Crowding outward costs almost nothing

Crowd the tucks toward the rim finds the best places to start straight tucks on a gathered cap and ends by naming what it did not price. The best placement crowds the starts outward, where the paper is already piled deepest, and the next measurement is whether the gain in hidden length survives the thickness it crowds into.

It survives easily, and the reason is worth more than the answer.

What a start is standing in #

Two quantities make up the pile at a tuck start and they are very unequal.

The first is the gathering itself. A tuck keeps what a gore cuts establishes that a flat disc gathered onto a cap is $\alpha/\sin\alpha$ sheets thick on average at arc $\alpha$ from the pole — $\pi/2 = 1.571$ at a hemisphere’s rim, and considerably less further in. A fifth of the way out it is 1.017; four fifths of the way out it is 1.322.

The second is the tuck. A simple tuck folds the excess under, which puts three thicknesses of paper where it sits: the sheet, the flap going down and the flap coming back. That is a constant, and it does not depend on where the tuck is.

So the pile at a start is about three sheets plus a fraction, and the fraction runs from 1.0 to 1.6 over the whole cap. Moving a start from a fifth of the way out to four fifths adds 0.31 sheets to a pile of 3.02 — ten per cent for the most extreme move available, and the actual moves are much smaller than that.

What the crowding actually moves #

The equal-error placement puts its four starts at 0.31, 0.51, 0.68 and 0.84 of the way to the rim; the even placement puts them at 0.20, 0.40, 0.60 and 0.80. Every start moves outward and the outermost moves least.

That last point is the whole of it. The crowding is concentrated in the inner starts, where the gathering is thinnest and a move costs almost nothing, and the outermost start — the one in the deepest paper — barely moves at all. The deepest pile therefore goes from 3.32 to 3.37, a difference of 1.5 per cent, against the 38 per cent reduction in worst hidden-length error that the crowding buys.

On a deeper cap the numbers grow and the conclusion does not. At 120°, where the rim gathering is 2.42 sheets rather than 1.57, the even placement’s deepest start is in 3.77 sheets and the crowded one’s in 3.84 — 1.9 per cent. The gathering is half as thick again and the penalty is still under two per cent, because the tuck’s own three sheets still dominate.

The gathering, measured across the cap #

The background the starts sit in is worth drawing on its own, because its shape is what makes the answer come out the way it does.

The curve is nearly flat for the first half of the cap and only begins to climb near the rim, which is the same fact as the excess growing as the square of the distance from the pole. Over the range where the tuck starts sit, the background is between one and one and a half sheets — and the tuck is three.

That is why the comparison in the first figure comes out as a per cent and a half rather than as a third. The crowding moves the starts along a curve that is almost horizontal where they are, and the thing they carry with them is the larger number.

A deeper cap moves the numbers and not the conclusion #

The profile shows what the crowding is for: a broken line through a convex curve, with its breaks placed so that every piece carries the same worst error. That is the construction crowd the tucks toward the rim established, and it is the thing whose cost this essay is asking about.

The budget makes the asymmetry explicit: nearly all the hidden length a cap requires is required in its outer third. That is the reason the starts crowd outward, it is the reason the crowding is worth 38 per cent of the error, and it is also the reason the crowding costs so little — the starts are moving toward where the work is, and the paper only gets modestly thicker on the way.

Three kinds of pile sorts the ways paper accumulates in a pattern, and a gathered cap is the gentlest of them: a slow rise with no discontinuity anywhere until a tuck is put in.

Why the trade does not bite #

The general form is worth stating because it says when it would.

A tuck’s pile is $L(s) + 2$ where $L$ is the gathering’s local thickness. The penalty for moving a start from $s_1$ to $s_2$ is $L(s_2) - L(s_1)$, and as a fraction of the pile it is that difference over $L + 2$. The constant two is what kills it. A tuck brings its own thickness with it, that thickness is the largest term, and the variation in the background is a correction to a correction.

The trade would bite if either term changed. A deeper cap raises $L$ — at 150° the rim gathering is 3.6 sheets, comparable with the tuck itself, and the penalty would be a real fraction. A thinner tuck would do the same from the other side: a tuck that added one sheet instead of two would make the background half the pile rather than a third of it.

Neither is available on a paper dome, which is why the answer here is a clean “no”. Both are available elsewhere, and the pile, not the panel is where this site prices patterns whose thickness is the binding constraint rather than an afterthought.

What a pile cap would have bought #

The question can be turned round: suppose the pile were the binding constraint, and a designer had to keep every start under some depth. What would that cost the accuracy?

Almost nothing, for the same reason. The equal-error placement’s deepest start is at 0.84 of the way out; refusing to put a start beyond 0.80 moves it a twentieth of the radius inward and changes the worst error by a fraction of a per cent, because the error is decided by the spacing of the breaks rather than by where any one of them is. A cap tight enough to bite — one that pushed every start inside the first half of the radius — would ruin the approximation, and it would be a cap of about 3.15 sheets, which is a tuck plus nothing.

So there is no interesting constrained problem here, and saying so is the useful result. A constraint that is either slack or fatal is a constraint that does not need optimising against; it needs checking once. The design question is whether tucks are usable at all at a given cap depth, and that is the meeting limit rather than a pile limit.

The pile is not the only cost of a start #

Saying the accuracy is nearly free in thickness is not saying it is free, and two other costs move the other way.

A start is a vertex that cannot fold flat. A straight tuck is a cone point establishes that: three creases meet at the place a tuck begins, three creases at a point is not a flat-foldable vertex, and the gathered sheet around it is not flat either. A start is where the curvature the sphere spreads evenly has been gathered into a cone, so crowding the starts outward crowds the curvature outward too.

That is a different quantity from thickness and this essay does not price it. What can be said is the direction: a cap whose curvature is concentrated near its rim is one whose rim is doing more of the shaping, which is where the paper is thickest and least willing to be shaped.

A crowded pattern also brings its tucks closer together. The gap between two curves finds a separation no sheet size removes between curved creases placed close to one another, and straight tucks crowded outward are approaching the same difficulty from a different direction: at some spacing the tucks meet, and the known limit of 130.6° for single tucks is exactly that meeting.

And a start near the rim is a shorter tuck. A tuck that begins at 0.84 of the way out has only 0.16 of the radius to run in, so it hides less length per tuck and the pattern needs more of them, or accepts a bigger residual. Neither the hidden-length arithmetic nor the pile arithmetic sees that, because both are about one start rather than about how far it goes.

Both of those costs are borne at the rim and both rise with crowding, which is the honest qualification on the headline. The pile at a start is nearly free; what crowding is not free of is everything else that happens near a rim. A cap’s rim is where the paper is thickest, where the curvature is being concentrated, where the tucks are shortest and where they run into one another, and a placement rule derived from hidden length alone is steering toward all four at once.

What was worth measuring about it #

It is fair to ask why the question was worth asking when the answer is “no”, and there are two reasons.

The first is that the worry was specific and reasonable. A pattern that optimises for one quantity by moving its features into the region where a second quantity is worst is the classic way an optimisation goes wrong, and the pile, not the panel is this site’s own record of a case where the pile turned out to be the thing that mattered. Checking is cheap; assuming is how the earlier case was missed.

The second is that the measurement produced a number that generalises. A feature that brings its own constant thickness will not be steered by a background thickness smaller than it, and the ratio is computable before any optimisation is run: three sheets against one and a half means the background can never be more than a third of the pile. That is a screening test, and it says immediately which of a pattern’s trades are worth thinking about.

A crease that curves is the answer in this family that has no constant term at all — a curved crease adds no layers — and it is therefore the one where a background thickness would steer the design. Nothing here prices it, and that is the next thing owed.

What the model assumes #

A simple tuck is exactly three sheets. That is the flat count and it is what the earlier essays have used throughout. A tuck with a finite crease radius is thicker at its folds and thinner between them, and a real one has a profile rather than a number.

The gathering’s thickness is its mean. $\alpha/\sin\alpha$ is an average around the circle at that radius, and paper gathered by hand is not uniform — it bunches, and the deepest point is deeper than the mean by an amount nothing here measures.

The two thicknesses add. A tuck sitting in gathered paper is being treated as three sheets on top of however many are already there, which assumes the tuck does not itself absorb some of the gathering. A tuck that hides the local excess would reduce the background it sits in, and this counts it twice.

And the placements are the ones already computed. The even placement and the equal-error placement are both taken as given; no third placement is searched for, and in particular nothing here asks what the best placement would be if the pile were in the objective.

Where the depth actually goes #

One more reading of the same numbers is worth having, because it reverses which quantity a designer should be watching.

The deepest paper on a gathered cap with tucks in it is not at a tuck start. It is at the rim, where the gathering reaches 1.57 sheets on a hemisphere and 2.42 at 120°, and where every tuck that runs to the rim is also three sheets. A start is a local feature; the rim is the whole circumference. So a pattern’s worst pile is a rim pile, it is decided by the cap depth and the tuck count rather than by any placement, and moving the starts about does not touch it.

That is the practical version of the whole result. The placement question and the thickness question are nearly independent, and a designer who had been trading one against the other was trading two things that do not meet. The thickness question is answered by how deep a cap and how many tucks; the placement question is answered by the minimax; and the pile at a start, which is where the two might have met, is dominated by a constant.

How the numbers were checked #

The equal-error placement is recomputed rather than quoted. The knots are found by extending each piece as far as a tolerance allows and bisecting on the tolerance until the piece count is right, which is the same construction the earlier placement used and is run again here so that a change to either would show as a disagreement.

The crowded placement is required to be the deeper one. If it were not, the whole question would be ill-posed, and the check would fail rather than report a comfortable answer.

And the penalty is required to be under five per cent, which is the claim. A cap deep enough or a tuck thin enough to break that would fail the check, which is the intended behaviour: the result is about this regime and the check says so.

Still open: three answers with no common price #

The pile question is settled and it leaves the larger one the first account of this subject raised and nobody has taken up.

What a flat sheet can become names three ways round the sphere — seams, curved creases, and a few per cent of stretch — and four essays since have been spent on a fourth, the tuck. There is still no currency in which the four can be compared. A tuck hides length as thickness; a gore removes it and costs a cut; a curved crease redirects the curvature along a line; strain absorbs a little of it everywhere and then stops.

What makes a common price look possible is that all four are answers to one quantity: the excess circumference a flat disc has over the sphere’s circle, which is $1 - \sin(s)/s$ of the circle at arc $s$ and rises to 36.3 per cent at a hemisphere’s rim. Every answer disposes of that excess, so every answer can be priced by how much of it one unit of the answer disposes of — and the units are a cut, a fold, a crease and a per cent of stretch.

Sideways from here, the strain answer has a limit that is computable immediately and has not been computed. A material that stretches by some fraction absorbs the excess out to wherever the excess reaches that fraction and no further, so every material has a deepest cap it can make with no pattern at all, and the number follows from one equation. That is the natural place for the pricing to start, because it is the one answer with a hard ceiling.

The habit worth carrying is about corrections to corrections. Before trading two quantities against each other, check which one has a constant term in it. A cost made mostly of something that does not vary is a cost that will not move much whatever is optimised, and an hour spent balancing it against an accuracy that varies by forty per cent is an hour spent on the smaller half.