A spacing costs one plus the other side

The width is charged in grid found that a grafted strip has a second price. The paper is exactly width times length, as the first of these essays proved, but a width that is not a whole number of the design’s grid spacings moves every vertex past the cut onto a finer grid — twenty-five times finer for a strip four per cent over a spacing. The remedy it recommended is the one every box-pleating designer already uses: round the width up to the next whole spacing and pay for the extra paper. It priced that remedy one strip at a time, at a spacing of paper per strip at worst.

It then left two things standing. The first was a worry: a strip in one direction changes the spacing along one axis only, and a rectangular grid might bend the design’s forty-five-degree diagonals. The second was an arithmetic nobody had run: with strips in both directions, the rounding goes into both families’ totals, and the crossing term is the product of the two totals, so the rounding should be charged twice.

It is charged twice, and the worry turns out to point the wrong way. The rounding bill on a real mixture of features is not the sum of the per-strip roundings, it does not fall steadily as the grid gets finer, and a strip in one direction does not produce a rectangular grid in any sense a designer can use — it divides the square grid in both directions at once.

Five features, and what they cost as asked #

The design used throughout is small enough to follow by hand and mixed enough to show everything. A unit square sheet, on a grid, receives three strips across it — features that need 0.08, 0.14 and 0.2 of extra width — and two down it, needing 0.3 and 0.06 of extra height. Call the first family’s total $X = 0.42$ and the second’s $Y = 0.36$.

At exactly those widths the bill in paper is the one the earlier essays derived:

$$A = X + Y + XY = 0.42 + 0.36 + 0.1512 = 0.9312$$

The last term is the crossing: six rectangles where a strip of one family crosses a strip of the other, summing to the product of the totals. It is sixteen per cent of the bill, paper both features are charged for and neither uses.

At exactly those widths the design also leaves its grid. Every one of the five is a whole number of fiftieths and none is a whole number of tenths except 0.2 and 0.3, so on a ten-by-ten grid the grafted design ends up on a grid of fiftieths. That is the grid bill, and the obvious alternative is to pay in paper instead: round each width up to the next spacing.

Two neighbouring grids, measured #

The claim that rounding costs a spacing at most a strip is true of each strip. It says nothing about a mixture, and the mixture is what a design actually receives, so the first thing to do is perform the grafts rather than add up bounds.

On a grid of tenths the three strips across become 0.1, 0.2 and 0.2 and the two down become 0.3 and 0.1. The sheet grows by $0.5 + 0.4 + 0.5 \times 0.4 = 1.1$, measured over the panels the creases enclose after the grafts rather than substituted into the formula, and the grafted pattern is on a grid of tenths in both directions. The rounding cost 0.1688, eighteen per cent of the bill as asked.

On a grid of elevenths, one division finer, the same five features become 0.091, 0.182 and 0.273 across and 0.364 and 0.091 down. The sheet grows by 1.2479. The rounding cost 0.3167, thirty-four per cent — nearly twice the coarser grid’s.

Nothing about this is subtle once it is written out. The five widths as asked are close to tenths: 0.2 and 0.3 are tenths exactly, and 0.08, 0.14 and 0.06 are each within a few hundredths of the tenth above. Against elevenths the same widths land badly — 0.2 is just over two elevenths and has to go to three, 0.3 is just over three and has to go to four. A width’s rounding cost depends on where it falls against the spacing, not on how small the spacing is, and the widths a design needs were never chosen with any spacing in mind.

A bill that does not fall steadily #

Run the same rounding over every grid from four to twenty and the unevenness is the whole of the picture.

The rounding bill runs 121.5 per cent of the features’ cost on a four grid, 67.5 on five, 54.4 on eight, 33.9 on nine — and then 18.1 on ten, 34.0 on eleven, 20.8 on twelve, 21.4, 22.7, 23.1 and 23.3 on thirteen to sixteen, 34.1 on seventeen, 25.3 on eighteen, before nineteen and twenty finally beat ten at 17.5 and 10.6. Eight consecutive finer grids all cost more than the ten.

The rule of thumb is not wrong, it is answering a different question. A spacing at most per strip is a bound, and the bound does fall as the grid gets finer: on a twenty grid it is 0.05 a strip, on a ten grid 0.1. What the bound hides is that the actual rounding of a given width is anywhere between nothing and that bound, and where in that range it falls is decided by the width’s position against the spacing. On a fine grid the bound is small and the rounding is somewhere under it; on a coarser grid that happens to suit the widths, the bound is large and the rounding is nearly zero.

So the rounding bill as a function of the grid is a sawtooth under a falling ceiling, and a designer choosing the grid for the flaps and then rounding the features to it has chosen a point on the sawtooth by accident. For this design the accident is worth a factor of two between neighbouring grids.

There is a limit to how far the unevenness goes, and it is worth stating so the result is not overread. The ceiling does fall: at a hundred divisions no rounding can cost more than a hundredth a strip, and the sawtooth shrinks with it. The finding is that it is not monotone, not that fineness stops helping. What a designer loses by ignoring it is the difference between neighbouring points on the saw — which on this design, at grids anybody draws, is the difference between a feature costing a fifth more paper and costing a third more.

Paid in grid or paid in paper #

The rounding is one of two ways to pay for a width the grid cannot hold, and the two are worth setting side by side, because they are the same fact read in two currencies.

Slid in at exactly the widths asked, the five features divide a ten grid by five: its spacing of a tenth has a greatest common divisor of a fiftieth with the widths. They divide an eight grid by twenty-five, since an eighth and a fiftieth share only a two-hundredth. Twelve and sixteen are divided by twenty-five as well; twenty by five.

The two columns rank the grids the same way, and the reason is that they measure one thing. A width the grid nearly holds is cheap to round and divides the grid only a little; a width the grid holds badly is dear to round and divides it a lot. Both are functions of how the widths sit against the spacing, and the ten and twenty grids are the ones against which these five widths sit well. A designer who has already chosen to round is choosing between grids, and the column on the left is a quick way to see which ones the features were already close to.

That also gives the non-monotonicity an explanation that does not need the arithmetic. The features came in fiftieths. Any grid whose spacing shares a large factor with a fiftieth — tenths, twentieths, twenty-fifths — holds them nearly, and any grid that shares only a small factor holds them badly. Eleven, thirteen, seventeen and nineteen share none; eight and sixteen share only a factor of two. Among grids of similar fineness, the ordering by rounding cost follows the common factor with the widths, which is number theory and not geometry, and it is why neighbouring grids can be so far apart. Nineteen shares nothing with fifty and still beats ten, because at nineteen divisions the ceiling has fallen far enough to cover a bad fit; eleven, thirteen and seventeen share nothing and have not.

Where the rounding is paid #

The second thing the earlier essay predicted is the one with an exact form. Rounding raises both families’ totals, and the crossing term is their product, so rounding one family changes what the other family’s strips cost as well.

Write the rounded totals as $X' = X + \delta_x$ and $Y' = Y + \delta_y$. The rounded bill minus the bill as asked is

$$R = \delta_x\,(1 + Y') + \delta_y\,(1 + X)$$

which is exact, not an approximation, and which says what the price of a spacing is. A unit of width added across is paid once against the sheet’s own height and once against every strip running down, so it costs $1 + Y'$ rather than one. A unit added down is paid against the sheet’s width and against every strip running across, at $1 + X$. The asymmetry — one family’s rounded total and the other’s as asked — is only the order the two are applied in; the total is the same either way.

The split has a limit that can be read off the formula. When both roundings are small, the crossing’s share of the rounding bill is a weighted average of $Y/(1+Y)$ and $X/(1+X)$ — here 26.5 and 29.6 per cent. On every grid of ten or finer the crossing takes between 27.5 and 29.5 per cent of the rounding, and on the coarse grids it takes more, up to 36 per cent on a four grid, because there the two roundings are large enough to make a rectangle of their own.

That share is the answer to the question left open. The rounding is charged twice, and the second charge is not a small correction: on this design it is more than a quarter of everything the rounding costs, it is invisible in either family’s own bill, and it grows with the other family. A design whose features across total half the sheet charges every spacing of rounding down at one and a half spacings of paper. Rounding one family is priced by the other.

It also changes what the one-strip rule of thumb says. A spacing at most per strip, on a sheet of side one, bounds each strip’s rounding at a spacing of width; the paper that width costs is not a spacing but a spacing times one plus the other family’s total. The bound on the rounding bill for a design with $p$ strips across and $q$ down is therefore

$$R \le \frac{p\,(1 + Y') + q\,(1 + X)}{n}$$

on an $n$ grid, which for this design at $n = 10$ is 0.704 — and the actual 0.169 is under a quarter of it. The bound is honest and loose, and the looseness is exactly the sawtooth: how far under the bound a real design sits depends on its widths.

Two families that each want their own grid #

The crossing has one more consequence that is not visible with a single family, and it is sharpest on a design built to show it. Put two strips of a third across the sheet and three down it at a quarter, a quarter and a half.

Each family alone has an obvious grid. The thirds are exact on any multiple of three and the quarters on any multiple of four, and on its own grid each family’s rounding costs nothing. Together, a grid that suits one family still charges the other, and it charges it through the crossing as well as against the sheet: on a six grid the thirds are exact, the two quarters round up to thirds and the half stays, costing 11.9 per cent of the features; on an eight grid the quarters are exact and the thirds round to three eighths, costing 7.1. Only twelve — the least common multiple of three and four — holds both, and on the grids drawn it is the only one where the rounding bill is zero.

Two things in that result are worth drawing out. The first is that the ranking of the half-suited grids is not decided by which family is exact but by which family is larger and which rounds worse: the eight grid, suiting the three quarters, costs less than the six, suiting the two thirds, because a third is closer to three eighths than a quarter is to a third, and because the thirds’ rounding is multiplied by one plus the quarters’ total rather than the other way about. The second is that the least common multiple is not a luxury. A design with two families is on the lowest grid either family would choose only by coincidence, and a design extended later, whose second family arrives after the first was fitted, is extended into exactly that coincidence or out of it.

That is the design reading of the crossing term. Grafted features in one direction can be fitted to the grid one at a time. Features in both directions are coupled — not through their widths, which are independent, but through the product that prices them — so a grid good for one direction is charged against the other.

A strip in one direction, and the grid in both #

The worry left by the earlier essay was that a strip in one direction produces a rectangular grid, and that on a rectangular grid a design’s forty-five-degree diagonals would stop being at forty-five degrees. That would make a one-direction graft the dangerous kind, since every vertex a diagonal touches would have to be re-derived.

It cannot happen, and the reason is the same one that makes a graft possible at all. A strip is slid in only along a line every crossed crease meets square, and a diagonal meets nothing square, so no diagonal crosses the cut. Every diagonal lies wholly to one side of it and is carried across by a translation, and a translation turns nothing.

The figure is the plainest flat-foldable pattern that has diagonals and a line across that none of them crosses: a preliminary base, whose diagonals end on the rim, beside a band of vertical pleats. A strip of 0.1 goes into the band. Every crease comes out with the direction it had, every vertex keeps the angles it had, and the local conditions are satisfied from scratch on the grafted pattern. Nothing needs re-deriving.

What changes is the grid, and it changes in both directions. The spacing across goes from an eighth to a fortieth, as the earlier essay’s table predicted for a width four fifths of a spacing. The spacing down stays at a quarter, since no vertex moved up or down. So the pattern’s own coordinates really are on a rectangular lattice, a fortieth by a quarter.

But a box-pleated design is extended by adding creases at forty-five degrees, and a diagonal rises exactly as far as it runs. A new diagonal from a vertex on one side of the grafted strip to a vertex on the other runs across the strip’s 0.1, so it has to rise 0.1 as well, and the lattice down has nothing at 0.1. The grid a later diagonal can use is the spacing common to both directions — a fortieth, both ways. A strip in one direction divides the square grid in both, which is the opposite of the worry: the diagonals already drawn are safe, and every diagonal not yet drawn pays for the strip’s width in the direction the strip was not slid.

For the rounding this makes the case for rounding stronger. A width a whole number of spacings keeps the lattice square as well as coarse, and a single awkward width in one direction is enough to put every future forty-five-degree crease on the finer grid.

What the picture rests on #

The sheet is a unit square and the grid is square. On a rectangular sheet the formula carries the sheet’s own sides — a spacing across is charged against the height plus the strips down — and the shares change accordingly; nothing else does.

Rounding is up. A feature needs at least its width, because a strip narrower than asked makes a flap shorter than asked, so rounding down is not available. A design that tolerates some latitude in its features — a scale may be a little smaller — has a genuinely different problem, closer to rounding in the cheap direction for a uniaxial base, where each rounding is a small reallocation with a price and the direction of each is a decision.

Every strip is rounded separately. A family could instead be rounded as a total, the extra width assigned to whichever strip suits it — but every vertex between two cuts is shifted by the sum of the strips before it, so the grid survives only if every partial sum is a whole number of spacings, and that forces each strip to be one.

The patterns are plain grids and a preliminary base beside a band. A real box-pleated design has fewer clear lines, which is the constraint that binds first; a design with no line clear in one direction takes no strip in that direction and has no crossing term to pay. What the figures establish is the price once the lines exist.

And the figures cannot show the folding. A grid that suits the features is cheaper in paper and keeps the pattern regular; whether a finer grid is harder to fold, and by how much, is a separate cost paid in the hand and not in the sheet. Twenty divisions can beat ten in paper and lose in folding, and nothing here weighs the two.

Why the crossing couples what the widths do not #

The unexpected thing in all of this is where the coupling comes from. The two families’ widths are chosen independently — one set of features across, another down — and each family’s own bill is independent of the other’s. The coupling is entirely in the crossing term, which exists because a strip slid in across a sheet already lengthened by strips down has to be longer than the original sheet.

That is a statement about the order of operations becoming a statement about prices. The second term found the crossing rectangle as paper both features are charged for and neither uses; six rectangles and one term found it factorises into a product of totals. Read as a price, the product says that each family’s cost per unit of width is one plus the other family’s total — so a family’s marginal price is not a property of the family. A feature across gets dearer when features are added down, with no change to anything it does.

The same structure turns up whenever a quantity is an area built from two independently chosen lengths. It is why the circles are not the condition in a uniaxial base — every pair of flaps interacts through the sheet they share — and why the grid a design lives on is the one decision every later feature is charged against. What is distinctive here is only how cleanly it factorises: two totals, one product, and a price of one plus the other.

Still open: choosing the grid for the features #

The analysis prices a grid once the features are known, and the practical question runs the other way. A designer choosing a grid at the start chooses it for the flaps; the features that will be grafted later are not known yet.

What a grid should be, for features not yet chosen, is a question with an answer in expectation. If feature widths arrive from some distribution — the widths a designer tends to ask for, in some unit — the expected rounding bill on each grid is computable, and so is the expected crossing term. A grid whose spacing shares large factors with the widths designers ask for would be cheap on average, and a grid a prime number of divisions across would be dear. Whether the grids box-pleaters actually use — multiples of two and of three, overwhelmingly — are the cheap ones in that sense is a measurable question about practice.

The second direction is the latitude a feature has. A scale on a snake need not be exactly 0.14 wide; it needs to be about that, and between 0.12 and 0.16 would do. With latitude, some features can round down, and the problem becomes choosing a grid and a set of roundings together to minimise paper while keeping each feature inside its range — which is the grafting version of the question a price holds until the arrangement moves asked of limbs, and which the product term makes harder rather than easier, since a feature’s price depends on every other family’s total.

Sideways from here, the one-direction result deserves a place beside the width is charged in grid: a single awkward strip divides a design’s square grid in both directions, which makes the damage table there an underestimate for any design that will be extended at forty-five degrees.

The habit worth carrying is about bounds under a sum. A bound that holds for every term of a sum is a bound on the sum, and it is almost never the sum. A spacing a strip was true, and it predicted a bill that falls as the grid gets finer; the bill does not, because where each term sits under its bound is decided by something the bound does not contain.