A straight tuck is a cone point

A tuck keeps what a gore cuts gathers a flat disc into a spherical cap by folding the circumference the cap cannot use into tucks. It finds that a tuck following the sphere must widen as the cube of the distance from the centre, so its edges are curves, and that a tuck with straight edges — a wedge with its point at the centre — hides length in proportion to the distance and makes a cone.

Curved creases are hard to fold accurately and straight ones are easy. So the practical question is what can be done with straight tucks alone, and the answer is to start them at more than one radius: a first ring of straight tucks from the centre, a second beginning partway out, a third further out still, each starting where the tucks inside it stop keeping up with the sphere.

That works, and it has a law. It also has a price that appears at a point, and the point is where the curvature of the whole construction turns out to live.

A straight tuck hides in a straight line #

A tuck with straight edges that starts at some radius $s_0$ is a wedge whose point is at $s_0$. At radius $s$ beyond it, the wedge has opened to a width proportional to $s - s_0$, so the length it hides grows in a straight line from nothing at its start.

The length a sphere needs hidden inside the circle at radius $s$ is $2\pi\bigl(s - R\sin(s/R)\bigr)$, which starts flat and steepens as the cube of the radius. A straight line cannot follow a cubic. It can only be made to agree with one at its ends, and the choice of where to start the line is the choice of where it agrees.

Start every tuck at the centre and there is one straight line, from zero at the centre to the rim’s total at the rim. It hides too much everywhere in between: halfway out it hides half the rim’s total, where the sphere wants an eighth. Start a second ring of tucks partway out and the hiding becomes two straight pieces — gentle until the second ring begins, steeper after — and the broken line lies much closer to the curve.

The error falls as the square of the starts #

Evenly spaced starts turn the question into one about a curve and a broken line through it, and that question has a standard answer.

A broken line with $m$ equal pieces, agreeing with a smooth curve at the ends of each piece, is out by at most one eighth of the curve’s greatest curvature times the square of each piece’s length. The pieces are $1/m$ of the radius long, so the error falls as $1/m^2$: double the number of starting radii and the worst shortfall falls by four.

The census bears it out on a hemisphere. Straight tucks from the centre alone miss the sphere’s hiding by 36.9 per cent of what the rim hides. From two radii, 12.3 per cent. From four, 3.3. From eight, 0.8. From sixteen, 0.2. The ratios between successive doublings are 2.99, 3.73, 3.93 and 3.98 — approaching four as the pieces get short enough for the curve to look like a parabola across each one, exactly as the law says.

That is the same square law that governs a sphere covered in gores — the residual strain in a strip falls as the square of the number of strips — and the same law that governs every piecewise approximation of a smooth thing by straight pieces. What is particular to tucks is what the count counts. It is the number of starting radii that sets the error, not the number of tucks: a ring of eight tucks and a ring of twenty-four hide the same total length along the same broken line, and more tucks per ring only share it out more finely.

Where a tuck starts is three creases at a point #

Now look at a single tuck’s start, because the construction has a feature there that the smooth version does not.

A straight tuck starting partway out has three creases meeting at its point: the two edges that fold the flap under, and the centre line along which the flap folds in half. Around that point the paper’s angles still add up to a full turn — the paper has not been cut — so as paper the vertex is developable. But three creases at a point is a vertex of odd degree, and nothing meets at three: an interior vertex that folds flat has an even number of creases and at least four. A tuck’s start cannot fold flat.

That is not a defect of the construction. It is the construction working. The gathered cap is not supposed to be flat, and the start of a tuck is the one place in it where the flat-folding conditions fail — so it is the one place the gathered sheet is allowed to be anything other than flat.

The curvature goes to the starts #

What the sheet is instead, at a tuck’s start, has an exact description.

The visible surface around the start has lost the wedge folded under it. The paper’s full turn of angle is still there, but only $2\pi$ minus the wedge’s angle of it is on the visible surface. A surface whose angles round a point add to less than a full turn is a cone at that point, and the curvature concentrated there is exactly the missing angle — the angle deficit.

So a disc gathered with straight tucks is flat everywhere except at the points where tucks start, and at each of those it is a cone. All of the gathered sheet’s curvature sits at the starts, in lumps, where a sphere spreads the same curvature evenly over every point.

That reconciles two statements that could look contradictory. What a flat sheet can become says folding a flat sheet produces no curvature anywhere, concentrated or otherwise, because every vertex made by folding a flat sheet has zero angle deficit. And a gathered cap is visibly curved. Both are right: the paper’s own angles at a tuck’s start sum to a full turn, so the paper has no curvature; the visible surface’s angles sum to less, because part of the paper has been folded out of it. The curvature belongs to the surface the paper makes, not to the paper, and a tuck is a way of making a surface out of less than all of the paper.

The flat-folding conditions mark the curvature #

The last section can be turned into a general rule, and it is a surprisingly useful one.

On a sheet folded flat, every interior vertex satisfies the flat-folding conditions — an even number of creases, three more of one letter than the other, the alternating angle sums balanced. Two conditions at a point are what this whole subject checks. On a sheet folded into a shape, some vertices do not satisfy them, and the tuck’s start shows what those vertices are: places where the folded surface is curved, because a vertex that cannot fold flat is a vertex around which the surface is not flat.

So a crease pattern for a curved object can be read for where its curvature is by asking where it fails to be flat-foldable. The starts of straight tucks fail and are cone points; everywhere else the pattern is flat-foldable and the surface is flat. That is a reading of why the difference is two from the other side: the conditions that keep a vertex flat are exactly the conditions whose failure puts curvature there.

One cone, and many #

The simplest case makes the rule concrete and gives the whole construction a limit at both ends.

Start every tuck at the centre and all their points coincide. The centre is then a single vertex with every tuck’s three creases meeting at it, the visible surface round it has lost every tuck’s wedge, and the disc closes into one cone with all its curvature at one point. Straight tucks from one radius make a cone — the figure’s first row, 36.9 per cent away from a sphere’s hiding.

Start the tucks at more radii and the curvature is shared among more points in rings. As the number of starts grows, the rings crowd together, the points spread over the whole cap, and the lumps of curvature approach the even spread a sphere has. The broken line approaches the cubic as the square of the number of rings, and the surface approaches the sphere at the same rate.

The same picture explains the familiar pleated objects. A paper baking case has straight pleats from its base, all effectively starting at one circle, and it is a frustum of a cone. A lampshade gathered from one ring is a cone. A dome gathered in several rings of pleats, as a fabric dome or a paper model is, approaches round in proportion to the square of how many rings it has.

How many starts a given roundness needs #

The square law turns into a budget as soon as a shortfall is named, and the budget is small.

Multiply each shortfall in the census by the square of its number of starting radii and the product settles: 12.3 per cent times four is 49, 3.3 times sixteen is 53, 0.8 times sixty-four is 51, 0.4 times a hundred and forty-four is 58 — the last inflated by rounding a number that small to one decimal. So on a hemisphere, $m$ starting radii leave a worst shortfall of roughly half of what the rim hides, divided by $m^2$. A dome whose hiding must follow the sphere to within one per cent of the rim’s hidden length needs about seven or eight starting radii; to within a tenth of a per cent, about twenty-three.

The law also says which starts are worth folding. Going from two starting radii to four removes three quarters of the shortfall, 12.3 per cent of what the rim hides down to 3.3. Going from eight to sixteen would remove three quarters again, but of 0.8 per cent, saving about six tenths of a per cent. The first few starting radii buy almost all of the roundness, and every doubling after them buys a quarter of what the one before it did.

That number of rings is independent of how many tucks each ring has. The rings decide how closely the hiding follows the sphere; the tucks per ring decide, as the gathered cap found, how strained the visible pieces between them are. A designer choosing a pattern of straight tucks is therefore choosing two numbers against two separate errors, and the one this essay measures is the one that falls fastest.

The same trade appears wherever a curve is followed by straight pieces, and this subject has met it before in other guises. Exact is not accurate found that an exact division by repeated crossings grows less accurate in a folder’s hands the more parts it divides a strip into — from 0.31 to 0.56 millimetres between three parts and sixteen; here the error falls with the number of pieces a designer allows, and the difference is that a tuck pattern’s pieces are laid out once on the flat sheet rather than folded one after another. Error is folded too is the compounding case, where each fold carries every earlier error with it, and a pattern of starts is a case that does not compound.

It also bears on the gap between two curves, which found that the paper between neighbouring curved creases exists only as far as their rulings allow, so that a concentric pleat has a hole at its middle that no size of sheet removes. A curved tuck is three curved creases converging on the centre, which is where that kind of bound bites. A pattern of straight tucks sidesteps the question, since the paper between straight creases can lie flat and has no rulings to cross, and the starting radii it needs are what it pays for doing without a curve.

What the patterns cannot show #

The patterns show where creases go on the flat disc, not the gathered surface, and three things about the gathered surface are outside them.

The cone points are idealised. Paper cannot come to a true point; a tuck’s start is rounded over a few sheet thicknesses, and the curvature the construction concentrates there is spread over a small cap. The square law describes the pattern of starts and not what happens within a paper’s thickness of each one.

The visible pieces between starts are only nominally flat. In a real gathered sheet they bend under their own weight and under the forces the flaps exert, and the curvature the construction puts at points leaks into the pieces between them. The error measured here is in the length hidden, which the flaps fix exactly, and not in the shape the sheet settles to.

And the starts are evenly spaced. Spacing them to follow the cubic — close together near the rim, where it steepens, and sparse near the centre — would do better for the same number of starts, and nothing here computes the best spacing.

The tucks the census assumes #

The target is a spherical cap laid on the disc with its radii kept at full length, as in the gathering argument this one builds on, so the length to hide at each radius is fixed by the sphere and the tucks only share it out.

A tuck is straight, three creases, and hides length linearly from its start. Each ring’s tucks hide the increase in slope the broken line needs at that ring, shared evenly among the ring’s tucks, and each ring is offset from the one inside it.

The error is measured in hidden length, as a share of what the rim hides: the largest difference, over the whole radius, between the broken line and the sphere’s curve.

How the law was checked #

The broken line’s slope is required to rise at every start, so that each new ring adds tucks rather than having to take some away — which is true because the sphere’s curve is convex, and a construction in which it failed would not be a gathering.

Each doubling of the starting radii is required to divide the worst shortfall by between three and a half and four and a half, once there are at least two starts. The measured ratios, 3.73, 3.93 and 3.98, sit inside that and close in on four; the first doubling, from one start to two, is 2.99 and is outside the rule’s scope, because a single straight line is too coarse for a parabola to describe the curve across it.

Still open: the best places to start #

Evenly spaced starts are the obvious choice and not the best one. The sphere’s cubic curves hardest near the rim, so a broken line with its breaks crowded toward the rim would follow it more closely for the same number of starts. Placing starts to minimise the worst shortfall is a small optimisation with a definite answer — spacing the breaks so that each piece carries the same error — and it would say how many cone points a paper dome of a given accuracy actually needs.

The other direction is the combination the first account of this subject noted nobody appears to have asked about. A pattern of straight tucks puts curvature at points; a curved crease puts it along a line; strain spreads a little of it everywhere. Whether a mix of the three reaches shapes none of them reaches alone, and what the cheapest mix is for a given shape, is the question the three answers and the fourth have now set up.

The habit worth carrying is a way of reading a crease pattern for a curved thing. Find the vertices that cannot fold flat. They are not errors in the pattern; they are where the pattern keeps its curvature, and counting them is counting how finely the curvature has been shared out.