Gaussian curvature — where it appears
Named by 19 essays across 2 fields — each of them below, with the objects they name alongside it.
What a flat sheet can become
A sheet that cannot stretch cannot become a sphere. That much belongs to differential geometry; what belongs to folding is the three ways round it — seams, curved creases, and a few percent of stretch — and what each one costs.
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.
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 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 tuck keeps what a gore cuts
A flat disc gathered into a spherical cap has more circumference than the cap, and a gore removes the excess while wet-folding stretches it away. A tuck folds it under, which keeps the sheet whole and turns the excess into thickness. At the rim of a gathered cap the paper is α ⁄ sin α sheets thick on average — π⁄2 for a hemisphere — and a simple tuck is three, so single tucks reach a cap of 130.6° before they run into one another. And because a sphere's circles fall short of a plane's as the cube of the radius, a tuck that follows the sphere widens as the cube too: its edges are curves.
A straight tuck is a cone point
A tuck with straight edges hides length in proportion to how far past its start it has gone, which is a cone's law and not a sphere's. Started at the centre, straight tucks make a cone. Started at several radii, they hide length in a broken line that follows a sphere's cubic, and the worst shortfall falls as the square of the number of starting radii: 36.9 per cent of the rim's hiding from one start, 12.3 from two, 3.3 from four, 0.8 from eight. Every start is three creases at a point, which is a vertex that cannot fold flat — and it is exactly where the gathered sheet's curvature goes.
Crowd the tucks toward the rim
Straight tucks started at several radii follow a sphere's hidden length in a broken line, and evenly spaced starts leave a worst shortfall that falls as the square of their number. Evenly spaced is not the best spacing. A sphere's hiding bends hardest near the rim, so the best starts crowd outward — on a hemisphere, four of them at 0.36, 0.60 and 0.80 of the radius — and leave 38 per cent less error than four evenly spaced. As the count grows the saving closes on 42 per cent, a limit set by the square root of how the sphere's hiding bends; on a shallow dish it approaches five ninths. To follow a hemisphere within one per cent takes six rings of tucks instead of eight, and within a tenth of a per cent eighteen instead of twenty-four.
Crowding outward costs almost nothing
Placing the tuck starts for equal error crowds them toward the rim, where the gathered paper is already at its thickest, and the obvious worry is that the accuracy is bought with depth. Measured, it is not: on a hemisphere the crowded placement's deepest start sits in 3.37 sheets against the even placement's 3.32, because a start is three sheets of its own and the gathering beneath it is only one and a half.
Three answers, one count
Seams, curved creases and a few per cent of stretch are the three ways round the sphere, and a tuck is a fourth. All four dispose of one quantity — the excess circumference a flat disc has over the sphere's circle — and all four dispose of it by dividing the circle. So the number of divisions needed is the same whichever answer is chosen: nineteen for a hemisphere in a material that gives two per cent, eight at five, four at ten. What differs is what a division costs, and one of the four runs out.
Where a ring of divisions belongs
A pattern that divides the circle everywhere as finely as its rim requires is over-divided for most of its radius, because the excess grows from nothing. Putting a ring of new divisions in wherever the residual would otherwise pass what the material takes gives seven rings on a hemisphere at five per cent of stretch, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out — and the first of those sits where a completely different criterion put its first tuck start.
The paper reads strain
Two placements of a gathered cap's tuck starts agreed on their first radius and nobody knew why. Both were proxies. What the paper takes is the shortfall at a circle divided by the circle's length, and judged by that the first start must sit exactly where the excess reaches the material's give — and past it the starts spread outward, not in. Four radii placed for length leave a hemisphere 5.34 per cent strain; the same four placed for strain leave 1.81.
Crossing the curve moves no start
The broken line of hiding that straight tucks make on a gathered cap has always been drawn touching the sphere's curve at every start, so the paper between starts can only stretch. Let it cross, and the best line turns out to be the old one lowered by a cone: the starts stay exactly where they were at every count, the worst strain halves exactly — 1.81 per cent to 0.90 on a hemisphere at four starts — and a paper that stretches one amount and compresses another places its tucks exactly as a stretch-only paper giving their sum would.
A start is worth an inverse square
Straight tucks gathering a flat disc into a cap leave a worst strain that falls as the starts multiply, and every essay on them counted starts without saying what one buys. It buys an inverse square. A chord of the hiding curve misses it by its width squared times the curve's bend over eight, so starts placed for strain leave a worst strain of I² ⁄ 8m², with I one integral of the cap's shape — 1.464 for a hemisphere. A paper that gives ε therefore needs about I ⁄ √(8ε) starts, a tenth of the give costs three times the starts, and letting the hiding cross the curve saves exactly a factor of √2. The same argument says why the starts spread outward, and by how much.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Developable surfacePleatDifferential growthConeGrowthMetricAngle deficitDevelopabilityGoreMeasurementTrade-offConstraint