Growth as metric — the series
-
A sheet that grows cannot lie flat
A leaf does not decide to buckle. Growth changes the distances between a sheet's own material points, a set of distances determines a curvature, and a curvature that is not zero cannot be laid in a plane by anything — whatever the sheet is made of and however slowly it grew.
-
Which way the disc curves
A growing disc either domes or ruffles, and which one it does is not a matter of how much it grew. It is decided by where the growth was — more at the rim opens the sheet, more at the middle closes it — and one number in one formula takes it through both.
-
The excess does not choose its waves
A rim that has grown longer than its span has to put the extra length somewhere, and two large waves and eight small ones are the same metric — identical arc length, identical excess. Geometry fixes the family and is completely indifferent about the member.
-
A crease carries no curvature
A fold looks like the sharpest curvature a sheet could have, and intrinsically it has none at all. Developability — the first of the four conditions this site's checker runs — is exactly the statement that folding an uncut sheet creates no curvature anywhere, including at the creases.
-
The container picks the member
Two large waves and eight small ones are the same metric, and geometry has nothing to say about which. Bending has nothing to say either — it rises at every step of the family, so a least-bending rule always answers the fewest waves and never anything else. What decides is the container: amplitude times wave count is constant across the family, so a ceiling on the height is a floor on the number, exactly inversely.
-
The test measures the rim
Flattening the specimen is the right test and the measurement anybody actually makes on a flattened specimen is a boundary one — how far the margin overruns its chord. Total curvature is a boundary quantity too: it equals minus two pi R times the growth profile's slope at the rim, and nothing else about the interior survives into it. So a sheet can be curved everywhere and integrate to nothing, and the test reports it flat.
-
Three marks see nothing
The measurement a rim cannot make is an interior one, and the obvious interior measurement — two marks a known distance apart, measured again after growth — cannot detect curvature at all, because a uniformly enlarged sheet changes that distance and stays flat. Three marks cannot either: any three distances obeying the triangle inequality are the sides of a flat triangle. Four marks give six distances, and six distances are not free on a flat sheet. The growth profile a rim measurement reads as flat misses by three and a half per cent with four marks at the rim.
-
A cut reads a slope
A rim measurement returns one number for a whole grown disc, and a family of growth patterns share it. Cut the disc in a circle and the piece inside has a rim of its own, and its reading is minus two pi r times the slope of the log of the growth at the cut — so a cut reads a slope, a set of cuts reads the slope at each radius, and the slopes add up to the growth profile itself. The one thing no cut can recover is how much the whole sheet was enlarged, which is the one kind of growth that curves nothing.
-
A circle sees only the average
A cut through a grown disc was found to read a slope: the total curvature inside a circle is minus two pi r times the slope of the log of the growth there, and a set of circles gives the growth back up to one uniform enlargement. That rested on growth arranged in rings. Drop the rings and the integration by parts survives unchanged — a cut of any shape reads the flux of the log of the growth across it — but what a circle reads changes completely. A centred circle sees only the average of the growth's Laplacian round each ring, so a leaf grown at its margins reads, circle for circle, exactly like a disc grown at its rim. And every cut of every shape is blind to any part of the growth whose log is harmonic, which is precisely the growth that curves nothing: a sheet grown e to the ax, faster toward one end, changes area by a factor of three and a half across its width and lies flat.
-
A cut cannot tell which way a leaf grew
Let a sheet grow faster in one direction than the other, and a cut still reads a flux across its edge — but of a field whose parts are how fast the growth along each direction changes across it. Growth that varies only along its own direction curves nothing. And a whole family of growths shares one such field: a disc grown only along its midrib by 1 + a y², one grown equally every way by e^(a y²), and one grown more across than along all hold the same curvature in every piece, though one grows to twice the area of another. No cut, of any shape, can tell which way a leaf grew.
-
Growth that only turns
Grow a sheet by the same two stretches at every point — 1.4 along one direction, 1 across it — and let only the direction change from place to place. No rate varies anywhere, and the sheet still cannot lie flat. A cut reads two whole turns' worth of geometry: the number of times the direction turns round it, less the circulation of a field built from how fast the direction turns moving along itself and moving across itself. Directions radiating from a point put all the curvature at that point, a cone of 2π(1 − 1⁄1.4); directions running round it make an anticone; a spiral pitched forty degrees from the radius holds nothing at its centre; and a direction that turns steadily across the sheet curves it in stripes of alternating sign.