The split that is worst on a rectangle

A grid chosen before the features let every grafted strip come in short by its own latitude and found the whole bill moved down by that latitude, through the crossing term, with the sawtooth’s teeth left where they were. It ended on a question it thought was small. A unit of shortfall on a strip across saves one plus the down family’s total, and a unit on a strip down saves 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. The prices are correct. The conclusion holds on one kind of sheet at one limit of the grid, and on the other kind of sheet it names exactly the wrong family.

The tolerance has to be shared for any of this to be a question. If each feature carries a latitude of its own, nothing competes: on a sheet that may be any shape, every unit of every latitude saves paper, and all of them should be used. A shared allowance is a different thing, and it is what a design often has. A row of scales must still fill the body they sit on, so they may come in short between them by only so much. A model’s proportions drift by the total of what its features give up, and a folder who will accept a two per cent change in the length of a body has made one allowance, not several. So the setting here is a budget $\Delta$ of shortfall, shared across all the strips, on the five features the earlier essays used: 0.08, 0.14 and 0.2 across, 0.3 and 0.06 down. Those widths are chosen for illustration and belong to no published design.

A product with a fixed sum #

The first of these essays priced a strip as its width times the cut, and the second found the rectangle where two strips cross; together the sheet grows to a rectangle of sides $X = 1 + A$ and $Y = 1 + B$, where $A$ and $B$ are the families’ totals. Spending $a$ of the budget across and $b$ down, with $a + b = \Delta$, leaves sides whose sum is the same whatever the split:

$$X' + Y' = 2 + A + B - \Delta.$$

So the choice is the choice of a rectangle with a set perimeter, and the second book of Euclid’s Elements settled that one. Its fifth proposition is the identity

$$X' Y' = \left(\frac{X' + Y'}{2}\right)^2 - \left(\frac{X' - Y'}{2}\right)^2,$$

a level square less the square of half the gap. The area is largest when the sides are level and falls as they part. A designer trying to save paper therefore wants the two sides as unequal as the budget can make them, which means taking everything from the side that is already shorter. That agrees with the earlier essay’s prices: the shorter side’s strips face the larger total. It also says something the prices did not. Between the two extremes the bill is not a compromise. It rises to a peak and comes down again.

With a budget of 0.1 the split that levels the two sides puts 80% of it across and is the dearest split available, dearer even than spending all of it on the long side. A designer who evens things out from a sense of fairness, or who aims at the longer family because it looks like the place with the most to give, lands near the top of the arch.

The vertical axis of that drawing has been stretched until the arch shows, and it spans less than a hundredth of the original sheet. The whole budget saves 0.142 at best and 0.1356 at worst. Which family gets it moves the answer by under five per cent of what is being saved, and at a budget of 0.04 by about the same.

What aiming is worth #

The gain from aiming has a closed form, and the form says when it matters. Against an even split, putting all of $\Delta$ on the shorter side saves an extra

$$\frac{\Delta\,(Y - X)}{2} + \frac{\Delta^2}{4},$$

with $Y$ the longer side and $X$ the shorter, and an even split itself saves about $\Delta (X + Y)/2$. So the gain, as a share of the even saving, is close to the gap divided by the sum of the sides, whatever the budget.

The five features have a gap of 0.06 between sides of 1.42 and 1.36, and aiming adds 2.9 per cent to a saving of 0.055. A design has to be strongly one-sided before this is worth a thought: with all 0.8 of the families’ width split as 0.7 and 0.1, the gap of 0.6 makes aiming worth a little over a fifth. Most of the difference between the curves is at a gap of nothing, where the Euclid identity is exactly flat to first order and only the $\Delta^2/4$ remains.

So on this kind of sheet the choice of family is a real one and a small one. The reason it is small is in the identity itself: the gap enters squared. The five features’ gap of 0.06 takes 0.0009 off the level sheet, about one part in a thousand of the bill.

The grid eats the rule #

All of that treated shortfall as continuous, and a box-pleated design does not have continuous widths. A spacing costs one plus the other side put each strip on a whole number of spacings, and on a grid a budget is spent in lumps. Dropping a strip by one spacing saves a whole spacing of paper, priced by the crossing term, but it uses only as much budget as takes the strip below its asked width. A width just above a grid line gives up a spacing for almost no budget; a width just below one, for nearly a whole spacing of it.

So a strip’s price is one factor, the family’s crossing term, and how much budget it needs is another, the width’s position inside its spacing. On the five features the first varies by the ratio 1.42 to 1.36, four per cent. The second varies by as much as the positions do, often several times over. Trying every set of whole spacings on every grid shows which one wins.

The continuous rule would have every bar point down. It does on seven of the forty-five grids, and on seventeen the cheapest arrangement takes more from the longer family than the shorter. On eighteen divisions the search takes all of it from the strips down, and on nineteen and twenty most of it from the strips across, and nothing about the design has changed between them except where the lines fall. The bars have the same teeth the earlier essay’s saw had, because they come from the same place.

One design proves little, so the same measurement runs over four hundred designs drawn at random, three strips across and two down with widths from 0.02 to 0.3 of the sheet, all sharing a budget of 0.04. Three rules are compared with the best found by trying everything: spend only on the shorter family, spend only on the longer, and ignore the families altogether, always dropping whichever strip can give up a spacing for the least budget.

The rule that ignores the families is nearly perfect, and the rule the prices recommend starts at half. On eight divisions, where the budget is a third of a spacing, aiming at the shorter family keeps 49 per cent of the best saving, less than aiming at the longer family’s 67, because the longer family usually holds more strips, and wider ones: a strip already down to a single spacing has nothing to give, and on a coarse grid that describes much of the shorter family. The family-blind rule keeps 99.

The continuous answer does return. By the time the budget is several spacings, the lumps are small against it and the two family rules start to separate in the direction the prices say: at 3.84 spacings the shorter family keeps 81 per cent and the longer 74, and the family-blind rule has slipped to 96 as the crossing term starts to matter. At the 128 divisions where the drawing stops, the budget is five spacings and blind choice still leads at 94 against 84. On the grids anybody box-pleats on, then, the useful procedure asks nothing about totals. It lists each strip’s distance above the next line down and takes the short ones first. Rounding in the cheap direction ranked a tree’s limbs by their prices instead, and the two are not in conflict: a ranking by price wins where the prices differ by more than the positions do, and on these two families the prices differ by four per cent.

On a square sheet the same split is the best #

All of the above assumed the grown sheet could be any rectangle, and paper does not always come that way. A designer starting from a square, or wanting a model whose proportions stay square, pays for the longer side twice:

$$\text{paper} = \max(X', Y')^2 - 1.$$

In terms of the same two numbers, the half-sum $p = (X' + Y')/2$, which the split cannot change, and the half-gap $h = |X' - Y'|/2$, the two bills are

$$\text{rectangle: } p^2 - h^2 - 1, \qquad \text{square: } (p + h)^2 - 1.$$

Each is the level sheet moved by what is left of the gap, and they move it in opposite directions. On a rectangle a remaining gap is a discount and on a square it is a surcharge. So the split that levels the sides, the worst on a rectangle, is the best on a square, by the same identity read the other way. The level sheet is one sheet, priced the same on both, and every other split moves away from it, down on one and up on the other.

The difference in scale is the point of the drawing. A square is priced to first order in the gap, and a rectangle to second. The five features as asked have that gap of 0.06, and it takes 0.0009 off the level sheet if the paper may be a rectangle and adds 0.0843 if it must be square, ninety-four times as much. At a budget of 0.1, spending it all on the shorter side saves 0.142 on a rectangle, the best available, and on a square saves nothing at all; the levelling split saves 0.1356 on the rectangle, the least available, and 0.2208 on the square, the most.

That also changes what a latitude of its own is worth to each strip, where nothing was shared and on a rectangle no decision arose. With 0.02 on each of the five strips, all of it used, the bill on a rectangle falls from 0.9312 to 0.7952, and each strip contributes. On a square it falls from 1.0164 to 0.8496, and the two strips down contribute nothing: three strips across coming in short bring the long side to 1.36, level with the down family as asked, and any further shortening of the strips down only widens a margin the square has already been cut to. The strips across cannot give more. So two fifths of the design’s latitude is useless, on a design nobody would describe as one-sided.

Rules on a square, and the free side #

The grid has much less to say here, and that is a finding of its own, because it is the reverse of what happened on the rectangle.

On a rectangle the strip mattered and the family hardly did. On a square the family is nearly all of the decision: 91 to 96 per cent of the best saving lies in one family, a rule that spends on the other keeps at most 4, and the family-blind procedure that led everywhere on the rectangle falls to about three quarters. The residue the longer family misses is what the budget can do on a design close enough to level that spending past the gap pays.

The same reasoning runs in the other direction and gives a square sheet something for nothing. If the shorter family’s strips only ever widen a margin, then rounding them up costs nothing until they overtake.

On a rectangle, rounding the strips down to whole spacings costs paper on every one of the thirty-seven grids drawn. On a square it is free on twenty-nine. From twenty-two divisions up it is free on every grid drawn. From thirty-four up that is guaranteed, since two strips round up by less than two spacings between them and two spacings are then smaller than the gap; between those it is the positions that happen not to overtake. The shorter family can have any grid it likes at no charge, which matters because the width is charged in grid found a strip’s rounding to be the expensive thing about it: a square sheet’s margin is the place where an awkward width can be made a whole one, and the only thing it uses up is a slack that was doing nothing else.

The order of decisions, revised #

The earlier essay’s procedure started from the latitude and went on to the kind of width and the free lines. This argument puts one question in front of all of those: what shape may the finished sheet be?

If the sheet may be any rectangle, the shared budget hardly cares which family it comes from. It should go, strip by strip, to whichever width sits closest above a line, and only on a grid fine enough that the budget is many spacings does the shorter family earn a preference, worth a few per cent. If the sheet must be square, the budget goes to the longer family until the sides are level, and then to both. Everything on the shorter side is then free in the opposite sense, able to round up to the grid until it catches up.

The two procedures give different answers on the same five features. On a square sheet, 0.04 spent on the strips down saves nothing, and spent on the strips across it saves 0.112. On a rectangle those same two choices save 0.0568 and 0.0544, and a reader comparing them would be right to shrug.

None of this helps a feature with nowhere to go: a graft needs a square line, and a design that keeps its lines clear is what decides whether a strip can be slid in at all.

What the sheet is assumed to be #

A square sheet is modelled as a square cut to fit the longer side, with the margin beside the shorter side counted as paper paid for and unused. A designer could put that margin to work, and then it is not slack: a margin holding a second flap has a price of its own, and the free rounding above stops being free. A rectangle is modelled as available at any proportion, which holds for trimmed sheets and for anything cut from a roll and does not hold for a designer who was handed kami.

The budget is a hard total over absolute shortfall. Rounding a strip up does not refund any of it, though a design whose features trade against one another might well let it. A real allowance also falls off gradually: no folder rejects a body at 2.01 per cent short and accepts one at 1.99.

The random designs are stated, not observed. Three strips across and two down, widths evenly spread from 0.02 to 0.3, is the same model the earlier essay used and has no more evidence behind it now; a design whose family totals sit far apart would give aiming more to do, as the second figure shows. The picture cannot show the crease pattern: each of these figures is a price, drawn on the assumption that the strips can be inserted at all, and whether any given design has the square lines to take them is a separate count.

Two theorems in one identity #

The unexpected thing is how much of this was already in a proposition older than paper. Euclid’s identity was written for line segments, and it says an area with a fixed perimeter is a level square less a square of half the difference. Everything measured on the rectangle above is that identity, and the small size of the whole question comes from the single fact that the difference is squared.

The square sheet is the same two quantities in a different function. Replace the product of the sides with the larger side squared and the difference appears to the first power, and with that one change the choice grows ninety-four times as large and points the other way. Two designers can agree about every price per unit and still give opposite advice, because a price per unit is a derivative and says nothing about which sheet it is the derivative of.

Still open: a tree’s limbs, and a margin that earns #

A price holds until the arrangement moves priced a latitude on a tree’s limbs, and its bill is not a product of two sums, so it has no such identity. Whether its version has an arch at all, a split that is locally the dearest and that a designer’s instinct for even treatment would land on, is a direct computation on the trees already built.

The other lead is the margin itself. A square sheet’s slack is paper in the shape of a strip along one side, and the method of placing a flap on an edge says an edge is where a flap is cheapest. If a grafted design’s margin can hold a flap it would otherwise have paid for, then the square sheet’s surcharge is not all waste, and the true comparison between the two sheets is smaller than ninety-four to one by an amount that depends on what the margin can carry.

The habit worth carrying is about prices. A price per unit is a derivative, and before spending by it, ask what it is the derivative of. Each strip’s price was right on the sheet it was computed for, and the sheet was the one assumption that nobody had written down.