A grid chosen before the features

A spacing costs one plus the other side priced five grafted features on every grid from four to twenty and found the bill a sawtooth: ten divisions cheaper than every grid from eleven to eighteen, because the five widths happened to sit close to tenths. It priced a design whose features were known. It ended on the question that runs the other way, which is the one a designer actually faces: a box-pleating grid is chosen at the start, for the flaps, and the features that will be grafted into it later are not known yet. What does a grid cost before its features are chosen?

That question has an answer in expectation, and the answer depends on what kind of width is being asked for. It also has a second half, which the same essay raised: a feature usually has some latitude, and a width allowed to come in a little short changes the question more than any choice of grid does.

Widths nobody chose cost half a spacing #

Suppose the widths to be grafted are measured from a subject — a scale, a limb, a gap — and so are spread evenly across a range many spacings wide, with no preference for any fraction of the sheet. Rounding one up to the next whole spacing on a grid of $n$ divisions costs somewhere between nothing and a spacing, depending only on where the width falls inside the spacing it lands in. A width spread evenly over a whole number of spacings is equally likely to fall anywhere inside that spacing — the spacings are all alike, and each gets the same share of the range — so its rounding is spread evenly between nothing and a whole spacing. So on average it costs

$$\mathbb{E}[\text{rounding}] = \frac{1}{2n},$$

half a spacing, and nothing about $n$ other than its size enters.

The measured curve is flat. From ten divisions up, widths spread from 0.02 to 0.4 of the sheet cost half a spacing a strip on every grid to within a fortieth of a spacing — on eleven, thirteen and forty-seven exactly as on twelve, twenty-four and forty-eight. Below ten the range is only a few spacings long and the part-spacing at its ends shows; on four divisions the average is 0.555 of a spacing.

So the sawtooth the earlier essay found is real for any one design and absent from the average. Each design sits at some point on its own saw, and a different design sits at a different point on a different saw; averaged over designs nobody has chosen, the teeth cancel and what is left is the ceiling’s midline. A grid’s divisors buy nothing against widths that were not chosen with the grid in mind, and a prime grid is no dearer than its neighbours.

The same average prices the earlier essay’s bound. That bound charged every strip a whole spacing — times one plus the other family’s total, through the crossing term — and called itself honest and loose, the five-feature design sitting under a quarter of it on ten divisions. Averaged over designs, the looseness has a value.

On every grid from eight to twenty-four the average rounding bill of six thousand random designs is between 0.47 and 0.52 of the average bound. The bound is twice the expected bill, for the plain reason that a strip rounds up by half a spacing on average where the bound charges a whole one, and the crossing term inherits the same factor. Any one design can sit anywhere between nothing and the bound; the earlier five sat at a quarter of it on ten divisions.

Widths that are fractions of the sheet #

The other kind of width is derived rather than measured. A feature a third of a flap, a gap a quarter of the sheet, a band a fifth of the model’s length: widths like these are simple fractions, and against them a grid’s divisors matter a great deal.

With widths that are fractions up to eighths, a grid of twenty-four divisions rounds a strip by 0.15 of a spacing on average, because twenty-four is divisible by every denominator up to eight except five and seven, and most of those fractions land on it exactly. Forty-eight costs 0.18, forty 0.21. The powers of two sit well above: 0.42 on eight, 0.39 on sixteen, 0.34 on thirty-two, since they hold the halves, quarters and eighths and nothing with a three or a five in it. The primes pay close to the full half — 0.55 on eleven, 0.49 on thirteen — as they must, holding no fraction with a small denominator at all.

With denominators up to twelve the gaps narrow, because twelve denominators is more than any grid of this size can cover: forty is cheapest at 0.27 of a spacing, twenty-four and forty-eight cost 0.33, the powers of two 0.38 or 0.39, and the primes from eleven to forty-seven 0.43 to 0.60 — every prime above ten dearer than twenty-four and forty-eight. Twelve itself, whose divisors are two, three, four and six, costs 0.44: it holds the twelfths’ own denominators badly, because in lowest terms most fractions up to twelfths have denominators of seven, nine, ten, eleven or twelve.

The practical reading is narrow and definite. Box-pleaters draw overwhelmingly on powers of two, because each doubling of the grid is a fold in half, and against derived widths those grids are the middling ones of their size: twenty-four beats sixteen and forty-eight beats thirty-two by a sixth to a third of a spacing a strip, depending on the denominators in play. Against measured widths they are exactly as good as any other grid. Whether a designer’s future features will be fractions or measurements is therefore the whole of the grid question, and the answer is usually both: the widths a subject needs are measured, and the widths a design’s own geometry needs are fractions.

A latitude is a grid #

The earlier essay’s second open question was latitude. A scale on a snake need not be exactly 0.14 of the sheet; somewhere from 0.12 to 0.16 would do. With latitude a feature can round down, and the cheapest choice is the smallest whole number of spacings at or above the width less its latitude.

Rounding $w - \delta$ up instead of $w$ is the same rounding applied to a width moved by $\delta$, and for widths spread evenly the average rounding of a moved width is still half a spacing. So with a latitude $\delta$ each strip costs, on average,

$$\frac{1}{2n} - \delta$$

over its width as asked: the rounding-up curve moved down by exactly the latitude, on every grid. It crosses zero at

$$n^{*} = \frac{1}{2\delta},$$

twenty-five divisions for a latitude of 0.02 of the sheet, fifty for 0.01, ten for 0.05. On any grid finer than that the grafted features cost less paper, on average, than their widths as asked, down to a saving of the whole latitude a strip on a very fine grid; on any grid coarser, rounding still costs paper, but half a spacing less the latitude rather than half a spacing. The slider shows the whole family: the curve slides down as the latitude grows and its zero walks toward the coarse grids.

That turns the grid question into a statement about the features rather than about the grid. A designer who knows the latitude the features will have knows the grid at which grafting them breaks even, before knowing any of them, and the grid’s divisors play no part in it. The crossing term makes the saving larger rather than smaller: a strip across that comes in short lowers the family’s total, and every strip down is charged the product, so on a design with families both ways each unit of latitude used is worth one plus the other family’s total, exactly as each unit of rounding was.

One design keeps its teeth #

The expectation is a statement about designs in general. The five features the earlier essay used — 0.08, 0.14 and 0.2 across, 0.3 and 0.06 down — are one design, and a latitude does something less tidy to it.

On ten divisions the latitude buys nothing at all: every width less 0.02 still rounds to the same tenth, so the bill stays at eighteen per cent. On eleven it falls from thirty-four per cent to twenty, on thirteen from twenty-one to ten, and on sixteen it goes below the widths as asked for the first time, to minus four per cent. On twenty-five, the break-even grid for this latitude, the design saves nine per cent of its paper — more than the expectation would say, because three of the five widths less 0.02 — 0.12, 0.28 and 0.04 — land exactly on twenty-fifths. And on twenty-six it costs four per cent again.

Read across grids rather than at one, the design does what the expectation says. Over the twenty-five grids from sixteen to forty, rounded up, it costs 14.1 per cent more than its widths as asked, on average; with the latitude it costs 0.6 per cent less. The difference, 14.7 points, is almost exactly what the latitude is worth with no rounding at all — five strips each coming in 0.02 short, with the crossing term shrinking with them, 14.6 per cent of the features. The midline moves by the latitude, through the crossing term, and nothing else about the saw changes.

The latitude moves the saw down and keeps its teeth. The teeth come from the widths’ positions against the spacing, and a latitude moves every width by the same amount, so the positions that were bad stay bad, shifted. What latitude removes, for this design and in general, is the necessity of paying: past about sixteen divisions the five features cost less than their widths as asked on most grids, where without latitude they never did.

A procedure, in the order it is used #

Put together, the three results give a designer an order of decisions that does not need the features in advance.

First, the latitude. Before the grid, decide how far short a grafted feature may come: a scale that must read as a scale, a gap that must clear a flap. That number sets the break-even grid, $1/(2\delta)$, and any grid at least that fine will, on average, graft its features for no more paper than their widths as asked. A design with no latitude at all has no break-even grid, and every grid charges half a spacing a strip whatever its divisors.

Second, the kind of width. If the features will be measured from the subject, any grid of the chosen fineness is as good as any other, and the one that suits the flaps should be taken. If they will be derived from the design — halves and thirds of existing flaps — the grid should be a common multiple of the denominators they will come in, and a grid of twenty-four or forty-eight is worth more than the sixteen or thirty-two a designer doubling the sheet would reach first.

Third, the lines. None of this helps a feature that has nowhere to go: a strip can be slid in only along a line every crossed crease meets square, and a fine grid with its lines spent on diagonals has no room for any latitude to be used on.

The procedure is the grafting analogue of what rounding in the cheap direction found for a tree’s limbs: a rounding with room to move is a choice, and a choice spent where it is priced highest is worth more than the same choice spent evenly. There the room was which way to round a limb; here it is how far a feature may fall short.

What the averages assume #

The distributions are stated, not observed. Widths spread evenly over a range, and widths that are fractions in lowest terms up to a denominator, are two models of what a designer asks for; nobody has published the widths grafted into real box-pleated designs, and the expected bills are only as good as the model that produced them. The evenly spread model is the one the half-spacing result needs, and it is exact only when the range is a whole number of spacings long — which is why the coarsest grids wobble.

Latitude is modelled as a hard interval. A feature between $w - \delta$ and anything above is accepted, and one below is not; a real feature’s acceptability falls off rather than stopping, and a designer might accept a feature a little short to save a great deal of grid. Rounding up is always allowed, as in the earlier essays: a feature larger than asked is a larger feature, not a wrong one.

Everything is one sheet of side one. The features and their latitudes are fractions of the sheet, and the rounding costs paper the sheet has to grow by, charged against the sheet’s own sides and against the other family through the product, exactly as the first of these essays set up.

An average over a sawtooth #

The unexpected connection is between the number theory of the earlier essay and the averages here. The sawtooth was number theory: which grids hold which fractions, decided by common factors. Averaged over widths nobody chose, the number theory vanishes, because every tooth of a sawtooth has the same mean, and what remains is the size of the spacing and nothing else. The divisors come back only when the widths themselves are number-theoretic — fractions of the sheet — and then they come back as exactly the discount a common factor would predict.

The width is charged in grid found that a width nearly right is the expensive mistake, because it divides the grid most. The expectation says the same thing from the paper’s side: a designer can do nothing about a width’s position against a grid it was not chosen for, and should expect to pay the midline. The only lever that moves the midline is the latitude, and it moves it by exactly its own size.

Still open: the latitude used where it pays most #

A latitude is used best where the crossing term is largest. The saving from coming in short on a strip across is one plus the down family’s total a unit, and on a strip down it is one plus the across family’s total, so a designer with a fixed tolerance to spend would spend it on the family facing the larger total. How much that beats spending it evenly, and whether it interacts with the grid — a latitude spent on one family might put that family on a coarser grid than the other — is a small optimisation with the formula already in hand.

Real widths would settle the model. Measuring the grafted features in a handful of published box-pleated designs would say whether they behave like measurements or like fractions of the sheet, and therefore whether the powers of two designers use are costing them anything. A design that keeps its lines clear already asked for those files for a different count.

Sideways from here, a price holds until the arrangement moves asked the same question of a tree’s limbs, where a latitude on a limb’s length trades against the rest of the tree; the grafting version is simpler because its prices are linear in each family, and it would be worth knowing whether the tree’s version also moves its whole bill down by the latitude and keeps its teeth.

The habit worth carrying is about sawtooth costs. Before choosing a setting because it sits low on a sawtooth, ask whether the sawtooth belongs to the setting or to the thing being measured. The grid’s saw belonged to five particular widths; for widths not yet chosen there is no saw, only a line, and the one thing that moves the line is how much the widths are allowed to give.