Three answers, one count

What a flat sheet can become names three ways round the sphere — seams, curved creases, and a few per cent of stretch — and says what each one costs. A tuck keeps what a gore cuts adds a fourth and prices the thickness it makes. What none of them does is put the four on one scale, and every essay since has been about the fourth alone.

There is a scale available, and it comes from noticing that all four are answers to a single quantity.

The one quantity #

A flat disc of radius $s$ has a circumference of $2\pi s$. The circle at arc distance $s$ from a sphere’s pole has a circumference of $2\pi R \sin(s/R)$. The disc has too much edge, and the fraction it has too much by is

$$f(s) = 1 - \frac{R\sin(s/R)}{s}$$

which is nothing at the pole, grows as $s^2/6$ near it, and reaches $1 - 2/\pi = 36.3$ per cent at a hemisphere’s rim.

That is the whole problem. Every one of the four answers is a way of disposing of $f$, and nothing else about the sphere enters: not its curvature as such, not the theorem that forbids the isometry, only this one fraction of surplus edge at each radius.

Every answer divides the circle #

Here is the step that makes the four comparable, and it is the same step in each case.

A gore cuts a wedge out of the disc and rejoins the edges. With $m$ gores the disc is divided into $m$ panels, and each panel now has to absorb only $f/m$ of its own share of the excess.

A tuck folds a strip under along a radius. With $m$ tucks the disc is again divided into $m$ panels, each absorbing $f/m$.

A curved crease puts a fold along a line and takes up the excess by redirecting the surface. With $m$ of them, $m$ panels, each absorbing $f/m$.

Stretch divides nothing and absorbs a fixed fraction $\varepsilon$ everywhere.

There is a fifth answer the list never includes and it belongs here for completeness: give up, and accept that the sheet approximates the sphere to within $f$. A flat disc laid over a hemisphere is a perfectly good approximation to within 36.3 per cent of its rim circumference, which for a small enough cap is an approximation nobody would question. Every one of the four real answers is a way of making that residual smaller, and the count measures how much smaller.

So a pattern works when the residual inside a panel is inside what the material gives: $f/m \le \varepsilon$, and therefore $m = \lceil f/\varepsilon \rceil$. The count does not depend on which answer is chosen. A hemisphere in a paper that gives two per cent needs nineteen divisions, whether they are nineteen cuts, nineteen tucks or nineteen curved creases.

What a division costs, which is where they differ #

The three columns of the table are the same number with three different descriptions attached, and the descriptions are the entire content of the choice.

A gore costs a cut and a seam. Nineteen gores on a hemisphere is nineteen cuts and 59.7 radii of seam to rejoin. The sheet is no longer one sheet, which is not a small thing — what a flat sheet can become treats a seam as the answer that gives up the premise.

A tuck costs thickness. Nineteen tucks is nineteen radial strips at three sheets each, on a gathering already 1.57 sheets thick at the rim, and it costs no cut at all.

A curved crease costs neither — no cut, no pile — which is why it sounds like the free answer and why nobody uses it for domes. What it costs instead is not in this table, and that is the honest gap in the comparison: a curved crease is a crease the sheet has to be persuaded into, its shape has to be solved rather than drawn, and a crease that curves is where this site says what is actually known about one.

The three answers, drawn #

Each of the three is worth a picture, because the arithmetic above flattens objects that look nothing alike.

A gore is a removal. The wedge that comes out is the excess, the two edges are rejoined, and the result is a sphere to within whatever the panels cannot take. Nothing is hidden and nothing is stored — the surplus paper is simply gone, which is why a gored dome is the thinnest of the three and why it is not one sheet.

A curved crease is a redirection. Nothing is removed, nothing is stacked, and the surface either side stays developable — which is exactly why it belongs in this comparison and exactly why it is the hardest of the three to design with.

And stretch is an absorption, with no geometry in it at all. It requires no pattern, leaves no mark, and stops.

What stretch reaches on its own #

The fourth answer behaves differently from the other three because it does not divide anything, and its limit is exact.

Stretch alone works out to wherever $f(s) = \varepsilon$ and no further. Since $f \approx s^2/6$ near the pole, the reach goes as $\sqrt{6\varepsilon}$ — a square root, not a proportion. Two per cent of give reaches 20°, five per cent reaches 32°, ten reaches 45°, and twenty — more than any paper — reaches 65°.

So doubling what a material will stretch buys about forty per cent more cap, and no material in this range reaches a hemisphere with no pattern at all. That is the hard ceiling the other three answers do not have, and it is why wet-folding is a finishing technique rather than a method: it makes the shallow part of a dome disappear and leaves the rest exactly as it was.

What it does do, and this is the useful half, is divide the count. Nineteen divisions at two per cent become eight at five and four at ten. A material’s stretch is worth a reciprocal in the pattern, so a paper that gives twice as much needs half as many tucks — which is a far better return than the square root it gets when used alone.

Why the reach is a square root and the count is not #

The two ways a material’s give enters are worth separating, because they behave so differently that a maker could reasonably believe only one of them.

Used alone, stretch has to cover the whole excess at the deepest point it reaches, and the excess grows as the square of the distance from the pole. So the reach is the square root of the give: four times the stretch buys twice the cap, and a paper that goes from two per cent to eight per cent goes from a 20° dish to a 40° one.

Used with a pattern, stretch has to cover only the residual $f/m$, and the count adjusts to make that true. So the count is inversely proportional to the give: four times the stretch quarters the divisions, and the same paper goes from nineteen tucks to five.

The second is four times the leverage of the first, and the difference is entirely about whether there is a pattern to divide the work with. Paper that stretches on purpose prices what dampening actually buys a sheet; read against these two, the answer is that its value depends completely on what it is used with, which is not how a material property usually behaves.

A deeper cap, and the same arithmetic #

A cap of 120° has a rim excess of 58.7 per cent, so every count rises by about the ratio of the two excesses. At one per cent of stretch it needs fifty-nine divisions and 247 radii of seam if they are gores — which is a number that says plainly why nobody makes a deep dome out of one flat sheet.

The excess curve for the deeper cap shows where the divisions have to do their work: the inner half is nearly free and the outer half carries almost everything. That is the same shape the tuck essays have been working with throughout, arriving now as a statement about all four answers at once rather than about tucks.

What the count is worth knowing #

Three consequences follow from the count being shared, and none of them is obvious before it is computed.

Efficiency is not the axis. A designer choosing between a gored dome and a tucked one is not choosing the method that needs fewer divisions, because they need the same number. Arguments about which approach is “more efficient” are arguments about a quantity that does not vary, and the real disagreement is about whether a cut is acceptable.

A material improvement is worth a reciprocal. Every column halves when the material’s give doubles, so anything that raises $\varepsilon$ — dampening, a different fibre, a coating — is worth as much to a gored design as to a tucked one, and worth much more than it is worth to a design relying on stretch alone.

And the deep cap is hopeless for all three equally. Fifty-nine divisions on a 120° cap is fifty-nine cuts, fifty-nine tucks or fifty-nine curved creases, and there is no answer among the three that scales better with depth. A tuck keeps what a gore cuts finds a separate limit for tucks at 130.6°, where single tucks meet one another; the count limit bites well before that, and it bites all three at once.

Where the model is generous to each #

It is fair to say which way the assumption errs for each answer, since they are not treated identically by the arithmetic.

It is generous to gores, because a gore’s residual really is $f/m$ within a panel and the panel really is developable once the wedge is out. The gore column is the one the model describes best.

It is conservative for tucks, because a tuck hides length continuously along its run rather than only sharing out a surplus, and the earlier essays measure exactly how much: a broken line following the sphere’s cubic, with the error falling as the square of the number of starts. A tucked pattern can do better than $f/m$ and this table does not credit it.

It is barely applicable to curved creases, as the assumptions section says. A curved crease changes the surface’s rulings rather than dividing a surplus, and putting it in the same column is an analogy rather than a computation. The honest reading is that the curved-crease column says how many creases a pattern of that shape would need if a curved crease behaved like the other two, and nobody knows whether it does.

What the common price does not settle #

The table makes the four comparable in one respect and leaves them incomparable in every other.

It does not price a seam against a pile. Nineteen cuts and nineteen tucks are the same count and completely different objects, and choosing between them is choosing what a maker values — one sheet, or a flat result, or a shape that holds. No arithmetic decides that and this one does not pretend to.

It does not handle a mix. A pattern using tucks in some places and stretch everywhere is exactly what a wet-folded dome is, and the table treats stretch as reducing the count rather than as a separate answer used in some regions. A mix that varied by radius would be a different and better pattern, and the table has no way to express it.

And it treats a division as uniform along its length. A tuck runs from a start to the rim, a gore runs the whole radius, and the residual each leaves is different at every radius because $f$ is. Taking the rim’s value for the whole pattern is the conservative choice and it over-divides everything inside.

What the model assumes #

A division reduces the residual in proportion to the count. With $m$ divisions each panel absorbs $f/m$, which assumes the excess is shared equally between panels — true for a rotationally symmetric pattern and not for any other.

The material’s stretch is a single number and is available everywhere. Paper’s give is neither isotropic nor uniform, and paper that stretches on purpose is where this site says what wet-folding actually does.

A tuck is three sheets and a gore is a clean cut. Both are the flat counts used throughout and both are optimistic about what a real one is.

And a curved crease is counted as a division. That is the weakest of the four treatments, because a curved crease does not divide the circle in the same sense a cut or a tuck does — it redirects curvature along its length rather than sharing out a surplus — and putting it in the same column is a modelling choice this essay makes and cannot justify from anything drawn.

How the numbers were checked #

The rim excess is checked against its closed form. A hemisphere’s is exactly $1 - 2/\pi$, and the figure fails if the computed value differs by more than a part in a trillion, so the quantity everything else is a fraction of is verified rather than assumed.

The counts are required to fall as the material gives more, and to fall in proportion — the stretchiest material must need at least four times fewer divisions than the stiffest across the range drawn.

The reach is required to be under a hemisphere at every strain level drawn, which is the claim that stretch alone never gets there, and it would fail immediately if the reach were being computed from the wrong equation.

And the reach values must increase with the strain, which catches a solver that had converged to the wrong root of $f(s) = \varepsilon$ on some row.

Still open: a pattern that changes with the radius #

The table’s own worst assumption names what comes next, and it is a good one.

The count is computed at the rim, where the excess is largest, and applied to the whole pattern. But $f$ rises from nothing, so most of a cap needs far fewer divisions than its rim does — a hemisphere in a five per cent material needs eight divisions at the rim and none at all for the first third of its radius, where the excess is inside what the material absorbs.

A pattern that divided only where it had to would put a ring of new divisions in wherever the residual would otherwise pass what the material takes, and the radii of those rings follow from one equation: the crossings of $f(s)$ with $\varepsilon$, $2\varepsilon$, $3\varepsilon$ and so on. Those radii are computable immediately, they are unevenly spaced, and they should crowd toward the rim — which would be the same placement crowd the tucks toward the rim reached from a minimax on hidden length, arrived at from a completely different criterion.

Whether the two criteria agree is a check worth making, because they have no reason to. One minimises the worst error of a broken-line approximation and the other keeps a residual inside a material limit, and if they put the rings in the same places that is a fact about the sphere rather than about either method.

The habit worth carrying is about comparing methods. Find the one quantity every method is disposing of, and count how much of it one unit of each disposes of. Methods that look unrelated often turn out to need the same count and to differ only in currency — and when they do, the choice between them is a choice of values rather than a calculation, which is worth knowing before spending an afternoon on the calculation.