A cut reads a slope

The test measures the rim found the blind spot of a boundary measurement and named two ways round it. One was marks inside the sheet, and three marks see nothing works that one out: it takes four, and four marks see the curvature a rim reading loses. The other was cutting the disc into annuli and seeing which of them lie flat on their own, on the grounds that the integral over a piece is a boundary quantity of that piece, and moving the boundary inward is how the interior is interrogated.

That second suggestion turns out to be more than a way of seeing the blind spot. A cut does not merely detect that a sheet is curved. It reads a particular number about the growth at a particular radius, and a set of cuts read together gives back the growth profile — everything the growth did except one thing, and the one thing turns out to be the part of growth that does not curve anything.

What a cut reads #

Take a grown disc and cut it along the circle that was at radius $r$ on the flat sheet. The piece inside is a smaller grown disc with a rim of its own, and a rim measurement on that piece — the margin’s overrun, the angle at which it leaves a table, any of the boundary quantities the earlier argument found equivalent — returns that piece’s total curvature.

The integration by parts that made the whole disc’s total a rim quantity works for the piece as well. The growth is a factor $\Omega(r)$ on the flat sheet’s radius, the curvature is minus the Laplacian of $\ln\Omega$ divided by $\Omega^2$, and integrating over the inner disc collapses to its boundary:

$$\int_{\text{inside the cut}} K\,dA = -2\pi r\,(\ln\Omega)'(r)$$

So a cut at radius $r$ reads the slope of the log of the growth at $r$, multiplied by $-2\pi r$. Nothing else about the growth enters: not what happened further in, not what happened further out. A cut reads a slope.

That is the same growth-to-curvature rule a sheet that grows cannot lie flat started from, and the same sign rule which way the disc curves read off it: growth that rises toward the rim has a positive slope and a cut through it reads a negative total, growth that falls toward the rim reads a positive one.

The first figure draws the reading for four growth profiles as the cut moves from the centre to the rim. Uniform growth reads zero everywhere, because its slope is zero everywhere. Growth concentrated at the rim reads a total that grows steadily more negative to −3.26 at the rim. The spherical cap reads a positive total growing to +1.12. And the profile whose curvature cancels reads −0.26 at a quarter of the radius, −0.82 at a half, −1.10 at $R/\sqrt{2}$, −0.68 at nine tenths — and zero at the rim, which is where a margin measurement looks.

The largest reading is where the sign changes #

The cancelling profile’s readings have a shape worth reading closely, because it answers a question the rim argument left open: where in the disc was the growth that the rim could not see?

The reading is most negative at $R/\sqrt{2}$, and there it is exactly $-\pi a R^2$, which for the drawn strength is −1.0996. That radius is not arbitrary. It is the radius at which the profile’s curvature changes sign, and it is also the radius that divides the flat sheet into halves of equal area.

The two facts are one fact. The reading at $r$ is the total curvature inside $r$. Moving the cut outward adds a thin ring whose curvature is negative inside $R/\sqrt{2}$ and positive outside it, so the reading falls while the cut is inside and rises once it is past. The extreme of the reading is at the sign change of the curvature, for any profile, and the value there is the most curvature of one sign the disc holds in one piece.

So a single cut at the right radius exposes the whole hidden curvature: the inner piece holds $-\pi a R^2$ and the outer ring holds $+\pi a R^2$, and the rim — which sees only their sum — sees neither.

A set of cuts reads the profile #

One cut reads one slope. Several cuts at several radii read the slope at each, and a slope known at enough radii is a function known well enough to integrate.

The reconstruction is short. From a reading $c$ at radius $r$, the slope of $\ln\Omega$ there is $-c/2\pi r$. Starting from the centre, where the slope is zero for any smooth radial growth, add up slope times radial step from one cut to the next. The running total is $\ln\Omega(r) - \ln\Omega(0)$ at each cut’s radius: the log of the growth relative to the centre.

The second figure does that for the cancelling profile with three, five and nine evenly spaced cuts and draws the result on top of the profile itself. Three cuts miss the true curve by 0.019 in $\ln\Omega$ at their worst. Five miss by 0.0070. Nine miss by 0.0022. The error falls by about three for each near-doubling of the cuts, which is the trapezoid rule’s second-order behaviour — a set of cuts is a quadrature of the growth’s slope.

The same reconstruction on the profile grown more at its rim is better still, because that profile’s slope is smoother: three cuts miss by 0.0042, five by 0.0015, nine by 0.00046. The cancelling profile’s slope turns over in the middle of the disc, and a quadrature needs more cuts to follow a turn.

The one thing no cut can see #

The reconstruction recovers $\ln\Omega$ relative to its value at the centre, and the constant is not a detail. It is $\ln\Omega(0)$ — how much the centre grew — and no cut reading contains it.

The reason is that every cut reads a slope of $\ln\Omega$. Multiply the growth everywhere by a constant factor, so that $\Omega$ becomes $c\Omega$, and $\ln\Omega$ becomes $\ln\Omega + \ln c$. The slope is unchanged at every radius, so every cut reads exactly what it read before.

The fourth figure checks that on the cancelling profile enlarged by one, one and a half and two times: four cuts, three enlargements, and not one reading moves, to a part in a million.

And uniform growth on its own reads zero at every cut, whatever its factor. The one part of growth cuts cannot recover is uniform enlargement, and uniform enlargement is exactly the part of growth that leaves a sheet flat. A measurement of curvature is blind to the one thing that has no curvature, which sounds tautological and is in fact the useful form of the statement: a set of cuts recovers everything about the growth that matters to the shape, and the one extra number — the overall size — is measured with a ruler across the specimen.

The cut as a moving surface #

The structure of this measurement has a well-known relative in physics, and the resemblance is exact rather than loose.

In electrostatics the charge inside a closed surface is read from the flux through the surface alone — Gauss’s law — and a spherical surface round a spherically symmetric charge reads the total inside its radius. Move the sphere outward, and the change in the reading is the charge in the shell it swept through. Take readings at a series of radii and difference them, and the charge density is recovered as a function of radius, from boundary measurements only.

A circular cut is the same instrument. Its reading is the total curvature inside, its change between two radii is the curvature of the ring between them, and a series of cuts recovers the curvature — or, integrated once more, the growth — ring by ring. What the rim argument presented as a limitation of boundary measurements is the working principle of one: a boundary quantity is blind to the interior only while the boundary stays where it is.

The two cases differ in one way worth noting. A Gaussian surface can be moved without disturbing the charge. A cut cannot be moved without cutting the specimen again, so a series of cuts at increasing radii needs either a series of specimens or cuts made from the outside in — and a leaf cut at nine tenths of its radius cannot then be cut at one quarter without first being trimmed at every radius in between.

What cuts and marks each see #

The two interior measurements now both exist, and they are different enough to be worth setting side by side.

Four marks detect curvature without destroying the specimen, and their reading is a weighting of the curvature along the shortest paths between them. They say that a sheet is curved and which way, and they are indifferent to the growth’s radial symmetry in the sense that six distances among any four marks can be compared with a flat sheet. What they do not do easily is say where the curvature is.

Cuts locate curvature. A cut’s reading is exactly the curvature inside it, and a series of cuts places it ring by ring. But cuts destroy the specimen, and circular cuts decompose the growth usefully only when the growth is organised in circles.

So they answer different questions, and the questions arrive in order. Marks say whether a sheet’s metric is flat. Cuts say how its growth was distributed. A specimen that passes four marks folded rather than grew, and there is nothing for cuts to find; one that fails them grew, and cuts are how to read what it did.

A creased specimen reads nothing at every cut #

The cut readings separate the two halves of this field as cleanly as four marks do, and the reason is the same fact seen from a boundary.

A leaf that folded rather than grew has a flat metric, because a crease carries no curvature: its paper is flat between creases and its vertices have no angle deficit. So every piece of a creased specimen, however it is cut, has a total curvature of zero, and a rim measurement on any piece reads nothing. A corrugated lamina cut into rings gives a column of zeros. A lamina that grew unevenly gives the profile’s slopes.

That is worth saying because the rim measurement on a whole specimen could not make the separation. The excess does not choose its waves and the container picks the member both concern a margin that has more length than its span, and a corrugated leaf pressed at its margin and a grown leaf whose growth has levelled off can both lie flat there. Cut both in rings and the corrugated one reads zero on every ring while the grown one reads its slope on each.

The cut also makes a point about cutting that the rest of this subject usually makes in the other direction. In the study of cut paper, a cut is not local: it changes what the whole sheet can fold into. Here a cut changes nothing about the metric of either piece — distances within the paper are exactly what they were — and it is precisely that indifference that makes the cut an instrument. A cut that does not disturb what it measures measures the thing it separates. And the two readings a cut produces, inner piece and outer ring, add back to the whole disc’s total, as the rim contributions of adjoining pieces add wherever a boundary is shared.

What cannot be cut is what was never cut out of anything. A grown lamina has no seam, and the rings a cut makes are pieces nothing in the organism ever separated — so the measurement is a question put to the specimen, and not a reading of any structure the specimen itself has.

What the readings cannot show #

The readings are exact for the model and the model is a disc grown in rings. Three things about a real specimen are outside it.

A real cut changes what it measures. Cutting a grown lamina releases the stresses that held it in its buckled shape, and the piece inside a cut relaxes into a different buckled shape from the one it had as part of the whole. The metric of the piece — distances within it — is unchanged by the cut, which is what the reading depends on, but a rim measurement on a relaxed piece has to be taken from the piece’s own margin, and the relaxation changes how easy that is.

The reading needs a rim measurement that can be made. The rim argument’s measurements — overrun, ripple count, the angle off a table — each estimate the total curvature with an error, and a reconstruction from nine cuts inherits nine such errors. The figures reconstruct from exact readings and do not model the error, which in a real sweep would dominate the quadrature error shown.

And the growth is radial. For growth that is not organised in circles, a circular cut reads a total that mixes growth from different directions, and the reconstruction’s premise — that one slope describes each radius — fails. The cut still reads the curvature inside it; what fails is recovering a profile from the readings.

The disc the cuts are made in #

The sheet is a disc of flat radius R whose growth is a positive function of radius, and the curvature at every point is computed from that growth alone, as it has been throughout this field. A cut is a circle of the flat sheet — a set of material points that were at one radius before growth — and the piece inside it is everything that was closer to the centre.

The reading is the piece’s total curvature, computed from the growth’s slope at the cut by the integration by parts that already served for the whole disc. At the rim the reading is checked against the total computed over the disc directly, so the cut and the rim measurement are the same measurement at R, and the cut is that measurement moved inward.

How the readings were checked #

A cut at the rim is required to read the whole disc’s total, computed separately over the disc by integrating the curvature, on every profile drawn. They agree to within 0.00006 on the cancelling profile and exactly on the uniform one.

The cancelling profile’s reading at $R/\sqrt{2}$ is required to equal $-\pi a R^2$, a closed form the reading never uses, to a part in a million.

The reconstruction’s error is required to fall as cuts are added, and a uniform enlargement is required to leave every reading unchanged, so a figure in which cuts could see scale would refuse to draw.

Still open: growth that is not arranged in circles #

Both interior measurements have now been worked out for the case this field has used from its beginning, a disc grown in rings, and both depend on that case more than the rim argument did. Four marks were arranged about a centre; cuts were made in circles about it. A leaf is not grown in rings.

The general case is where the next question lies. A growth field that varies with direction as well as radius has a curvature that is still determined by its metric, and still integrates to a boundary quantity over any piece — the integration by parts does not need symmetry. What changes is which pieces to cut. For a growth field organised along some other family of curves, the useful cuts follow those curves, and whether a small set of cuts along the growth’s own directions recovers it as circles recover a radial profile is a question about the field, not about the measurement.

The other continuation is the one this field keeps meeting and cannot close from a keyboard. Every interior measurement here is a statement about what a specimen would read; none has been read. The instrument — a set of cuts, or four marks — is now specified closely enough that the reading of one grown leaf would test it.

The habit worth carrying is the one the Gaussian surface makes obvious. When a quantity is a boundary integral, the boundary is a parameter. A measurement fixed at one boundary sees a total and a kernel of things it cannot see; the same measurement swept across boundaries sees the density, and the kernel shrinks to what no boundary can register at all.