Origami & Folding
what one sheet can do
Series

Growth as metric — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 00.20.40.60.8100.511.5radius on the flat sheetgrowth factor Ωhow much each ring grew00.20.40.60.81-2-112radius on the flat sheetcurvature Kclosed formthe curvature that forcesgrown more at the rim · K(0) = −4a = -2.40

    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.

    part 1 · biology
  2. a = 0.6K(0) = -2.40opens — a rufflearea ×1.72a = 0.25K(0) = -1.00opens — a rufflearea ×1.27a = 0K(0) = 0.00stays flatarea ×1.00a = -0.25K(0) = 1.00closes — a domearea ×0.77a = -0.4K(0) = 1.60closes — a domearea ×0.65the growth factor is the same function throughout — only the sign of one number changesΩ = 1 + a r² · rim-heavy growth opens the sheet, centre-heavy growth closes it

    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.

    part 2 · biology
  3. one excess of length, 4 ways to spend it2 wavesheight 0.0398∫κ² ds = 173 wavesheight 0.0266∫κ² ds = 395 wavesheight 0.0159∫κ² ds = 1088 wavesheight 0.0100∫κ² ds = 276amplitude × waves is constant, so these are one shape at four scalesexcess 6.0% of the span · every profile below has exactly that excess

    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.

    part 3 · biology
  4. a crease — concentrated at a pointdeficit 0.5236 rad, all of it hereflat everywhere elsegrowth — spread over the areatotal 0.5236 rad, none of it anywherecurved at every pointa 30° wedge removed, against Ω = 1 − 0.0400 r² · same total curvature, 0.5236

    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.

    part 4 · biology
  5. wavesamplitudefits inside2∫κ² = 170.03980.043∫κ² = 390.02660.045∫κ² = 1080.01590.04, 0.028∫κ² = 2760.01000.04, 0.02, 0.0112∫κ² = 6210.00660.04, 0.02, 0.0120∫κ² = 17240.00400.04, 0.02, 0.01depth 0.04 needs 2 waves or more · depth 0.02 needs 4 waves or more · depth 0.01 needs 8 waves or moreexcess 6.0% · amplitude × waves = 0.0797 throughout · floor = that constant ⁄ depth

    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.

    part 5 · biology
  6. growth profilelargest |K| it carries∫K dAgrown by the same factor everywhereflat — it can be laid in a plane0.0000.0e+0 — nothinggrown more at the rimnot flat at any radius1.400-3.258grown more at the centrenot flat at any radius7.8426.767the growth of a spherical capnot flat at any radius0.3911.118the growth of a hyperbolic discnot flat at any radius0.391-1.360grown so that the curvature cancelsnot flat at any radius1.4004.8e-5 — nothing∫K dA = −2πR (ln Ω)′(R) — the total is decided at the rim, so the interior cancels out of it

    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.

    part 6 · biology
  7. 00.20.40.60.81-11234radius of the three marks, on the flat sheetmiss against flat (%)grown by the same factor everywhere · −0.00%grown more at the rim · +4.27%the growth of a spherical cap · −1.71%grown so that the curvature cancels · +3.52%a centre mark and three a third of a turn apart · on a flat sheet the three sit √3 times their radius apart

    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.

    part 7 · biology
  8. 00.20.40.60.81-3-2-11radius of the cut, on the flat sheet∫K dA inside the cutR ⁄ √2grown by the same factor everywheregrown more at the rimthe growth of a spherical capgrown so that the curvature cancelsa cut at radius r reads the total curvature of the disc inside it, −2πr (ln Ω)′(r) — the rim measurement moved inward

    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.

    part 8 · biology

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