Three marks see nothing

The test measures the rim finds that pressing a grown leaf and looking at its margin returns a boundary quantity. Total curvature is one: it equals minus two pi R times the growth profile’s slope at the rim, and everything the interior does cancels out of it. So a sheet can be curved everywhere and integrate to nothing, and a margin measurement reports it flat.

The essay ends by naming the measurement that would see it. Two material points, a known distance apart on the flat sheet, measured again after growth — the metric where it matters, and invisible to everything a rim can report.

That is the right direction and the wrong number of points. One distance cannot show that a sheet is curved, and neither can two, and neither, surprisingly, can three. The smallest set of marks that can refuse flatness has four points in it, and working out why says exactly what an interior measurement has to compare.

Why one distance says nothing #

Mark two points on a flat sheet a centimetre apart, let the sheet grow, and measure them again: they are now a centimetre and a half apart. What has been learnt is that the sheet grew by half along that line.

Nothing has been learnt about curvature, because a sheet that grew by half everywhere — a photocopier enlargement, the one kind of growth that leaves a sheet flat — produces exactly that measurement. Any single distance is consistent with a uniform enlargement by the right factor, and a uniform enlargement is flat.

The point generalises in a way that is worth stating carefully. A measurement can only refuse flatness if there is no flat sheet at all that would give the same numbers. A single distance is matched by a flat sheet in which the two marks are that far apart, and there always is one.

Why three marks cannot see it either #

Two marks give one distance. Three marks give three distances: from the first to the second, the second to the third, and the third back to the first.

Three distances look like more than enough. They are not, and the reason is a theorem everybody learns without noticing it is a theorem. Three lengths are the sides of a flat triangle whenever each is shorter than the other two together. That is the triangle inequality, and a set of distances measured along a surface — any surface, curved or not — always satisfies it, because the shortest path from one mark to another cannot be longer than a detour through the third.

So any three marks on any grown sheet can be laid out on a flat table with exactly the distances they have in the sheet. No curvature can be inferred from them, because there is always a flat triangle that agrees.

The figure’s rows make this concrete. On the disc grown by the same factor everywhere, the centre is 0.9546 from each mark and the two marks are 1.6534 apart. On the disc grown more at its rim, 0.7483 and 1.3281. On the spherical cap, 0.6959 and 1.1953. On the profile whose curvature cancels, 0.7437 and 1.3169. Four different metrics, four different triangles, and every one of them a triangle that exists on a flat table.

Four marks, and the one relation among them #

Four marks give six distances, and six distances among four points are not free on a flat sheet.

The reason is counting. Four points in a plane are fixed, up to moving the whole figure, by five numbers: place the first anywhere, the second along a line from it (one number), the third anywhere (two numbers), the fourth anywhere (two numbers). Six distances among them are therefore one more than the plane allows, and on a flat sheet the sixth distance is determined by the other five.

A curved sheet is under no such obligation. Its six distances can be any six the surface happens to produce, and in general they will not satisfy the flat relation. The test for flatness with four marks is whether the sixth distance is the one a flat sheet would have given, and the size of the miss is a measurement of curvature the way the triangle’s angle sum is in a textbook.

Written out in general, the relation among six distances is a determinant — the Cayley–Menger determinant, which vanishes exactly when four points lie in a plane. For a radially grown disc there is a much simpler arrangement that makes the relation obvious.

The centre and three marks a third of a turn apart #

Put one mark at the centre of the disc and three at a single radius on the flat sheet, a third of a turn apart. By symmetry, all three are the same distance $\rho$ from the centre in the grown sheet, and all three are the same distance $d$ from one another.

On a flat sheet, three points equally far from a fourth and equally far from one another are the corners of an equilateral triangle with the fourth at its centre, and that fixes their spacing: $d = \sqrt{3}\,\rho$. Nothing else is possible.

So the four-mark test reduces to one ratio, $d/\sqrt{3}\rho$, and its departure from one is the miss. The first figure plots that miss against the radius the three marks are placed at, for four growth profiles.

On uniform growth the miss is zero at every radius — to one and a half parts in a billion, which is the accuracy of the distances themselves. The test has no false alarm.

On growth concentrated at the rim the miss is positive: the three marks are further apart than a flat sheet allows, reaching +4.27% at the rim. That is what negative curvature does. Shortest paths spread apart in a ruffled sheet, so points equally far from a centre are further from one another than they would be on a plane.

On the spherical cap the miss is negative, −1.71% at the rim. Shortest paths converge on a dome, and the three marks are closer together than a plane allows. The sign of the miss is the sign of the curvature enclosed, which is the reading the triangle cannot give.

What the rim reads and what four marks read #

Now the case the rim could not see.

The test measures the rim built a growth profile that is curved everywhere and integrates to nothing: negatively curved in an inner disc, positively curved in an outer ring, balanced so that the total is zero. A measurement at the margin returns zero for it, exactly as it would for a sheet that never grew unevenly at all.

Four marks at the rim do not. The cancelling profile misses by +3.52%.

The reason it does is that the four-mark test is not an integral of curvature over the disc. The shortest path between two marks at the rim runs inward, through the negatively curved middle, and the spreading of paths there is not undone by the positive ring outside. The marks read the curvature the paths actually pass through, weighted by where they go, and a profile that balances its total does not balance that.

That is the whole gain of the interior measurement, stated precisely. A rim reading is one number, and a whole family of growth fields share it; four marks read a different weighting of the curvature, and the cancelling family does not cancel in it.

A stronger cancellation, and the other sign #

The cancelling profile has a strength, and the rim’s blindness does not depend on it: every member of the family integrates to nothing. The four marks do depend on it, and in the direction that matters.

At the strength drawn above the marks miss by 3.52% at the rim. Grow the same family harder — a half rather than a little over a third, so that the curvature at the centre reaches minus two — and the miss at the rim rises to 4.72%, while the rim reading stays at zero. The measurement the rim cannot make grows with exactly the quantity the rim cannot see.

The constant-curvature discs are the control for the sign. A disc curved negatively by the same amount at every point misses by +1.55% with the marks at the rim; a cap curved positively by the same amount misses by −1.71%. Equal and opposite curvatures give nearly equal and opposite misses, which is what a measurement of curvature should do, and the small asymmetry is the difference between how a dome and a saddle bend the paths near them.

What the cancelling profile looks like inside #

It helps to see what the marks are responding to, because the positive miss at the rim is not obvious from the profile’s balanced total.

The cancelling profile’s curvature runs from −1.4 at the centre, through zero at $R/\sqrt{2}$, to about +1 at the rim. The negative inner disc and the positive outer ring each hold half the flat sheet’s area, and their totals are equal and opposite.

Marks at small radius sit wholly inside the negative disc and read a positive miss that grows with their radius, just as the rim-grown profile does. Marks further out enclose some of the positive ring as well, and the miss still keeps growing all the way to the rim: 0.23% with the marks at a fifth of the radius, 1.73% at three fifths, 3.52% at the rim itself. The shortest paths between marks at the rim run inward and spend most of their length in the inner, negatively curved half. The four-mark reading at the rim is therefore dominated by the curvature the rim reading throws away.

Why a creased leaf passes the test #

The four-mark test separates the two halves of this field, and it does it the way the rim argument hoped an interior measurement would.

A leaf that folds rather than grows unevenly has a flat metric. A crease carries no curvature: at every vertex the paper’s angles sum to a full turn — the developability condition — and away from the creases the paper is flat. So distances measured along the paper between marks on a corrugated leaf are distances in a flat sheet, however folded the leaf is in space, and four marks on it meet $\sqrt{3}$ exactly.

A leaf that grew unevenly does not, and the figures above say by how much. So the test run on a pressed specimen whose shape is ambiguous — a corrugated lamina or a ruffled one, which can look alike — gives a verdict that does not depend on how the specimen is held, pressed or photographed: a zero miss is a fold and a nonzero miss is growth. The rim reading, which cannot tell a completed growth pattern from no growth at all, cannot make that separation.

That sharpens what the corrugated leaf is claimed to be doing. The corrugation packs the leaf, and the metric of a corrugated leaf is the metric of the flat sheet it grew as, and a measurement that reads the metric can tell the two claims apart where a picture of the leaf cannot.

Gauss’s triangle, and what it was for #

A story is told about exactly this kind of measurement, and it deserves its caveat.

In the 1820s Gauss directed a geodetic survey of the Kingdom of Hanover, and among its triangles was a large one joining three hilltops — the Hohenhagen, the Brocken and the Inselsberg. It is often said that he measured its angles to test whether space itself is flat, and found the sum equal to two right angles within the error. Historians of mathematics have doubted that this was the survey’s purpose, and the angles were measured for the map; the idea that the triangle was an experiment on space is at least partly a later reading.

The story survives because the measurement is the right kind. An angle sum is a comparison among several measured quantities that a flat plane constrains and a curved surface need not satisfy, which is what four distances are here. A single measured distance would not have told Gauss anything about the flatness of anything, and neither would the lengths of the triangle’s three sides on their own. What makes a measurement a curvature test is that it over-determines a flat figure, and the triangle’s angles, like the four marks’ six distances, do.

What the marks cannot show #

The figures draw an idealised measurement on an idealised sheet, and three things about a real one are missing.

The marks’ distances are geodesic. They are the shortest paths within the grown sheet, and measuring one on a real leaf means laying a flexible tape along the surface between two marks, which is feasible on a sheet that can be pressed and awkward on one that cannot. A straight-line distance through the air between two marks on a curved leaf is a different number and the argument does not apply to it.

The misses are a few per cent. A leaf lamina grows unevenly by far more than that and a measurement of a few per cent is not difficult in principle, but a mark has a size and a tape has a stiffness, and the figures say nothing about whether a particular specimen’s miss is above its measurement error.

And the growth is radial. Four marks arranged about a centre are the natural test for a radially grown disc, and a leaf’s growth is not radial — it has waves the geometry does not choose and a container that does, and neither is symmetric about a centre. The general test is the six distances among any four marks, and the figures only evaluate its symmetric special case.

The sheet the marks are placed on #

The sheet is a disc whose growth is a positive function of the flat sheet’s radius and nothing else: the conformal metric this field has used throughout, with curvature determined entirely by how much each ring grew.

Distances are computed within that metric. A radius is a shortest path by symmetry, so the centre-to-mark distance is the integral of the growth along it. The distance between two marks at one radius is found by shooting along the one conserved quantity a radially symmetric metric has — Clairaut’s relation — for the path that turns at the bisecting ray, which is the shortest by symmetry for the arrangements drawn.

Nothing about the material enters. The misses are properties of the metric, and a sheet with that metric has them however it is made, whether or not it can be pressed flat, and whatever it does when it cannot.

How the distances were checked #

On uniform growth the shortest path between two marks is required to be the straight chord of the enlarged disc, which the shooting method reproduces to 1.5 parts in a billion. A method that curved a path on a flat sheet would fail there first.

Uniform growth is required to give no miss at every radius, to a part in a million, so the test has no false alarm on the one profile that must not raise one.

The cancelling profile is required to read nothing at the rim and to miss by more than one per cent with four marks there. The first half is the rim’s blindness, computed from the same growth profile; the second is the marks seeing through it. A figure in which either failed would not draw.

Still open: what a cut would read #

Four marks test flatness and say little about where the curvature is. The sign of the miss says which way the enclosed curvature leans, and its size grows with how much is enclosed, but a single set of four marks cannot say that the cancelling profile is negative in its middle and positive in its ring.

A different interior measurement can, and the rim argument named it: cut the disc. A cut makes a new boundary, the piece inside it has a rim of its own, and the rim measurement that fails on the whole disc succeeds on the piece. Whether a set of circular cuts reads the growth profile itself, ring by ring, rather than only the total, is the next computation, and it turns the rim’s blindness from a limitation into an instrument.

The habit worth carrying is a check on any proposed measurement of shape. Ask whether some flat configuration would give the same numbers. If one always does, the measurement cannot be a curvature test however precise it is; it becomes one only when it over-determines the flat figure, and the smallest over-determined figure is the one to build.