A circle sees only the average

A cut reads a slope found what a circular cut through a grown disc measures. The disc grows by a factor $\Omega$ that depends on the radius alone; a cut along the circle of radius $r$ leaves an inner disc with a rim of its own, and a rim measurement on that piece reads its total curvature, $-2\pi r\,(\ln\Omega)'(r)$. A set of cuts reads the slope at a set of radii, the slopes add up to the growth profile, and the one thing no cut recovers is a uniform enlargement of the whole sheet — which is the one kind of growth that curves nothing.

It ended on the obvious weakness. Four marks were arranged about a centre and every cut was a circle about it, because the growth was arranged in rings. A leaf is not grown in rings. It grows from a base, along a midrib, faster at the margins or faster toward the tip, and the question left was which cuts read a growth like that, and whether a small set of them recovers it as circles recovered the radial profile.

The integration by parts behind the circular result never used the symmetry, so the first answer is general and short. What changes is everything that answer says about circles.

What any cut reads #

Growth that is the same in every direction at a point — a factor $\Omega(x, y)$ on lengths, wherever on the flat sheet the point started — gives the grown sheet the metric $\Omega^2(dx^2 + dy^2)$. Its curvature is $K = -\Delta(\ln\Omega)/\Omega^2$, where $\Delta$ is the flat sheet’s own Laplacian, and the grown area is $\Omega^2$ times the flat. So over any piece $D$ of the flat sheet the total curvature is

$$\int_D K\,dA = -\int_D \Delta\ln\Omega\;dx\,dy = -\oint_{\partial D} \partial_n \ln\Omega\;ds.$$

The second equality is the divergence theorem: the integral of a Laplacian over a region is the flux of the gradient out through its boundary. A cut of any shape reads the outward flux of the gradient of $\ln\Omega$ across it, and nothing else. On a circle round the centre of a radial growth the gradient points straight out with magnitude $(\ln\Omega)'(r)$ all the way round, and the flux is $2\pi r$ times that — the slope the earlier essay found. Every figure below computes each reading twice, as a flux along the cut and as an integral of the Laplacian over the piece, and the two agree.

That one line is the whole of the general case. The rest is what it implies for the cuts a person would actually make.

Two sheets the circles cannot tell apart #

Take two discs grown by the same small amount, $a = 0.35$. One is grown at its margins: the growth depends only on the distance $y$ from a midrib through the centre, $\ln\Omega = a y^2$, as a leaf grows more toward its edges than along its rib. The other is grown at its rim: $\ln\Omega = a r^2/2$, as the earlier essays’ discs were. Neither can lie flat; both carry curvature of the saddle sign everywhere.

The maps show two different surfaces. The margin-grown sheet’s curvature is constant along every line parallel to the midrib and changes across it; the rim-grown sheet’s is constant round every circle. At the same point of the flat sheet the two curvatures differ by as much as 0.19, against a largest value of 0.70. A specimen of one would never be mistaken for a specimen of the other by anyone who could measure curvature at a point.

Both are the sign which way the disc curves assigned to growth that rises toward the outside: more at the edge opens the sheet, and the sheet ruffles rather than domes. A margin grown more than the lamina inside it has more edge than a flat outline can hold, and the excess does not choose its waves found that the metric fixes how much extra edge there is and says nothing about how many waves it is spent in; the container picks the member found what does. On the margin-grown disc the extra is laid out along lines parallel to the midrib rather than round circles, so a kale leaf and a ruffled disc spend the same total of curvature in different places — and the question here is which instruments can tell.

Every centred circle reads them identically. The three cuts drawn read $-0.2694$, $-0.9291$ and $-1.9847$ on the margin-grown sheet and exactly the same on the rim-grown one. The reason is in the formula: a cut reads $-\int \Delta\ln\Omega$ over the piece inside it, and here $\Delta(a y^2) = 2a$ and $\Delta(a r^2/2) = 2a$. The two growths have the same Laplacian everywhere, so every piece of the flat sheet — every disc, centred or not, and every other shape — holds the same total curvature on both.

Four growths, two readings #

The coincidence above is exact for a reason particular to quadratics. The circles’ blindness is more general, and the second pair shows it on its own.

A sheet grown more sharply toward its margins, $\ln\Omega = a y^4$, has a Laplacian $12 a y^2$ that is zero along the midrib and largest at the edges. Its average round the circle of radius $r$ is $6 a r^2$, and the radial growth with exactly that Laplacian is $\ln\Omega = 3 a r^4/8$. The two are different growths with different Laplacians — one is banded, the other ringed — and a centred circle cannot tell them apart, because the integral of the Laplacian over a disc about the centre sums each ring of the disc whole, and summing a ring keeps only its average.

That is the general statement, and it is the leaf’s problem. Write the log of any growth in polar coordinates as a sum of angular harmonics, $\ln\Omega = u_0(r) + \sum_k u_k(r)\cos(k\theta - \phi_k)$. A centred circle reads only $u_0$: every harmonic with $k \ge 1$ integrates to zero round the circle, in its flux as in its Laplacian. A leaf’s growth is almost all angular — more along one axis than the other, more toward the tip than the base — and a set of circles about any centre reads the leaf as the radial average of itself. The reconstruction the earlier essay made from circles would return a disc grown at its rim and report it accurately.

Cuts that follow the growth #

The cure is in the same formula. A cut reads the flux of the gradient across it, so a cut laid along the direction the gradient points reads it in full, and a cut laid across that direction reads nothing of it.

On the sheet grown toward its margins, a thin rectangle laid along the midrib, its long sides parallel to it, reads the change in the gradient between its two long sides — and as it slides from the midrib toward a margin its reading runs from almost nothing, −0.0002, to −0.1237. That is the margin growth read row by row, the leaf’s version of a circle reading a slope. A rectangle laid across the midrib reads −0.0076 wherever it is slid, because the growth does not change along the midrib and the cut sees the same rows wherever it sits. On the radial average both kinds of cut vary as they slide, and the two directions are no longer distinguishable from each other.

So a set of cuts recovers a margin growth exactly as circles recovered a ring growth, provided they follow the growth’s own lines: long cuts parallel to the midrib at a series of distances from it, each reading the slope of the log of the growth at its two edges, integrated outward from the rib. The general rule is the one the radial case concealed. Circles were the right cuts for ring growth because circles are the level lines of ring growth. For any growth, the cuts that read it are the ones that follow the lines along which it is constant, and the cuts that read nothing of it are the ones that cross them square.

That makes the instrument something a person could use on a specimen whose growth is unknown. Cut a small rectangle, turn it, cut another: the orientation at which the reading stops changing as the cut slides is the direction the growth is constant in.

The growth no cut can read #

The first pair, margin and rim, is worse than the second, and the reason is worth a figure of its own. The second pair differs in its Laplacian, so cuts laid along the midrib separate them. The first pair has the same Laplacian everywhere, and so does any pair whose logs differ by a function with no Laplacian at all.

The difference between the two growths is $\tfrac{a}{2}(y^2 - x^2)$, more growth along one axis than the other, and its Laplacian is zero at every point. A cut of any shape, anywhere on the sheet, reads its flux, and a function with no Laplacian has no net flux out of any closed curve: what enters across one side leaves across another. No cut, however placed, can tell these two sheets apart, though their curvature differs from point to point.

The earlier essay found exactly one growth that no cut could see — a uniform enlargement, a constant added to the log — and observed that it was the one kind of growth that curves nothing. In the plane that kind is far larger. A function with no Laplacian is a harmonic function, and the harmonic functions are the real parts of analytic ones; $\ln\Omega$ is harmonic exactly when $\Omega$ is the stretching factor $|f'|$ of an angle-keeping map $f$ of the plane. Growth like that is a conformal map: the grown sheet is an image of the flat one under a map that keeps every angle, and it lies flat. Uniform enlargement is the only member of the family that is the same at every radius, which is why the radial argument met only that one.

Grown unevenly, and flat #

The plainest member is growth that rises exponentially along one direction: $\Omega = e^{a x}$, a leaf growing faster toward its tip at a constant proportional rate. Its log, $a x$, is a linear function and has no Laplacian, so the grown sheet is flat — the image of the disc under $(e^{a z} - 1)/a$, drawn exactly. At $a = 0.35$ the cells nearest the growing end have 1.88 times the area of the flat sheet’s and those at the other end 0.53 times: a factor of three and a half from end to end, every crossing of the grid still square to within a ten-thousandth of a degree. Three cuts anywhere on it read zero to fourteen decimal places.

A sheet can grow very unevenly and stay flat, provided it grows the same amount in every direction at each point and the log of that amount has no Laplacian. A linear rate of growth along the leaf, $\Omega = 1 + bx$, does not qualify: its log has Laplacian $-b^2/(1 + bx)^2$, and the sheet domes. The exponential is the borderline that a gradient of growth along one axis has to hit exactly to leave the sheet flat, and any departure from it in either direction is read by cuts laid across the gradient.

What the cuts recover, and what they do not #

Put together: cuts of every shape read $\Delta\ln\Omega$ as a density on the flat sheet, piece by piece, and nothing more. From that density the growth is determined up to a harmonic function — the growth that curves nothing — and the earlier essay’s “up to a uniform enlargement” is the radial shadow of that statement. A small set of cuts recovers the density only along the lines it follows, as circles recovered it ring by ring and as midrib-parallel strips recover a margin growth row by row.

The harmonic part is not invisible to everything. Three marks see nothing found that four marks on a sheet see curvature a rim misses, because marks measure distances on the grown sheet, and distances are the metric itself rather than an integral of its Laplacian. The margin-grown and rim-grown discs, indistinguishable to every cut, differ in curvature at a point and differ in the distances between marks. Cuts and marks measure different things once the growth is not round: a cut reads how much curvature a piece holds, and marks read where it is. For radial growth the two coincide closely enough that either recovers the profile; in the plane only the marks reach the harmonic part.

A formula on a disc is not a leaf #

A leaf. Every growth here is a formula on a disc, and none is a measurement of a specimen. The leaf the figures stand for is a disc with a midrib drawn through it, grown by a stated function; a real leaf has an outline, veins that grow differently from the lamina between them, and a growth field nobody has written down.

Growth that is different in different directions. The whole argument assumes the growth at a point is the same factor in every direction, so that the metric is $\Omega^2$ times the flat one. Leaves are known to grow anisotropically, faster in some directions than in others at the same point, and an anisotropic growth has a metric that is not a single factor times the flat one. The total curvature of a piece is still a boundary quantity of the piece, by Gauss and Bonnet, but it is no longer the flux of a single gradient, and which cuts read it would have to be worked out again from the metric’s own connection.

The cut itself. A cut is taken to release the piece without disturbing its curvature, and its rim measurement to read the piece’s total exactly. The earlier essays on rim measurement made the same assumption and named it.

Isotropic, smooth, and cut perfectly #

Isotropic growth, small and smooth, on a flat disc of radius one, read by perfect cuts. The growth factor is the same in every direction at a point and differentiable everywhere; the curvature is the curvature of the grown metric, whether or not the sheet could actually take that shape in space; and a cut reads its piece’s total curvature with no error. Growth strengths are a single parameter, $a = 0.35$, chosen to make the maps legible.

Every reading taken twice #

Every reading is computed two ways: as the flux of the gradient of $\ln\Omega$ along the cut, by quadrature at 720 points round a circle or 200 along each side of a rectangle, and as the integral of the Laplacian over the piece on a fine grid. The two are required to agree, and they agree to two parts in ten thousand on a circle and one in a million on a rectangle.

The identical readings are required to agree to $10^{-9}$ on every centred circle drawn, the curvatures to differ by more than a tenth of the largest at some point, and the harmonic difference’s Laplacian to vanish at every cell of the map, with three off-centre cuts reading less than $10^{-9}$.

The flat growth is drawn as the exact image of the disc’s grid under its conformal map, with nothing relaxed; its grid crossings are required to be square to a thousandth of a degree and three cuts on it to read zero.

Still open: anisotropic growth, and a cut that reads direction #

The isotropic assumption is the one the leaf most obviously breaks, and it is the one to remove next. A leaf is also not one uniform material: nothing grown was cut out of anything looked at the pleats some leaves are folded in, which run along the veins rather than round any centre, and a growth that produces pleats like that is organised along the same lines. An anisotropic growth has two stretch factors and a direction at every point, and its curvature is no longer minus a Laplacian over a factor. Each piece’s total is still a boundary quantity — the turning of its rim on the grown surface — but it is the circulation of the metric’s connection rather than the flux of a gradient, and there is no reason to expect the circles, the strips or anything else to stay blind to the same things. Whether a family of cuts can recover the direction of growth as well as its amount is the question that would make cuts a field instrument for real leaves.

The other continuation is the one this field keeps meeting. The test measures the rim and a cut reads a slope specified instruments closely enough to read a specimen, and the specification here adds an instruction: turn the cut until its reading stops changing as it slides, and that is the direction the growth is constant in. A single grown leaf, cut that way at a few orientations, would test it.

Sideways from here, the harmonic blind spot has a counterpart in a crease carries no curvature, which found that folding along a crease changes how a sheet sits in space and nothing about its intrinsic geometry. Conformal growth is the opposite case: it changes the intrinsic geometry everywhere — areas by a factor of three and a half — and leaves the total curvature, and so every boundary reading, exactly where it was. A sheet that grows cannot lie flat was true of every growth this field had drawn; the exponential is a growth that can.

The habit worth carrying is about symmetric instruments. When an instrument shares the symmetry of the thing it was designed for, ask what it averages away on anything else. A circle about a centre reads everything a ring growth does and only the ring-average of every other growth, and the reading gives no sign that anything was averaged.