Generator

Growth is a change of metric, and a metric decides a curvature

A generator in the folding nobody designed library, called 51 times across 12 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

growth-curvature is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

`sub: "plane", show: "pair", a: 0.35`

`sub: "plane", show: "circles", a: 0.35`

`sub: "plane", show: "strips", a: 0.35`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

the sheet cannot lie flat: |K| reaches 2.400 on a disc of radius 1 ×9
2 of 4 profiles integrate to nothing, and only 1 of them is actually flat ×4
the computed curvature reproduces the closed form to 4.5e-6 — K(0) = −4a = -2.40 ×4
the curvature is the same at every radius, to 1.5e-5% across the disc ×4
the integral and its by-parts form agree on all 4 profiles to 1.0e-4, so the total really is a rim quantity ×4
at the rim the boundary reading of the cancelling profile is -5.5e-5 and its four-mark miss is +3.52% ×2
one profile carries curvature reaching 1.40 and integrates to 4.8e-5 — curved everywhere, and invisible to the rim ×2
the centred circle's reading is the same computed as a flux across the cut and as the sum of the angle defects of a triangulation measured in the grown sheet: 0 and 0; -2.2515 and -2.2500 ×2
the computed curvature reproduces the closed form to 1.6e-7 — K = 1/ρ² = 0.391 everywhere ×2
the computed curvature reproduces the closed form to 2.0e-7 — K = −1/ρ² = -0.391 everywhere ×2
the computed curvature reproduces the closed form to 4.4e-6 — K(0) = −4a = -1.40, and ∫K dA = 0 ×2
the profile a rim measurement reads as flat — total curvature -7.8e-5 — misses by +4.72% with the marks at the rim ×2
the profile recovered from 3, 5, 9 cuts misses the true one by 1.9e-2, 7.0e-3, 2.2e-3 — more cuts, less error ×2
5 growth profiles are compared, with rim-heavy and centre-heavy cases both present ×1
a circular cut's reading is the same computed as the flux of the log of the growth across it and as the integral of its Laplacian inside: -1.9847 and -1.9849 at radius 0.95 ×1
a cut at the rim reads the whole disc's total, and on grown by the same factor everywhere that agrees with the rim measurement to 0.0e+0 ×1
a cut at the rim reads the whole disc's total, and on grown more at the rim that agrees with the rim measurement to 6.9e-6 ×1
a cut at the rim reads the whole disc's total, and on grown so that the curvature cancels that agrees with the rim measurement to 5.5e-5 ×1
a cut at the rim reads the whole disc's total, and on the growth of a spherical cap that agrees with the rim measurement to 1.1e-6 ×1
a direction turning steadily across the sheet curves it in stripes: the cut where the direction runs along its own turning reads -0.8078, the cut where it runs across it reads 0.5404, and the circle reads -1.3174, by the formula and by angle defects alike ×1
a rectangular cut reads the same as a boundary flux and as an area integral: -0.06321 for a strip along the midrib at half the radius ×1
a sheet grown only along its midrib and a sheet grown equally in every direction read the same on every cut drawn — -1.8095, -0.6157, -0.8000 — because their two growths have the same flux field ×1
a spiral of directions holds 2π(1 − k sin²α − cos²α ⁄ k) at its centre, by the formula and by angle defects at every pitch drawn, and at a pitch of 40.2 degrees from the radius it holds nothing: the defects there sum to 0.0000 ×1
and its grid keeps every right angle — the worst crossing is 1.4e-4° from square — while the cells at the growing end are 1.88 times the area of the flat sheet's and those at the other 0.53 ×1
and uniform growth alone reads zero at every cut, whatever the factor ×1
enlarging the whole sheet by 1, 1.5, 2 times changes no cut's reading — the largest difference is 1.8e-11 ×1
every centred circle reads the margin-grown sheet exactly as it reads the rim-grown one, and the sheet grown toward its margins exactly as its own average round each circle — two pairs of different growths, one curve each ×1
every centred circle reads the same on the two sheets — -0.2694, -0.9291, -1.9847 at radii 0.35, 0.65, 0.95 ×1
every member of the family, from growth along the midrib alone to growth across it at twice the isotropic rate, reads the same on each of the 3 cuts ×1
growth along the midrib that varies along the midrib curves nothing — every cut reads 0, 0, 0 — while the same growth varying across it reads -2.2515, -0.8230, -1.3271 ×1
growth by e^(0.35x) has no Laplacian in its log, so every cut reads zero — 2e-14, -5e-15, -6e-15 for three circles anywhere on the disc — and the grown sheet lies flat ×1
on a sheet grown uniformly the geodesic between two marks is the straight chord, to 1.5e-9 ×1
on every profile the three distances between the centre and two of the marks obey the triangle inequality, so there is a flat triangle with those sides and three marks cannot refuse flatness ×1
on the sheet grown toward its margins, a cut laid across the midrib reads -0.0076 wherever it is slid, while a cut laid along it reads from -0.1237 to -0.0002; on its radial average both directions vary ×1
on uniform growth the four marks close up exactly at every radius — the test has no false alarm ×1
slid along the midrib, a square's reading does not change on either sheet; slid across it, the reading changes identically on both — so the direction the cut finds is the one the readings are constant in, on a sheet grown in one direction and on one grown in every direction alike ×1
the cancelling profile's inner disc at R ⁄ √2 reads −πaR² = -1.0996, the largest any cut on it reads, while the whole disc reads nothing ×1
the case with no differential growth comes out flat, which is the control the comparison needs ×1
the centre's reading is 2π(1 − across ⁄ along) at every ratio from 0.5 to 2, by the formula and by the angle defects alike ×1
the computed curvature reproduces the closed form to 0.0e+0 — K = 0 at every radius ×1
the difference between the two growths has no Laplacian anywhere, so every cut reads nothing of it — off-centre circles included: -3e-16, -1e-15, -1e-15 ×1
the four sides' readings add to -1.3096, and the angle defects inside the rectangle add to -1.3077 ×1
the off-centre cut reads -0.6157 on all 4 discs, whose growths range from wholly along the midrib to more across it than along ×1
the off-centre cut's reading agrees with the angle defects on every member, while the grown area runs from 1.200 to 2.464 times the flat disc's and the length of a margin line rises with it ×1
the sign of the curvature is the opposite of the sign of a in Ω = 1 + a r², at every profile drawn ×1
the two sides along the midrib read 0.1727 and -0.9047: the stretch along the midrib changes across it, and only these sides see that ×1
uniform growth leaves the sheet flat — every sampled curvature is zero to 1e-9 ×1
veins radiating from a point inside the blade put 1.7952 of curvature at that point; radiating from a point on the rim they curve nothing — every cut inside the blade reads zero — and the base's straight angle grows to 128.6 degrees ×1
with the same stretches 1.4 and 1 at every point, a cut round the centre reads 1.7952 when the long stretch is along the radii and -2.5133 when it runs round them — 2π(1 − 1⁄1.4) and 2π(1 − 1.4⁄1) — and a cut that misses the centre reads nothing; the angle defects of a triangulation measured in the grown sheet agree ×1
yet the two grown sheets are not the same surface: their curvature differs by as much as 0.190 at a point, against a largest curvature of 0.699 ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

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 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.

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.

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 leaf ends its pattern

A hornbeam leaf's corrugation does not stop at the margin by being cut off: the pleats narrow until there is nothing left of them, and the margin is where the pattern reaches zero rather than where it was interrupted. The same is available to a drawn pattern and costs nothing — a corrugation tapered by a factor of eighty-two across its columns folds with a Kawasaki residual of 4×10⁻¹⁶, exactly as an untapered one does, because the column widths never enter the condition.

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.

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.

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.

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.

What one cut buys

A fold moves paper about and cannot change how much of it surrounds a point. A cut can, and that one difference is the whole of what this site's founding rule is worth. Take a wedge out and the sheet closes into a cone; let one in and it has more paper than the plane will accept.

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.

All the figures · The folding nobody designed field · The patterns a reader can fold