A cut cannot tell which way a leaf grew

A circle sees only the average found what any cut through a grown sheet reads, under one assumption: that the growth at a point is the same factor in every direction. A cut of any shape then reads the flux of the gradient of the log of the growth across its edge, and nothing else. The essay ended on the assumption it could not keep. Leaves grow faster in some directions than in others at the same point, and an anisotropic growth has two stretch factors and a direction at every point, not one factor. It asked whether a family of cuts could then recover the direction of growth as well as its amount, and it offered an instruction that might: turn the cut until its reading stops changing as it slides, and that is the direction the growth is constant in.

The answer is no, and it is no for a reason that can be written in one line. A cut still reads a flux across its edge. The field it reads the flux of is built from the growth in a way that loses the direction, and many growths that point in entirely different directions build the same field.

The field a cut reads #

Keep the directions fixed, as a leaf’s midrib fixes them: the sheet stretches by $e^{p}$ along the midrib, $x$, and by $e^{q}$ across it, $y$, where $p$ and $q$ can vary from point to point. The grown sheet’s metric is then $e^{2p}\,dx^2 + e^{2q}\,dy^2$. When $p = q$ that is the isotropic growth of the earlier essays; when they differ, a small circle on the flat sheet grows into an ellipse, wider along the midrib if $p > q$ and taller across it if $q > p$. The figures draw those ellipses at a grid of points, which is exactly the picture cartographers draw over a map projection to show its distortion: Tissot’s indicatrix, named for Nicolas Auguste Tissot, who used it in the nineteenth century to compare projections. A grown leaf is a map from the flat sheet to the grown one, and its ellipses say how each small piece was stretched.

The classical formula for the curvature of a metric with perpendicular coordinates gives, for this one,

$$K = -e^{-(p+q)}\left[\partial_x\!\left(q_x\, e^{q-p}\right) + \partial_y\!\left(p_y\, e^{p-q}\right)\right],$$

and the grown area of a small piece is $e^{p+q}$ times its flat area. Multiplying the two cancels the exponential in front, and the divergence theorem turns the integral over a piece $D$ into one round its edge:

$$\int_D K\,dA = -\oint_{\partial D} \mathbf V\cdot\mathbf n\;ds, \qquad \mathbf V = \left(q_x\, e^{q-p},\; p_y\, e^{p-q}\right).$$

When $p = q = u$ the field $\mathbf V$ is the gradient of $u$, and the earlier essay’s result comes back. When they differ it is not a gradient of anything. Its first component is how fast the stretch across the midrib, $q$, changes as one moves along it; its second, how fast the stretch along the midrib, $p$, changes as one moves across it. Each component is the change of one direction’s growth in the other direction.

Two things follow at once, and the rest of the essay draws them.

Growth that varies along itself curves nothing #

The first is that the change of a direction’s growth along that same direction does not appear in $\mathbf V$ at all. Growth along the midrib that is faster toward the tip than the base — $p$ depending on $x$ alone, $q$ zero — has $\mathbf V = 0$ everywhere.

The sheet on the left grows by nearly five to one from base to tip and stays flat. Every cut reads zero. The reason is visible without the formula: a growth that lengthens each row of the sheet along itself, by amounts that depend only on where along the row a point is, lengthens every row in the same way. The rows stay the same length as each other, lie side by side as before, and the sheet is the flat sheet stretched — an affine change in each row, with nothing to make any row disagree with its neighbour.

The sheet on the right lengthens its rows by different amounts at different distances from the midrib, so the outer rows come out longer than the inner ones. Rows that lie side by side and disagree in length cannot all lie flat, and the cuts read $-2.25$, $-0.82$ and $-1.33$. That is the familiar mechanism of a ruffled leaf margin. Which way the disc curves found the same sign for growth rising toward an edge, and it is the model physicists use for the ripples on a torn plastic sheet and the frills of a lettuce leaf: the edge has grown longer than the lamina inside it can accommodate flat.

So the curvature of a grown sheet comes from growth that changes across its own direction, never along it. Isotropic growth hid this, because there the growth along every direction is the same function and its change across any direction is its change across every direction.

Each side of a cut reads one direction #

The second consequence is how a cut reads $\mathbf V$. A rectangular cut with its sides along and across the midrib has two kinds of side, and each sees only one component.

On this disc the stretch along the midrib grows toward the margins and the stretch across it grows toward the ends. The rectangle’s bottom side, along the midrib, contributes $+0.173$ and its top side $-0.905$: between them they read how much faster the growth along the midrib is at the top of the rectangle than at the bottom. Its two sides across the midrib contribute $-0.131$ and $-0.446$, and read the same thing for the growth across. The four add to $-1.310$, and the angle defects inside the rectangle, computed from nothing but the lengths of a fine triangulation’s edges in the grown sheet, add to the same total to within three parts in a thousand.

That is the generalisation of the earlier essay’s instruction, and the right form of it. A side of a cut reads how fast the growth along that side changes across it. A cut whose sides run along the midrib reads the midrib growth’s variation across the leaf; a cut whose sides cross the midrib reads the cross growth’s variation along it.

A family no cut can separate #

Now the main result. Two growths with the same $\mathbf V$ hold the same curvature in every piece of the sheet, whatever else differs between them, because every cut reads $\mathbf V$ and only $\mathbf V$. And the condition for two growths to share a $\mathbf V$ leaves a great deal free.

Take the isotropic growth of the earlier essay’s margin-grown disc, $\Omega = e^{a y^2}$, whose field is $\mathbf V = (0,\, 2ay)$. Any growth with no change of its cross-growth along the midrib, $q = q(y)$, has a first component of zero, and its second component, $p_y\,e^{p-q}$, equals $2ay$ exactly when

$$e^{p} = 1 + \int_0^{y} 2at\, e^{q(t)}\,dt .$$

Every choice of the growth across the midrib has a growth along it that makes the pair indistinguishable, by any cut, from the isotropic disc. With no growth across at all, $q = 0$, the partner is $e^{p} = 1 + a y^2$: a sheet grown only along its midrib, by a factor rising toward the margins. With $q = a y^2$ it is the isotropic disc itself. With $q = 2ay^2$, growth more across than along.

The three cuts — a centred circle, an off-centre circle and a rectangle — read $-1.8095$, $-0.6157$ and $-0.8000$ on all four members. The off-centre circle’s reading agrees with the angle defects on every member. Meanwhile the grown disc’s area runs from 1.20 times the flat disc’s, for growth along the midrib alone, to 2.46 times, for growth across at twice the rate along. A line across the sheet near its margin, 1.200 long when flat, grows to 1.81 on one member and 2.27 on another.

These are not subtle differences. One sheet has twice the paper of another, its ellipses point the other way, and a ruler laid across it reads a quarter more. Every cut of every shape reads them identically, because the one thing a cut can read — the total curvature of the piece it frees — is the same on all of them, piece by piece.

That family happens to hold a curvature of $-2a$ per unit of flat area everywhere, the margin-grown disc’s, so every cut reads $-2a$ times the flat area it encloses; but the construction does not depend on that. Any isotropic growth $u(x, y)$ whose field has no first component — one that varies only across the midrib — has the same kind of family, built the same way, and for growth that varies in both directions the same argument with both components free gives families larger still. The direction is not merely hard for cuts to read. It is not in what they read.

Three functions in, one out #

The family is not a coincidence of the margin-grown disc, and counting shows why it had to exist.

A growth that can be faster one way than another is described at each point by three numbers: the two stretch factors and the direction of the faster one. Over a sheet that is three functions. Its curvature is one function, and the cuts read less than that — the curvature’s totals over pieces, from which the curvature itself can be recovered by taking ever smaller pieces, and nothing more. So the cuts can at best recover one function out of three. Two functions’ worth of growth are left over for every pattern of curvature, and the family above, with its free choice of the growth across the midrib, is a one-function slice of that remainder.

Isotropic growth fitted the instrument almost exactly. It is one function per point, the factor $\Omega$, and its curvature is one function, so cuts could recover it up to the growths that curve nothing — a uniform enlargement in a cut reads a slope, and the larger harmonic family in the plane. That near-fit was a property of the assumption, not of the instrument. The cut was never measuring growth. It was measuring the one combination of growth that the test measures the rim identified as a boundary quantity, and isotropy made that combination nearly the whole story.

The same shape of result has turned up before in this subject at the next level up. The excess does not choose its waves found that a grown margin’s metric fixes how much extra length the edge has and says nothing about how many waves it is spent in: the metric is one level of description and the shape in space another, and the first does not determine the second. Here the curvature is one level and the metric another, and again the first does not determine the second. Each instrument reads one level, and the level below it carries more than it shows.

What the instruction finds #

The earlier essay’s instruction was to turn a cut until its reading stops changing as it slides. That still finds something, and the family shows what.

On the sheet grown only along its midrib, a square slid along the midrib reads the same everywhere and a square slid across it does not, so the instruction returns the midrib’s direction, which is the direction of growth. On its isotropic twin it returns the same direction, by exactly the same readings, and that sheet has no direction of growth at all — every small circle on it grows into a circle. The instruction finds the direction along which the curvature density is constant, which is a property of the field $\mathbf V$, and it coincides with a direction of growth only when there happens to be one aligned with it.

For the family above, whose curvature density is the same everywhere, the instruction is worse than ambiguous: a cut slid in any direction reads the same, so it reports every direction alike, on a sheet grown only along its midrib.

What reads the direction instead #

Cuts measure curvature, and curvature is an integral over the metric that has thrown its directions away. The measurements that keep them are the ones that read the metric itself: lengths, areas and angles on the grown sheet.

The family table already contains two such measurements, and both separate every member. The area of the grown disc is a single number; the length of one line across the sheet is another. A line along the midrib and a line across it, measured on the grown sheet and compared with their flat lengths, read the two stretch factors directly, and a small circle drawn on the sheet before it grows and measured after would read the whole ellipse. Three marks see nothing and its sequel found that four marks see curvature that a rim reading misses, because marks read distances; for anisotropic growth the same instrument reads the direction too, because a set of distances between marks is the metric, sampled.

So the division of labour between the instruments sharpens. A cut reads how much curvature a piece holds. Marks read how the piece was stretched. For growth that is the same in every direction, the stretching is determined by the curvature up to a growth that curves nothing, which is why the earlier cut experiments could recover a profile. For growth that is faster one way than another, the stretching contains a direction the curvature never had, and only the marks can return it.

The model under the figures #

Directions fixed along and across a midrib. The growth here stretches along two perpendicular directions that are the same at every point of the sheet. A real leaf’s principal directions of growth turn — they follow the veins, which fan from the midrib — and a growth with turning directions has a field with more terms. The conclusion that cuts read only a field built from the growth, and that different growths share fields, does not depend on the directions being fixed; the explicit family does.

Growth as a metric. Every growth is drawn as the metric it gives the flat sheet, and the curvature is the curvature of that metric whether or not the sheet could take that shape in space. As in the earlier essays, a cut is assumed to free its piece without disturbing it, and to read its total curvature exactly.

A leaf. The midrib, the margins and the four growths are formulas on a unit disc. They are chosen to separate the possibilities cleanly, not measured on any leaf, and the growth rate $a = 0.8$, larger than the 0.35 of the earlier essays, is chosen so that the ellipses in the drawings visibly differ from circles.

Every reading taken two ways #

Each cut’s reading is a boundary flux of $\mathbf V$, computed from numerical derivatives of the two growth functions at 400 points along each side or 720 round a circle, so that two growths sharing a field are compared through their own formulas rather than through the field written down once. Each is checked against the sum of the angle defects of a fine triangulation of the piece whose edges are measured in the grown metric: at each interior vertex, $2\pi$ less the angles of the six triangles round it, each angle found from its triangle’s three grown side lengths by the law of cosines. That uses no curvature formula, and it agrees with the flux to within a few parts in a thousand, the error of a triangulated boundary.

The family’s agreement is required to four decimal places on three cuts, its areas and margin lengths to rise strictly from member to member, and the flat growth’s readings to vanish. The sliding square is required to read a constant along the midrib on both twins and the same values across it on both.

Still open: directions that turn, and the marks that read them #

The family here was built with the midrib fixed. A leaf’s growth directions follow its veins, and the natural next case is growth whose principal directions turn across the sheet — radially from a point, or along curves fanning from a midrib. The curvature is then a flux of a field with a term for the turning itself, and there should be growths whose directions alone, with both stretch factors constant, curve the sheet: stretching by a fixed ratio along a direction that rotates from point to point. Whether such growth can produce the pleats nothing grown was cut out of anything looked at, with no difference in growth rate anywhere, is a question with a computable answer.

The other direction is the measurement the cuts cannot make. A sheet that grows cannot lie flat and its successors treated curvature as the thing to read; for anisotropic growth the thing to read is the ellipse, and how few marks fix an ellipse at every point of a sheet — three distances fix a small triangle’s shape, so three marks per region, read before and after growth — is the instrument a leaf would actually need.

Sideways, a crease carries no curvature found that folding changes how a sheet sits in space and nothing intrinsic. The family here is the dual surprise: growths that change the intrinsic geometry very differently — areas by 1.2 or 2.5 times — leave every intrinsic total the cuts can see exactly where it was.

The habit worth carrying is about scalar instruments. When an instrument returns one number per piece, ask how much of what produced the number it could have returned. A cut’s reading is a curvature, a scalar; the growth that made it had a direction, and no scalar keeps a direction.