Where a ring of divisions belongs

Three answers, one count computes how many divisions of the circle a cap needs and then applies that number to the whole pattern. It is computed at the rim, where the excess is largest, which makes it the right number in exactly one place.

Everywhere else it is too many. The excess grows from nothing at the pole, so a hemisphere in a five per cent material needs no divisions at all for the first third of its radius, and the pattern that recognises that has rings rather than a uniform count.

The rule, which is one line #

The residual a pattern leaves inside a panel is the excess divided by the number of divisions. With $m$ divisions in force at arc $s$, the residual is $f(s)/m$, and the material takes it if

$$\frac{f(s)}{m} \le \varepsilon$$

So a division has to be added wherever $f(s)$ reaches $m\varepsilon$ for the next $m$. The ring radii are the crossings of the excess curve with $\varepsilon$, $2\varepsilon$, $3\varepsilon$ and so on, and they follow from one equation with no optimisation in it at all.

On a hemisphere at five per cent that gives seven rings, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out. The first third of the radius carries no divisions, the next third carries two, and the outer third carries five.

They crowd toward the rim, and they must #

The gaps between consecutive rings shrink all the way out: 0.35, then 0.15, then 0.12, 0.10, 0.09, 0.09, 0.08. That is not a choice and not a result of any optimisation — it follows directly from the shape of $f$.

$f$ rises as $s^2/6$ near the pole and its slope keeps increasing, so equal increments of $f$ correspond to shrinking increments of $s$. A curve that steepens crosses equally spaced levels at shrinking intervals, and the ring radii are exactly those crossings.

A stiffer material gives more rings and the same shape. At two per cent a hemisphere needs eighteen, starting at 0.22 and ending at 0.99, with the outer half carrying eleven of them. The pattern is finer everywhere and the distribution is the same distribution.

Two criteria, one placement #

Here is the reason this comparison is worth having, and it was not expected.

Crowd the tucks toward the rim places tuck starts to minimise the worst error of a broken line following the sphere’s hidden length, and for four starts on a hemisphere it puts them at 0.31, 0.51, 0.68 and 0.84 of the way out. The construction here places rings so that a material tolerance is never exceeded, and its first four are at 0.35, 0.50, 0.62 and 0.72.

Those are not the same numbers and they are close — the first two almost exactly, the outer two diverging. The two criteria have nothing in common. One is a minimax on an approximation error with no material in it; the other is a tolerance on a residual with no approximation in it. That they place their first features within four per cent of one another is a fact about the sphere rather than about either method.

The divergence further out is informative too. The minimax placement spreads its four across the whole radius because it has only four to spend; the tolerance construction does not stop at four — it keeps adding rings until the rim — so the two are answering slightly different questions past the third ring. Where they agree is where both are asking “how far can one piece run before something is exceeded”, and near the pole that question has one answer whichever quantity is being bounded.

A stiffer material, ring by ring #

The plain inner region is the part of the answer a maker would notice first, and it shrinks quickly. At five per cent it reaches 0.55 radians, a third of the way to the rim; at three per cent, 0.42; at two per cent, 0.35. The region that needs nothing is the square root of the material’s give, exactly as the reach of stretch used alone is, because it is the same equation.

So a stiffer paper does not merely need more divisions; it needs them further in, and the plain disc at the middle of the pattern — the part a maker would think of as the easy part — is the first thing to disappear. That is a different sensitivity from the count, which falls in proportion, and a maker choosing a paper is choosing both at once.

The pattern drawn is what a ringed construction looks like as a crease pattern: tucks that begin at different radii, so the outer part of the disc carries more of them than the inner part. That is the same object this essay derives from a tolerance and the minimax placement derived from an error, and it is worth seeing once as a drawing rather than as a list of radii.

What a uniform pattern is actually doing wrong #

It is worth being precise about the fault, because “over-divided” is a vague complaint and this one is exact.

A uniform seven-division pattern on a hemisphere leaves a residual of $f(s)/7$ at every radius. At the rim that is 5.2 per cent, which is just inside a five per cent material — the number the count was chosen for. A third of the way out it is 0.7 per cent, which is seven times inside the material’s limit. The uniform pattern is working the material to its limit in one place and to a seventh of it everywhere else, and the creases doing that under-work are real creases in real paper.

Put the other way: a uniform pattern spends its divisions where they are needed least and needs the same number where they are needed most. The ring schedule spends nothing where nothing is needed and reaches the same count at the rim, so it leaves exactly the same residual there and much more slack inside. Nothing is given up.

What the rings cost #

A ring is not free and the cost is the one these essays have been tracking.

Each ring is a set of new tuck starts, each start is three creases meeting at a point, and a straight tuck is a cone point establishes that three creases at a point is a vertex that cannot fold flat. So the rings are where the cap’s curvature is being gathered, and a construction that crowds them outward is concentrating curvature near the rim.

Crowding outward costs almost nothing prices the thickness half of that and finds it negligible — a start is three sheets of its own on a gathering that is only one and a half. The curvature half is not priced anywhere, and the ring construction makes it sharper: seven rings with five of them in the outer third is a cap whose outer third carries most of its cone points.

What the construction saves #

Against a uniform pattern, the saving is worth stating because it is larger than it looks.

A uniform seven-division pattern puts seven tucks along the whole radius. The ring pattern puts none in the first third, two in the second and seven only at the rim. Counting crease length rather than divisions: the ring pattern uses about half the crease length of the uniform one for the same residual everywhere, because most of its tucks are short.

On a 120° cap at eight per cent the pattern is seven rings again — at 0.33, 0.48, 0.60, 0.70, 0.79, 0.88, 0.97 — which is nearly the same shape as the hemisphere’s at five per cent. That similarity is the practical content: the ring positions, expressed as fractions of the radius, are almost independent of the cap depth and the material, and only the count changes.

So there is a pattern shape here rather than a pattern, and it is one a maker could use without recomputing anything: rings at roughly a third, a half, three fifths, seven tenths, four fifths, nine tenths and the rim, with as many divisions at each as the material requires.

The reason the shape holds is that both figures are drawing the same curve at different scales. The excess is a function of the arc from the pole and nothing else, so a deeper cap is the same curve followed further along; a more generous material is the same curve read at coarser levels. Those two changes move the rings in opposite directions and, over the range drawn, very nearly cancel. That cancellation is a coincidence of the two parameter choices rather than an identity, and the essay claims no more for it than that two particular caps in two particular materials agree — which is the sort of agreement worth testing on a third before relying on.

The area the outer rings serve #

There is an easy misreading of the ring list and the arithmetic of area corrects it.

Five of the seven rings sit in the outer third of the radius, which sounds like a pattern concentrated in a small part of the cap. It is not: the outer third of a disc’s radius is more than half its area, since area grows as the square. The outer third of a hemisphere’s surface — measured on the sphere, where area grows as $1-\cos s$ — is more than half of it too.

So the ring construction is not crowding its work into a corner. It is putting the divisions where the paper is, and the inner region that needs nothing is a small disc rather than a third of the dome. A third of the radius is a ninth of the flat disc’s area, which means the plain middle of a five per cent hemisphere pattern is about eleven per cent of the sheet and the rest is divided.

That reframes what the construction saves. It is not saving two thirds of the work; it is saving the difference between a uniform seven and a schedule that reaches seven only at the rim, which in crease length is about half and in area is less. A tuck keeps what a gore cuts prices the thickness that follows from the tucks actually present, and the ring schedule reduces it by the same fraction.

What the construction cannot show #

The rule is one line and it hides several things.

It cannot say what happens between rings. A ring is the radius at which a division becomes necessary, and a real tuck has to start somewhere and go somewhere — it runs from its start outward, so the pattern past a ring has that ring’s tucks and every inner ring’s tucks still running. The count at the rim is the sum, which the construction gets right, and the geometry of how they nest is not addressed at all.

It cannot say whether the tucks fit. The known limit is that single tucks on a hemisphere run into one another at 130.6°, and a pattern with seven rings at the rim is a pattern with a great many tucks in a small circumference. Whether they collide is a question about widths, and nothing here has a width in it.

And it assumes the residual is what matters. A material takes $\varepsilon$ of strain, so a residual under $\varepsilon$ is absorbed — but absorbed strain is not free, and a sheet worked to its limit everywhere is a sheet at its limit everywhere. A pattern with more divisions than necessary would leave the material working well inside itself, which may be worth the extra creases.

A schedule rather than a pattern #

The most useful way to hold the result is as a schedule, because that is the form a maker can use and the form that transfers between caps.

Written as fractions of the radius, the hemisphere-at-five-per-cent schedule is 0.35, 0.50, 0.62, 0.72, 0.81, 0.90, 0.98 and the 120°-at-eight-per-cent schedule is 0.33, 0.48, 0.60, 0.70, 0.79, 0.88, 0.97. Those agree to within two per cent at every ring, on two caps of different depths in two materials of different give.

That is worth more than either list. It says the shape of the schedule is a property of the sphere and the counts are a property of the material, so a maker needs one drawing and a number. The drawing is where the rings go; the number is how many divisions each ring adds.

Whether the agreement survives further — a shallower cap, a much stiffer material, a cap deep enough for tucks to meet — is not shown here and would be a short computation. Crowd the tucks toward the rim found a placement whose saving closes on a limit of 42 per cent as the count grows, which is another quantity that stopped depending on the count; two such results in one subject suggest the sphere’s cubic is doing most of the deciding and the pattern very little.

What the model assumes #

The excess is shared equally among the divisions in force. That is true of a rotationally symmetric pattern and of nothing else.

A ring adds one division. The count goes 1, 2, 3 and so on, which is the finest possible schedule; a real pattern doubling its divisions at each ring would need fewer rings and would over-divide in between.

The material’s limit is a single number. Paper that stretches on purpose is where what that actually means for a damp sheet is set out, and it is not a single number.

And every ring is a ring of tuck starts. The construction is indifferent to which of the answers the divisions are — it is arithmetic about counts — and the costs quoted here are tucks’ costs.

How the numbers were checked #

The rim excess is checked against its closed form, exactly as the count figure checks it, so the quantity the whole schedule is derived from is verified.

The rings are required to number at least three, so a parameter combination that produced a trivial pattern would fail rather than draw one.

The gaps between consecutive rings are required to be non-increasing, which is the crowding claim and is checked on the computed radii rather than read off the curve. A gap that widened anywhere would mean the excess curve had been computed with the wrong sign of curvature somewhere.

And each ring radius is found by bisection on the excess function itself rather than from the small-angle approximation, so the positions are right near the rim where $s^2/6$ is no longer accurate.

Still open: whether the rings should be the same answer #

The construction says how many divisions a radius needs and is silent about what they should be, which is the question there are now pieces enough to ask.

A pattern could use different answers at different radii. The inner third needs nothing, so it should be plain sheet. The middle needs two or three divisions where the paper is thin and the tucks would be long — which is where a curved crease is most plausible, since it adds no thickness and has a long run to develop in. The outer third needs five and sits where the paper is thickest, which is where a tuck’s three sheets hurt most and where a gore’s cut would hurt least.

Whether a mixed pattern of that shape is better than a uniform one is a question with no answer yet, because the three answers have a common count and no common cost. Three answers, one count is explicit that the table says nothing about a seam against a pile, and a radius-dependent mix needs exactly that comparison at every radius.

Sideways from here, the agreement between the two placement criteria deserves following up. If a minimax on hidden length and a tolerance on residual excess put their features in nearly the same places, there may be a third quantity both are proxies for — and finding it would replace two constructions with one.

The habit worth carrying is about applying a requirement computed at the worst point. Ask where the requirement is actually binding before applying it everywhere. A quantity that grows from nothing has a region where it binds nothing at all, and a pattern uniform in a quantity that is not uniform is doing work in places that did not need any.