Crowd the tucks toward the rim

A straight tuck is a cone point gathers a flat disc into a spherical cap with tucks whose edges are straight. A straight tuck hides length in proportion to how far past its start it has gone, which is a cone’s law, so tucks from the centre alone make a cone. Tucks started at several radii hide length along a broken line that follows the sphere’s curve, and with the starts evenly spaced the worst shortfall falls as the square of their number: 12.3 per cent of what the rim hides from two starting radii, 3.3 from four, 0.8 from eight.

It ends by noting that evenly spaced is merely the obvious choice. The sphere’s hiding curves hardest near the rim, so a broken line with its breaks crowded outward should follow it more closely for the same number of starts — and since every start is a ring of cone points, a better placement is a rounder dome for the same number of points where the paper has to come to a peak.

The best placement is a small optimisation with a definite answer, and it is worth about two fifths of the error.

What each stretch between starts gets wrong #

A ring of straight tucks starting at radius $s_0$ hides length in a straight line from its start. Several rings together hide along a broken line through the sphere’s own curve,

$$h(s) = 2\pi\bigl(s - R\sin(s/R)\bigr),$$

agreeing with it at each starting radius and cutting across it between them. On any one stretch, from one start to the next, the broken line and the curve differ most somewhere in the middle, and by how much depends on two things: how long the stretch is, and how hard the curve bends along it.

The standard estimate is the one the evenly spaced census already relied on. A straight line through a smooth curve at the two ends of a stretch of length $\ell$ is out by at most $\tfrac18 \ell^2$ times the curve’s greatest second derivative on the stretch. For the sphere’s hiding that second derivative is $2\pi\sin(s/R)/R$: nothing at the centre, where the curve starts flat, rising to its greatest at the rim of a hemisphere.

So an evenly spaced stretch near the centre is almost wasted — the curve there is so nearly straight that a much longer stretch would have been as accurate — while a stretch of the same length at the rim carries the construction’s worst error. Evenly spaced starts give every stretch the same length, and the stretches bend unequally.

Every stretch the same worst error #

The fix is to make the stretches unequal in the opposite way: long where the curve is gentle, short where it bends. The best arrangement for a given number of starts is the one in which every stretch carries exactly the same worst error, because if one stretch had less, its end could move outward and relieve the stretch after it.

That arrangement can be found without guessing. Choose an error. From the centre, extend the first stretch as far as it can go without its worst shortfall exceeding that error; start the next ring there, and extend again; continue to the rim. A small error needs many stretches to reach the rim and a large one needs few. Halving the error back and forth finds the smallest error that a given number of stretches can reach the rim with, and the starts that reach it are the best starts. On a convex curve like the sphere’s hiding, a stretch’s error only grows as it lengthens, so extending each stretch as far as the error allows is optimal, not merely good.

The second figure draws the result for four rings on a hemisphere. Evenly spaced, the inner three starts sit at a quarter, a half and three quarters of the radius. Best spaced, they sit at 0.36, 0.60 and 0.80. The stretches shrink from 0.36 of the radius at the centre to 0.20 at the rim, and the worst shortfall falls from 3.3 per cent of what the rim hides to 2.0.

On the crease pattern the difference is visible as rings drawn closer together toward the edge of the disc. The third figure is the four-ring pattern with its starts at the best radii: eight tucks from the centre, then rings of eight beginning further out than even spacing would put them, the last ring close to the rim.

And the profile of hidden length shows why the placement works. The broken line through the best starts stays close to the sphere’s curve along the whole radius, where the evenly spaced one drifts furthest from it in the outermost stretch.

Two fifths, in the limit #

The census in the first figure gives the saving at each count: 31 per cent at two rings, 36 at three, 38 at four, 40 at eight, 41 at sixteen. It is growing more slowly each time, and it has a limit that can be written down.

With many stretches, each is short enough that the curve’s second derivative is nearly constant along it, and the error on a stretch is $\tfrac18\ell^2 h''$. Requiring every stretch to carry the same error $\varepsilon$ fixes each length at $\ell = \sqrt{8\varepsilon/h''}$, so the number of stretches needed to reach the rim is the integral of $\sqrt{h''/8\varepsilon}$ over the radius. Turned round, $m$ stretches reach the rim with an error of

$$\varepsilon_{\text{best}} \approx \frac{1}{8m^2}\left(\int_0^{s_{\max}}\sqrt{h''(s)}\,ds\right)^2,$$

while evenly spaced stretches, whose worst error is set by the steepest bend, give

$$\varepsilon_{\text{even}} \approx \frac{1}{8m^2}\,s_{\max}^2 \max h''.$$

Both fall as $1/m^2$, and their ratio is a number that does not depend on $m$ at all. For a hemisphere, where $h''$ goes as $\sin$ from zero to its peak at the rim, the ratio is $\bigl(\int_0^{\pi/2}\sqrt{\sin t}\,dt\bigr)^2/(\pi/2)^2 = 0.582$. In the limit the best starts leave 58 per cent of the evenly spaced error, a saving of 42 per cent, and the census closes in on it: 0.588 at sixteen rings.

The principle behind that number is older than any dome. When a smooth function is approximated by straight pieces, the pieces that minimise the worst error are spaced so that the square root of the curvature is shared out equally among them — dense where the function bends, sparse where it runs straight. Numerical analysts place the nodes of adaptive meshes by the same rule, so that a computation spends its resolution where the answer changes fastest. A folder’s accuracy is spent too, and exact is not accurate found it running out as divisions multiply.

The rule predicts the starts #

The limit is an argument about many short stretches, and it would be reasonable to doubt it at four. It holds up surprisingly well, and checking it says what the best starts are without any search.

If every stretch is to carry the same error, and a stretch’s error is an eighth of its length squared times the curve’s bending, then the starts must divide the integral of the bending’s square root into equal parts. For a hemisphere that integral, from the centre to a radius $\theta$ measured as an angle on the sphere, is $\int_0^\theta\sqrt{\sin t}\,dt$, and its value at the rim is 1.198. Four rings divide it into quarters, and the radii that do so are 0.377, 0.606 and 0.808 of the rim’s. The search found 0.363, 0.598 and 0.804. With eight rings the rule gives 0.236, 0.377, 0.497, 0.606, 0.709, 0.808 and 0.904, and the search found 0.226, 0.370, 0.491, 0.602, 0.706, 0.806 and 0.904 — agreeing to within a hundredth at every start and exactly at the last.

The rule is also easy to use near the centre, where the sphere’s circles have hardly begun to fall short. There $\sin t$ is very nearly $t$, the integral is very nearly $\tfrac23\theta^{3/2}$, and the first start sits where that reaches its share of the total. The first ring should begin at a radius that grows as the two-thirds power of the share it is given: for four rings, at $(\tfrac32\cdot 1.198/4)^{2/3}$, which is 0.373 of the rim’s radius, and for eight rings at 0.235. The first stretch is always the longest, because it spans the part of the disc where the sphere is still nearly a plane.

That is a construction a folder can carry out with a pencil and a table of four numbers rather than a search, and the table is the sphere’s own: where its circles fall short fastest, the rings of tucks go closest together.

A shallow dish gains the most #

The limit depends on the cap, because the shape of the curve’s bending does. The fifth figure runs the comparison at sixteen rings for caps from 20° — a shallow dish — to 150°, most of a sphere.

On a shallow cap the best placement is worth the most. The sphere’s hiding on a small cap bends in almost exact proportion to the radius, so its second derivative rises steadily from nothing at the centre to its greatest at the rim, and even spacing wastes the most on the inner half. The limit there is $\bigl(\tfrac23\bigr)^2 = \tfrac49$: the best starts leave four ninths of the evenly spaced error, and the 20° cap’s ratio of 0.468 at sixteen rings is closing on its limit of 0.450.

Past the hemisphere the advantage shrinks, because the bending peaks before the rim. On a 120° cap the second derivative is greatest at 90° and falls again after it, so the outermost stretches are not the worst ones and the even spacing is less wrong. The ratio rises to 0.675 at 120° and 0.679 at 150°, their limits 0.666 and 0.672.

Rings of cone points saved #

Every start away from the centre is a ring of points where the gathered sheet comes to a cone — three creases meeting at each, a vertex that cannot fold flat. A ring of starts is a ring of places the paper has to be persuaded to a point, and a dome with fewer of them is easier to fold and less likely to crack at its peaks — each peak carrying curvature the paper itself does not have. So the saving is best counted in rings.

The sixth figure asks how many rings a hemisphere needs to follow the sphere’s hiding within each of several accuracies. Within five per cent of what the rim hides: four rings evenly spaced, three best spaced. Within two per cent: six and five. Within one per cent: eight and six. Within half a per cent: eleven and nine. Within a tenth of a per cent: twenty-four and eighteen.

The saving in rings grows with the accuracy asked for, because the error ratio of 0.58 is a ratio of squares: the number of rings needed for a given error falls by its square root, about 0.76, so a dome that needs a hundred evenly spaced rings needs about seventy-six placed well. Placing the rings is worth about a quarter of them, and the quarter is the same whatever the accuracy, once the accuracy is fine enough for the limit to apply.

That also qualifies the earlier essay’s budget. About seven or eight rings for one per cent, about twenty-three for a tenth of a per cent was the budget for evenly spaced rings, and it was right for them. For rings placed where the sphere bends, the same roundness costs six and eighteen.

Kerfs, pleats and where the bend is tightest #

A trade outside paper folding has been placing its discrete bends by this rule for a long time, without the formula.

A woodworker who needs to bend a board saws a row of kerfs — narrow cuts partway through — across its inner face, so that the board bends at the cuts and stays straight between them. On a curve of one radius the kerfs are evenly spaced. On a curve that tightens, the practice is to set the kerfs closer together where the bend is sharpest, because each kerf closes by a fixed amount and a tighter bend needs more closings in the same length. The board is a broken line approximating a curve, each kerf a break, and the spacing follows the curvature.

Tucks are the same construction on the other side of the sheet. A kerf removes material on the inside of a bend so the board can close; a tuck folds away excess material on the outside of a gathering so the paper can close. Both put the curvature at discrete places — a straight tuck is a cone point because it concentrates the sphere’s curvature where it starts — and both do best with those places dense where the target curves hardest. The rule is about approximating a curve by pieces, and it does not care whether the pieces are wood, fabric or paper.

What paper adds is that the curvature being approximated is not visible on the flat sheet. A sphere’s hiding is a fact about circles laid on a disc, and the place it bends hardest is the rim of the cap, which on the crease pattern is simply the edge of the disc. The best tuck pattern is a pattern whose rings crowd toward the disc’s edge for a reason nothing on the disc shows.

What the placement assumes #

The tucks hide length exactly as a straight wedge would. Each ring’s tucks hide in a straight line from their start, shared evenly among the ring, and the error is measured in hidden length: the largest difference, over the whole radius, between what the rings hide and what the sphere asks. That is the quantity the evenly spaced census measured, so the two are compared on the same terms.

The broken line agrees with the curve at every start. That is what makes the rings nest — each ring starts where the tucks inside it leave off. A pattern allowed to hide slightly more than the sphere asks at some radii and slightly less at others, crossing the curve between starts, could reduce the worst error further for the same rings, at the price of a gathered surface that bulges and pinches alternately rather than lying consistently to one side of the sphere.

The cap is a sphere laid on the disc with its radii at full length, as in the gathering that started this. A different target surface has a different hiding curve and a different best placement; the principle — spacing by the square root of how the target bends — carries over, and the numbers do not.

What the figures cannot show #

They cannot show the gathered surface. The error is in hidden length, which the tucks fix exactly, not in the shape the sheet settles into; a real dome’s panels bend between the cone points, and whether crowding the points outward changes how the panels between them sag is outside the model, as it is for every way round the impossibility what a flat sheet can become lists.

Nor can they say whether the outer rings can be folded. The best placement puts rings close together near the rim, and the rim is also where a gathered cap is thickest — about $\pi/2$ sheets on average for a hemisphere. Rings of tucks crowded into the thickest part of the paper may be harder to fold cleanly than the same rings spread out, and a folder’s accuracy there is a number no figure here has, though error is folded too says errors in rings folded one after another compound.

And the optimum is for the worst error. A placement chosen to minimise the average shortfall, or the shortfall weighted by how much of the dome a viewer sees, would crowd the rings less, and the figures compute only the one objective.

How the placement was checked #

The best starts are found by halving the error, extending each stretch as far as the error allows and asking whether the stretches reach the rim; the resulting pattern is then measured over two thousand radii, the same way the evenly spaced patterns are.

The best placement is required to beat even spacing at every count, and its ratio to even spacing at sixteen rings is required to be within three hundredths of the limit the integral of the curvature’s square root gives, on every cap drawn. And the best starts are required to crowd outward, each stretch no longer than the one inside it, on the hemisphere whose starts are drawn.

Still open: straight tucks, curved creases and strain together #

Placing the rings finishes the question of what straight tucks alone can do well. The larger question the first account of this subject left is the one it said nobody seems to have asked. A pattern of straight tucks puts a sphere’s curvature at points; a curved crease puts curvature along a line; strain spreads a little of it everywhere, and curved creases close together meet a gap no sheet size removes. Whether a mix of the three reaches a given shape more cheaply than any one of them, and what the cheapest mix is, is now a question with a currency: this essay prices a point in hidden length per ring, and the same measure could price a curved ring of tucks against a straight one.

The nearer question is the dome’s outer rings. The best placement crowds them where the paper is already piled deepest, and the next measurement is whether the gain in hidden length survives the thickness it crowds into — a question the pile, not the panel asked of flat patterns and that a gathered cap asks again at its rim.

The habit worth carrying is a check on any evenly spaced construction that approximates something smooth. Ask where the thing being approximated bends hardest, and whether the spacing knows. Even spacing is the right answer only for a curve that bends the same everywhere, and very few of the curves a design has to follow do.