Growth is a change of metric, and a metric decides a curvature
growth-curvature is one function. Everything below came out of it during this
build, at arguments 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.
At its defaults
profile: "centre", a: 0.04
sub: "cuts", show: "reading", profiles: [uniform, rim, cap, cancel], a: 0.35
sub: "rim-blind", profiles: [uniform, cap, ruffle, cancel], a: 0.35
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim 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 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 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
- 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
- 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 uniform growth the four marks close up exactly at every radius — the test has no false alarm ×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 computed curvature reproduces the closed form to 0.0e+0 — K = 0 at every radius ×1
- the sign of the curvature is the opposite of the sign of a in Ω = 1 + a r², at every profile drawn ×1
- uniform growth leaves the sheet flat — every sampled curvature is zero to 1e-9 ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
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 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.
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.
Every generator · The folding nobody designed field · The patterns a reader can fold