvature is one sign across the whole sheet” from an observation into a proof.

With $\Omega = 1 + ar^{2}$, the flat Laplacian of $\ln\Omega$ is

$$\Delta \ln\Omega = \frac{1}{r}\frac{d}{dr}\!\left(\frac{2ar^{2}}{1+ar^{2}}\right) = \frac{4a}{(1+ar^{2})^{2}}$$

and dividing by $\Omega^{2}$ and negating gives

$$K(r) = -\frac{4a}{(1+ar^{2})^{4}}$$

At the centre that is $-4a$, which is the series result. Everywhere else the denominator is a fourth power of a positive quantity, so the sign is constant over the whole disc and cannot flip at any radius. The figure’s claim about the whole sheet is a theorem for this family rather than a reading of a plot.

And it says where the curvature concentrates #

The profile carries something the centre value cannot, and it runs against intuition in both directions.

For a rim-heavy sheet, $a > 0$, so $\Omega$ grows with radius and $K$ falls as $\Omega^{-4}$. The saddle is sharpest at the middle and flattens toward the edge that caused it.

For a centre-heavy sheet, $a < 0$, so $\Omega$ shrinks with radius and $|K|$ rises. The dome is gentlest where the extra growth is and sharpest at the rim.

In both cases the curvature is concentrated at the opposite end of the disc from the growth that produced it. That is not obvious from the sign rule, and it is exactly the sort of thing a formula supplies and a schematic column cannot.

The excess of edge, integrated #

One more thing falls out and it connects the sign rule to the ruffle directly.

Integrating $K$ over the grown disc, with area element $\Omega^{2} r\,dr\,d\theta$, everything cancels but one term:

$$\int K \, dA = -\frac{4\pi a R^{2}}{1 + aR^{2}}$$

By Gauss–Bonnet the boundary’s total turning is $2\pi$ minus that, so a rim-heavy sheet’s edge turns through more than a full circle — by exactly $4\pi aR^{2}/(1+aR^{2})$.

That excess is the ruffle, priced. It is bounded above by $4\pi$ however extreme the growth, which says something the sign rule does not: there is a ceiling on how much extra turning this family of growth profiles can put into a margin, and a leaf frillier than that ceiling needs a growth field of a different shape rather than a stronger one.

Why “more growth, more curvature” is wrong #

The intuition worth dismantling is the one that treats growth like a force: more of it, more effect.

A sheet grown uniformly by a factor of three has been enlarged enormously and its curvature is zero. A sheet grown by four percent at the rim and two percent at the centre has barely changed size and has a curvature that forbids flatness outright. The quantity that matters is not the growth but its variation, and specifically the second derivative of its logarithm — a quantity that can be large when the growth is small and zero when the growth is huge.

This is the same shape of error as reading a bank balance off a rate of change, and it is common in loose accounts of morphogenesis. The correction is not subtle once the formula is written down, which is a good reason to write the formula down.

The logarithm is the part of the formula that does the work, and it is worth a sentence of its own. Taking the logarithm before differentiating is what makes the rule scale-free: multiplying the whole growth field by a constant adds a constant to its logarithm, and a constant contributes nothing to a Laplacian. So doubling every growth factor on the sheet — genuinely doubling the amount of growth everywhere — leaves the curvature at every point exactly where it was. The curvature is a statement about the ratios between neighbouring growths and about nothing else.

That also disposes of a related intuition, which is that a sheet must curve more as it keeps growing. It curves more only if the unevenness increases. A sheet that grows for a long time with a fixed profile of relative growth arrives at a larger sheet with the same curvature at corresponding points, which is a different thing from a more strongly curved one.

There is a second, sharper consequence. Two sheets can have the same total growth in area and opposite curvatures, because the total is an integral of Ω² and says nothing about where Ω is large. The area factor is printed under each column in the figure above for exactly this reason: the columns at plus and minus the same parameter have nearly the same area ratio and opposite signs of curvature.

What each sign costs, if the sheet has to be made rather than grown #

The two signs are symmetric in the mathematics and are wildly asymmetric in practice, which is where this connects to the rest of the site.

A dome is the case a maker cannot have for free. A positively curved surface is not developable, so no flat sheet reaches it by bending, and everything on this site about what a flat sheet can become is about the three ways round that restriction and what each costs.

A ruffle is the case a maker gets whether or not it was wanted. Negative curvature is what an over-long edge produces on its own, and a sheet with too much material near its boundary will find some way to wave. That is why a badly cut hem ripples and a well cut one does not, and it is why the failure mode of a stretched membrane is a wrinkle rather than a dome.

The asymmetry is a fact about which sign a flat starting point is close to, not a fact about the geometry. Growth does not care and produces both.

The folding answer to the same question #

Folding has its own way of buying a sign, and it is worth putting beside the growth answer because the two are so nearly complementary.

A crease buys positive curvature by removing material from a neighbourhood — not physically, but in the sense that the paper’s angle around a point ends up less than a full turn. A cone is what that produces, and it is the folded equivalent of the centre-heavy growth profile: a dome with all of its curvature at one place.

Buying the other sign by folding is harder in a way that is worth noticing. A flat sheet has exactly 360° of paper around every interior point, and folding can only ever bring the surface to less than that around a vertex, never more. So a folded sheet reaches the negative sign only by using a boundary or by curving the crease itself, which is a different and much more interesting move.

That figure is the closest folding gets to a ruffle, and the mechanism is genuinely different from growth’s. The paper’s metric is untouched; every distance along the sheet is what it was. What happens is that the crease’s own curvature forces the developable surfaces either side of it to bend, and the result reads as a saddle to anybody looking at it from outside.

So there are two ways to get a surface that will not lie flat, and they leave different fingerprints. Growth changes the metric, and the flattened sheet will not lie flat without tearing. Curved creasing leaves the metric alone, and the sheet flattens perfectly once the creases are opened. Unrolling the thing is the experiment that tells them apart, and it is available to anybody with a leaf and a sheet of paper.

Where a leaf sits on this axis, and what is not being claimed #

Plants make both signs and often on the same organ, which is the honest complication.

A leaf whose margin grows faster than its interior has the rim-heavy profile and negatively curved edges — the frill on a kale leaf and the wave on a lettuce margin are usually described this way in the literature. A petal that cups, a pitcher, and the closing of a seed pod are the other sign. Some organs run both, with a domed centre and a ruffled edge, which in this model is a growth profile whose logarithm curves one way near the middle and the other way near the rim.

None of that is established here. No leaf was measured for this essay, no growth field was fitted to a specimen, and the profiles in the figures are chosen to make the geometry legible rather than to describe any species. What this site computes is the implication — given a growth profile, this is the curvature — and the antecedent is somebody else’s measurement. That division is the field’s standing rule and this essay is the one where it is most tempting to forget it.

What the picture cannot show #

The columns in the hero figure are schematics and the essay has said so twice, which is not excessive.

What is computed is a number per column — the Gauss curvature at the centre of the disc, and the whole radial profile behind it. What is drawn under each column is a parabola with that sign, chosen so a reader can see which way the number points. It is not a solved surface, it is not the shape the sheet takes, and its depth means nothing.

Solving for the shape is a genuinely hard problem. A prescribed metric may admit many isometric embeddings, or none that are smooth, and picking the one a physical sheet takes is a minimisation with a material in it. This site does not do that minimisation anywhere and does not have the standing to: elastic energy is illustrated-physics.com’s subject.

The other limit is that the model has one spatial variable. A growth field that varies with angle as well as radius is a different and much larger problem, and it is the realistic case for anything with a midrib, a vein pattern, or a direction it grows along. What survives is the sign rule, which is local: at any point, the curvature is set by how the growth varies in the neighbourhood of that point, whatever the global arrangement.

The idealisation, named #

The sheet is a surface with no thickness, so growth has one value per point rather than one per face. That is the same idealisation the first rung named, and this essay leans on it harder because the sign is the subject.

A sheet that grows more on its upper face than its lower face bends, and the direction of the bend has nothing to do with the argument here. It is not a curvature the metric forces — the metric of a bending bilayer can be perfectly flat — and it will not show up anywhere in these figures. That mechanism is what opens a pine cone in dry air and closes it in wet, and it is invisible to every computation on this page.

Untangling the two on a real organ needs a measurement rather than a model: whether the flattened organ lies flat. If it does, the curvature came from the thickness; if it will not lie flat without tearing, the metric is not flat and the growth was uneven in the plane.

That test is the one honest empirical claim this essay is in a position to make, and it is worth stating as a procedure rather than a principle. Take the organ, cut it radially from the rim toward the centre, and try to press it flat. A bilayer curvature relaxes and the piece lies down. A metric curvature does not relax, because there is nothing to relax — the extra material is still there — and the cut edges either overlap or gape, by an amount that grows with the curvature the sheet was carrying.

Anybody who has flattened a dried orange peel has run the positive-sign version of that experiment, and the gaps are the reason a peel cannot be pressed into a disc without tearing.

Where this ladder goes next #

The sign settles what kind of surface the sheet is asking for. It does not settle the shape, and the gap between those two is the next rung.

A sheet with too much edge has to spend the excess somewhere, and it can spend the same excess in a few large waves or many small ones with the metric completely indifferent between them. That is the point at which geometry runs out of things to say and hands the question over, and it is worth watching carefully because the handover is usually made silently.

Further along, the comparison with folding gets sharper. A crease is curvature too — concentrated rather than spread — and a cone and a grown disc can be made to carry exactly the same amount of it while looking nothing alike. The sign rule survives that comparison intact: a cone made by removing a wedge is the positive case, and one made by inserting a wedge, which a sheet of paper cannot do and a growing sheet can, is the negative one.