Growth that only turns

A sheet that grows cannot lie flat began with the idea every later essay on this line has used: growth changes the distances between a sheet’s own material points, a set of distances fixes a curvature, and a curvature that is not zero cannot be laid in a plane by anything. The essays after it asked which growths produce curvature and how to read it. Which way the disc curves found that a disc grown more at its rim than its middle ruffles and one grown more at its middle domes. A cut through a grown sheet reads the total curvature inside it as a flux across its edge.

A cut cannot tell which way a leaf grew let the growth be faster in one direction than another, and found that the flux a cut reads is of a field whose parts are how fast each stretch changes across its own direction. Growth that varies only along its own direction curves nothing. But every growth in that essay had its directions fixed — along a midrib and across it — and only the rates varied.

This essay does the opposite. The rates are fixed: a stretch of 1.4 along one direction and 1 across it, the same at every point of the sheet. Only the direction varies. Nowhere does the sheet grow faster than anywhere else; every small circle on it grows into the same ellipse, turned. The question is whether turning alone can curve a sheet, and the answer is that it can, and that where it puts the curvature is a matter of how the directions are arranged.

A cut reads turns and a circulation #

The geometry of a grown sheet is its metric: the rule giving the grown length of any short step on the flat sheet. Here a step $dx$ grows to length $\sqrt{\ell_1^2 (n\cdot dx)^2 + \ell_2^2 (m\cdot dx)^2}$, where $n$ is the growth direction at that point, $m$ is perpendicular to it, and $\ell_1$, $\ell_2$ are the two stretches. Write the direction as an angle $t(x, y)$.

The curvature of such a metric can be computed by the method of moving frames, which reads it off the way a pair of perpendicular unit steps rotates as it is carried across the sheet. Two rates come out of the calculation, and both have a plain meaning: $s_1 = n\cdot\nabla t$, how fast the direction turns when moving along it, and $s_2 = m\cdot\nabla t$, how fast it turns when moving across it. The frame’s rotation is carried by a field

$$W = \frac{\ell_1}{\ell_2}\, s_1\, n + \frac{\ell_2}{\ell_1}\, s_2\, m,$$

and the total curvature inside any closed cut $C$ is

$$\int K\, dA = \oint_C \nabla t \cdot dl \;-\; \oint_C W \cdot dl.$$

Two checks make the formula believable before any figure. When the stretches are equal, $\ell_1 = \ell_2$, the field $W$ is $s_1 n + s_2 m = \nabla t$ and the two integrals cancel: turning the axes of a growth that is the same every way changes nothing, as it must. And the first integral is $2\pi$ times the number of whole turns the direction makes as the cut is walked once round, which is zero for a smooth field of directions and is not zero round a point where the directions radiate or circle. So the formula already says where to look. A cut reads the turns the direction makes round it, less the circulation of a field built from how fast it turns.

Every reading below was also measured a second way, sharing nothing with the formula: the flat sheet is covered with small triangles, each triangle’s sides are measured in the grown metric, and the angle defects at the vertices inside the cut — $2\pi$ less the angles meeting there — are added up. The two agree on every cut drawn.

Radiating directions make a cone #

The simplest arrangement with a centre is directions that radiate from it. Away from the centre the growth turns as one moves across it — round the centre — at a rate of one over the radius, and does not turn at all as one moves along a radius. Put that in the formula and $W$ comes out as $\ell_2/\ell_1$ times the gradient of the polar angle, which has no curl anywhere but at the centre. So a cut that does not enclose the centre reads exactly zero, however close it comes. The sheet is flat everywhere but one point.

A cut round the centre reads one whole turn of the direction, $2\pi$, less $2\pi\,\ell_2/\ell_1$, and so the centre holds

$$2\pi\left(1 - \frac{\ell_2}{\ell_1}\right),$$

which at stretches of 1.4 and 1 is 1.795 — a hundred and three degrees of angle removed from round the centre, as if a wedge had been cut out of the paper and its edges joined. That is a cone. The reason can be said without the formula: every circle round the centre has its radius stretched by 1.4 and its circumference by 1, so in the grown sheet the circumferences are short for their radii, by exactly the factor a cone’s are.

Turn every ellipse a quarter turn, so that the long stretch runs round the centre, and the circumferences are long for their radii. The same formula gives $2\pi(1 - 1.4) = -2.513$: a hundred and forty-four degrees of extra angle at one point, an anticone, which cannot sit in space without wrinkling round its tip. The two discs grow by exactly the same amounts at every point. What distinguishes them is only which way the ellipses point.

One ratio between a cone and an anticone #

The cone’s size depends on one number, the ratio of the two stretches. At a ratio of one the growth is the same every way, its direction means nothing, and the centre is flat. Above one the centre is a cone point whose curvature rises toward $2\pi$ — a ratio of two gives $\pi$, half the angle round the point gone — and never reaches it, because no stretch along the radii can remove all of the circumference. Below one the centre is an anticone whose excess angle grows without limit as the stretch round the centre outruns the stretch along the radii: a ratio of one half gives $-2\pi$, a whole extra turn.

This is the same geometry Modes, Bhattacharya and Warner described in 2011 for sheets of nematic glass, a liquid-crystal solid whose molecules set a direction along which the sheet contracts when heated. A sheet patterned with circles of direction rises into a cone when heated, and one patterned with radii into an anticone; there the direction is written into the material and the change is a contraction rather than growth, but the metric is the same and so is the arithmetic. A straight tuck is a cone point found a cone’s law in folded paper, where a tuck hides length in proportion to its distance from the start. Growth reaches the same object by the opposite route, adding length unevenly by direction instead of hiding it.

A spiral can hold nothing #

Between radii and circles lie spirals: directions that leave the centre at a fixed angle $\alpha$ to the radius. A spiral turns both along itself and across itself, and the two turnings enter $W$ with opposite weights — the turning along the direction multiplied by the ratio $k = \ell_1/\ell_2$, the turning across it divided by it. Round the centre the circulation comes to $2\pi(k \sin^2\alpha + \cos^2\alpha / k)$, so the centre holds

$$2\pi\left(1 - k\sin^2\alpha - \frac{\cos^2\alpha}{k}\right).$$

At $\alpha = 0$ this is the cone and at ninety degrees the anticone, and in between it passes through zero. At $\sin^2\alpha = 1/(1+k)$ — 40.2 degrees from the radius when $k = 1.4$ — the spiral’s centre is flat. The growth is still anisotropic at every point, the directions still swirl round the centre with a winding of one whole turn, and a circle round it measures exactly the circumference its radius predicts. The defects of the measured triangulation inside a circle round that centre add to less than three thousandths.

The cancellation has a picture. Moving outward along a spiral, the direction turns; that turning is weighted by the long stretch and adds circumference. Moving round the centre across the spiral, it turns as well, weighted by the short stretch, and the two together make the circumference grow exactly as the radius does. A cone and an anticone, mixed in the right proportion, are a plane.

A steady turn curves the sheet in stripes #

A field of directions need not have a centre at all. Let the direction turn steadily as the sheet is crossed from left to right, by two radians per unit of width, and stay the same up and down: $t = 2x$. There is no point the directions radiate from, the first integral in the formula is always zero, and the whole reading is the circulation of $W$. Its curl can be worked out exactly, and the curvature it gives is

$$K = -\frac{(\ell_1^2 - \ell_2^2)\, q^2 \cos 2qx}{\ell_1^2\, \ell_2^2},$$

with $q$ the rate of turning. The sheet is curved everywhere, in stripes of alternating sign. Where the direction runs left to right — along its own turning, so that it bends as it goes — the curvature is negative, saddle-like. Where it runs up and down — across its turning, so that neighbouring directions fan apart — the curvature is positive, dome-like. The stripes repeat every half-turn of the direction.

The figure’s two squares read them: the one in the middle, where the direction lies along the turning, holds −0.808, matching the formula integrated across it, and the one to the right, where the direction lies across, holds 0.540. The circle spans parts of several stripes and reads what is left over, −1.317. Over a whole period of the stripes the curvature integrates to zero, so a cut wide enough to span whole periods in both directions reads almost nothing — the same blindness the test measures the rim found in a single reading of a whole sheet, here arising from a growth that never varied in rate at all.

So there are now two ways a growth with no change of rate can curve a sheet, and they are different in kind. A direction field with a centre puts a finite amount of curvature at a single point; a smooth field puts curvature everywhere, in proportion to the square of how fast it turns. Both vanish when the stretches are equal, and both change sign when the long and short stretches swap.

Veins from the middle and veins from the edge #

The lead a cut cannot tell which way a leaf grew left was that a leaf’s growth directions follow its veins, and veins turn. The radial case says what a fan of veins does in this model, and the answer depends entirely on where the fan starts.

Some leaves have their stalk joined to the middle of the blade’s underside, with veins radiating from that point in every direction; botanists call them peltate. If such a blade grew more along its veins than across them, the point where the veins meet would be inside the sheet, and the cone point above would be inside the blade: the blade could not lie flat, and would rise into a shallow cone or cup round the stalk.

Many more leaves have their main veins fanning from the base, a point on the blade’s edge. Then the centre of the fan is on the rim, and no cut inside the blade can go round it. Every cut inside reads zero, and the blade is flat. The fan’s geometry still has to go somewhere, and it goes into the rim: the straight edge through the base, an angle of 180 degrees, grows to $180 \times \ell_2/\ell_1$, or 128.6 degrees at these stretches. A fan from the edge kinks the edge; a fan from the middle curves the middle. The two leaves grow identically at every point of their blades.

That is a prediction of the model and nothing more. Whether a peltate leaf grows more along its veins than across, whether its cup shape has this origin or another, and whether the sharp point of the model survives into tissue that grows smoothly through its centre are questions about plants that geometry cannot answer. What the geometry says is that a leaf whose veins radiate from inside its blade has a choice a leaf whose veins fan from its edge does not: any anisotropy along the veins curves the first, and leaves the second flat.

What the turning cannot show #

Which shape the sheet takes in space. Every number here is intrinsic: the curvature a grown sheet must carry, not the surface that carries it. A cone point can sit as a cone or crumple, and an anticone must wrinkle; the excess does not choose its waves and the container picks the member found that geometry fixes the family of shapes and mechanics and confinement choose among them, and the same is true here.

A real centre. The cone point is a point only because the direction field is singular at one point and the growth there is undefined. Tissue that grows through its centre must have some region where the direction is not radial, and the curvature is spread over that region rather than concentrated; the total inside a cut round it is the same, because the formula only reads the cut.

Growth that is not fixed in rate. Real growth varies in amount as well as direction. The two effects add in the formula — the rate terms of the earlier essays and the turning terms here both enter the field a cut reads — but nothing here measured them together.

A metric and nothing else #

The sheet is idealised as a surface of no thickness whose growth is a smooth field of ellipses: at each point a small circle grows into an ellipse of fixed axes $\ell_1$ and $\ell_2$, turned by an angle that varies from place to place. Nothing about the material enters, and every result is a consequence of the metric alone. The moving-frame calculation is Cartan’s method applied to a metric of this one form; the cone and anticone of a radial pattern are the nematic-glass results of Modes, Bhattacharya and Warner; the spiral’s flat pitch and the stripes’ formula are derived here from the same field.

Two readings of one curvature #

Every number in the figures is computed twice. The formula’s reading is a line integral round the cut, of the direction’s gradient less the frame field $W$, taken with six hundred points to a side. The measured reading adds the angle defects of a fine triangulation whose every edge is measured in the grown metric by three-point quadrature, with no reference to $W$ or to any curvature formula. They are required to agree to a few thousandths on every cut drawn — to about a per cent on the squares, whose edges a lattice can only approximate. The centre figure further requires the radial and circular centres to read exactly $2\pi(1 - \ell_2/\ell_1)$ and $2\pi(1 - \ell_1/\ell_2)$, and cuts missing the centre to read zero; the spiral figure requires the measured defects at the flat pitch to sum to less than three thousandths; and the stripes figure requires the formula for $K$, integrated across the middle square, to give that square’s reading.

Still open: the marks that read an ellipse #

The measurement that matches this growth is not a cut. A cut reads a curvature, which is one number per region; the growth that made it is three numbers per point — two stretches and a direction. Three marks see nothing showed that three marks cannot detect curvature, because any three distances form a flat triangle. But three marks are exactly what fixes the ellipse at a point: a small triangle’s three sides before and after growth determine the metric there. So the instrument this growth calls for is a scatter of small triangles of marks, read once each, and the question is how many it takes to fix a field like the spiral’s or the stripes’ — whether a few triangles placed where the direction turns fastest can recover the whole field the way a few circular cuts recovered a ring-shaped growth in a cut reads a slope.

The second question is the leaf’s own. A real fan of veins is neither a full circle nor a straight-edged half-disc; veins curve, branch and meet the margin at angles of their own. Taking a vein pattern drawn from a real leaf as the direction field and computing where it puts its curvature — at the base, along the midrib, at the margin — would turn the model’s prediction into a number for one plant. And sideways, nothing grown was cut out of anything found a leaf’s corrugation ending at its margin because the plant stopped there; whether the pleats of a fan-veined leaf follow the directions this growth would curve the blade along is a measurable question.

The habit worth carrying is about what counts as a change. A field can be constant in every quantity a scalar measures and still not be uniform, because a direction is not a scalar. The growth here is the same at every point by every rate one could name, and it still holds a cone, an anticone, a flat spiral or a sheet of stripes, depending only on how its ellipses are turned.