The grid belongs to the subject

Rounding in the cheap direction treated each rounding of a subject onto a box-pleating grid as a small reallocation of length between its limbs, priced by what each limb costs the arrangement. Rounding the bird’s dearer edges down and its cheaper ones up gave the largest model of every rounding tried, on grids of four, six and eight units — 5.7 per cent larger than the nearest rounding on eight — and paid for it in proportion: the body came out a fifth short.

It ended on the obvious next thing. A rounding that weighs both — gain in size from the prices, loss in shape from the fractions — is a problem with two objectives, and its answer is a frontier: for each amount of shape error a designer will accept, the largest model available. The nearest rounding and the cheap-direction rounding are two points on or near it. On six units, the essay suspected, neither may be the point anybody wants.

Drawing the frontier across every grid at once changes what the question is about. It stops being a question about which rounding to choose on a grid and becomes a question about which grid, and the answer depends on the subject rather than on the grid.

Every rounding, placed twice #

The bird’s tree has a body of 0.5, wings of 1.6, a head of 0.9, legs of 0.8 and a tail of 1.3. On a grid of $g$ units across the wings each limb lies between two whole numbers of units, and a rounding picks one of the two for each of the five limbs — at most thirty-two roundings a grid, fewer where a limb is already whole. Over grids of three, four, five, six, seven, eight, nine, ten, eleven, twelve, fourteen and sixteen units that is 121 different roundings.

Each is placed by two numbers. Its shape error is the worst limb’s proportional error once the rounded units are scaled to match the true proportions as closely as possible. Its size is the measure the earlier essay settled on: the scale the rounded tree’s arrangement supports on the sheet, times the sheet length one unit of the subject’s own length receives — so rounding every limb up together changes nothing, and rounding only the cheap ones up gains. The unrounded bird has size 0.2651.

Nine of the 121 are on the frontier. Every other rounding is beaten by at least one of the nine: some rounding, on some grid, is at least as large with no more error. That is a strong filter. It says that for the bird, whatever a designer’s tolerance for distorted proportions, the choice is among nine roundings, and a designer who picks any other has picked a model that is smaller or less faithful than one on offer.

The nine, and where they come from #

The frontier starts at zero error, on sixteen units. The bird’s limbs are all tenths, and on a grid of sixteen units across the wings a unit is exactly a tenth: 5, 16, 9, 8, 13 spells the bird with no error at all. Its size is the unrounded bird’s, 0.2651, as it has to be — a rounding with no error is the subject. A designer who will accept no distortion has one choice, and it is sixteen units.

From there each step along the frontier trades shape for size. Nine units, 3, 9, 5, 5, 8, buys 1.4 per cent at 6.9 per cent error. Ten units supplies three frontier points between 7.5 and 12.3 per cent error, the largest 4.9 per cent bigger than the bird. Six units’ nearest rounding sits at 10.7 per cent and 2.7 per cent bigger, fourteen units’ at 10.8 per cent and 4.3 bigger, seven units at 21.0 per cent and 6.3 bigger, and five units ends the frontier at 30.4 per cent error and 8.4 per cent bigger. On average along the frontier, a point of worst-limb error buys 0.28 per cent of size.

Eight, twelve, eleven, four and three units contribute nothing. Every rounding on each of them is beaten on both counts by a rounding on another grid. That includes the rounding the earlier essay found best on eight units: 2, 8, 4, 4, 7, at 19.9 per cent error and size 0.2782, is matched — to two thousandths of a per cent, and on the larger side — by 3, 10, 5, 5, 9 on ten units, which is only 12.3 per cent out. The cheap-direction rounding was the largest on its grid, and its grid was the wrong one.

Two roundings that tie, read limb by limb #

The tie between eight and ten units is worth reading in the subject’s own measure, because it shows where a rounding’s size comes from. On eight units a unit is 0.2 of the bird’s length, and 2, 8, 4, 4, 7 gives a body of 0.4, wings of 1.6, a head of 0.8, legs of 0.8 and a tail of 1.4. On ten units a unit is 0.16, and 3, 10, 5, 5, 9 gives a body of 0.48, wings of 1.6, a head of 0.8, legs of 0.8 and a tail of 1.44.

The two agree exactly on the wings, the head and the legs. Both take a tenth off the head, which the price of a limb is not its length found to be the bird’s dearest edge — three times what a wing costs per unit — and both leave the legs and wings as drawn. That shared move, on the dearest edge, is the obvious source of the 4.9 per cent the two share. Where they differ is the body and the tail. Eight units can only give the body 0.4 or 0.6, and the cheap direction chose 0.4, a fifth short of the true 0.5; ten units can give it 0.48, four per cent short. Eight units’ tail is 1.4 and ten units’ 1.44, both long, and the tail is nearly free.

So the eight-unit rounding pays twenty per cent of error for a body it did not need to shorten, and the frontier shows it bought nothing measurable with it: the ten-unit rounding keeps the body nearly true and is, if anything, a hair larger. The frontier found what the prices alone could not say — that the dear move was available on another grid without the cheap move’s side effect. A price list says which direction each limb should round; only the grid decides how far a round step is, and a step of 0.2 on a limb of 0.5 is forty per cent of it.

A grid is measured against the frontier #

The frontier gives every grid a score that the earlier comparisons could not: how much a rounding on it gives up against the best available with no more distortion. The nearest rounding gives up nothing on sixteen and six units, a quarter of a per cent on twelve, and between 1.5 and 6 per cent on most of the rest — 5.4 on eight, 6.2 on nine — and 17 on three. On twelve units, the grid a designer who wanted more resolution than eight would reach for, the nearest rounding comes within a quarter of a per cent and still no rounding reaches the frontier.

This is the sense in which the nearest rounding is a coin toss. It is on the frontier on two grids of twelve for the bird, because rounding each limb by its fraction ignores both what the limb costs and what the other grids offer. And the cheap-direction rounding, which fixes the first of those, can still be beaten from another grid, because it does not know the second.

Exactness is a corner, not a peak #

The frontier’s first point deserves a second look, because it is the only point a designer can reach without trading anything. A subject has an exact grid only if its limbs are whole multiples of a common length, and then the exact grid is the one whose unit is that length or a divisor of it, counted across the longest limb. The bird’s limbs are all tenths, so its exact grids are sixteen, thirty-two and every multiple of sixteen; the lizard’s are fifths, so seven, fourteen and every multiple of seven.

A subject measured from the world does not have one. A real bird’s proportions are ratios with no common unit, and on every grid its frontier starts somewhere above zero error — at whichever rounding comes closest, on whichever grid gives it. The bird here is exact because its lengths were chosen to be round numbers, and what the condition does not decide is a reminder of how much of a tree’s arrangement already follows from such choices. The frontier for a measured subject is the same staircase with its first step lifted off the axis. Designing on a grid makes the case for box pleating as the price of creases that land where they should; the lifted first step is part of that price, paid before any crease is drawn.

That is also why the exact point is a corner and not a peak. Moving away from it along the frontier always buys size — that is what makes the other points worth listing — and the roundings that buy it are the ones that shorten a dear limb, which a subject drawn in its own proportions leaves at its true length. Exactness is where the shape is best and the size is worst, and the frontier is the list of every price at which the two can be exchanged.

The same question for the lizard #

A frontier drawn for one subject is a fact about that subject, and the obvious test is a second one.

The lizard has four groups of limbs and so at most sixteen roundings a grid, 82 in all. Its frontier holds thirteen, from eight grids: four, five, seven, eight, nine, eleven, twelve and sixteen units. Six, ten and fourteen contribute nothing.

The lists nearly swap. Eight and twelve, which contribute nothing for the bird, contribute three frontier points and one for the lizard; ten and six, which carry four of the bird’s nine, contribute nothing for the lizard. Only five, seven, nine and sixteen serve both. The lizard’s exact grid is seven — its limbs are all multiples of 0.2 and the tail is seven of them — and fourteen spells it exactly too, but seven gets there first with the same model. And the lizard buys size faster than the bird along its frontier: 0.40 per cent a point of error against the bird’s 0.28.

No grid is good or bad in general. A grid is good for a subject when its unit divides the subject’s limbs nearly, and in the right directions — and which directions are right is a fact about the subject’s arrangement, not about the grid. The designer’s grid is the dearest thing here and a grid chosen before the features both asked what a grid costs in general; the frontier says the question has an answer only once the subject is known, and then a different answer for each subject.

What rounding in the cheap direction was #

Set against the frontier, the earlier rule reads as the right-hand end of each grid’s own frontier — the largest model that grid can make — and nothing more. On ten units the rule’s rounding is also the right-hand frontier point for that part of the error range. On eight and six it is not: the largest model on eight is beaten from ten, and the largest on six, which the earlier essay called “a trade not worth taking” at 45 per cent error, is beaten from seven by a rounding at 21 per cent that is larger still. The instinct was right and the verdict now has a number behind it.

A tolerance chooses a rounding #

The price of a limb is not its length raised the case the frontier answers most directly: a subject whose proportions have latitude. If a bird’s head may be anywhere between 0.7 and 1.1, the designer has a tolerance on proportion, and a tolerance is an upright line on the plot. The rounding to fold is the frontier point just to its left — the largest model within the tolerance — and it can be read off without any further calculation. At a tolerance of twelve per cent that is fourteen units, 4, 14, 7, 7, 12, 4.3 per cent larger than the bird; at eight per cent it is ten units, 3, 10, 6, 5, 9; at five per cent it is sixteen units and the exact bird. Loosen the tolerance past 12.3 per cent and ten units takes the lead again with 3, 10, 5, 5, 9.

That is a different procedure from the one spelling a tree on a grid set out, which chose the grid first and the rounding second. Here the tolerance chooses the rounding, and the rounding chooses the grid. A designer who fixes a grid first has already given up whatever the frontier’s other grids offered, and for the bird fixing eight gives up all of it.

What the frontier cannot show #

The arrangement is re-solved for every rounding, and every solve is a search. The bird’s best arrangement at each set of lengths comes from sixty restarts of a packing search; a solve that fell short of the true optimum would put a rounding below where it belongs and could push it off the frontier. The frontier’s points are separated by fractions of a per cent in size, so a single poor solve could change which grids appear on it. The measurement is repeatable — the same solves give the same answers — but not certified.

Size is one number and shape error is one number. The worst limb’s error ignores the other four; a rounding that distorts one limb by ten per cent and one that distorts all five by nine count alike. A designer who cares about the head and not the tail would weigh them differently and draw a different frontier. The one here is the frontier of the measure the earlier essays used, not of every measure a designer might.

And the subjects are trees, not models. A uniaxial base is the paper arrangement a tree method produces before any shaping; a grid rounding that gives a larger base says nothing about whether the finished model looks like a bird, and every flap on one axis is the reminder of how much that method already assumes.

How the roundings were placed #

On each grid every limb is rounded both down and up from its exact number of units, never below one, and every combination is kept once. Each rounding is scaled to the subject by the least-squares ratio of units to true lengths; its shape error is the largest relative difference; its size is the arrangement’s scale for the rounded lengths, times one unit, times that ratio. A rounding is on the frontier if no other, on any grid, has at least its size and no more error. Every rounding with no error at all is required to reproduce the unrounded tree’s size, which checks that the size measure and the arrangement agree.

Still open: grids that are not whole numbers across the wings #

Every grid here measures its unit from the longest limb. A grid of $g$ units puts the wings on exactly $g$, which is how box-pleating grids are usually described, but a designer can equally put the wings on $g$ and a half and take the rounding from there. That doubles the grids available, and whether the frontier then fills in — so that some grid reaches every tolerance with a rounding that gives up nothing — is a question with an obvious experiment behind it.

The two subjects disagree about which grids serve them, and a third would say whether there is any pattern in the disagreement. A tree whose limbs are all multiples of a common length has an exact grid, and the frontier’s first point is always there. What decides the rest of the frontier — which inexact grids come next — is presumably how the limbs’ fractional parts fall on each grid against their prices, and the insect of three segments, with its two internal edges, would test that.

Sideways from here, a price holds until the arrangement moves spent a fixed total of limb where it was cheapest; the frontier is the same economy with the total no longer fixed, and whether its slope — a quarter of a per cent a point for the bird — is predicted by the prices at the unrounded tree is a comparison that would say how far a price list reaches.

The habit worth carrying is about choosing in order. When a decision has two stages, choose the second stage’s options before fixing the first. Fixing the grid and then rounding well gave the best model on a grid that was never in the running; placing every rounding on every grid first made the grid a consequence of the tolerance.