Six rectangles and one term

Paying in paper prices one graft: cut the crease pattern along a line every crossed crease meets square, slide in a strip, and the bill is the strip’s width times the length of the cut. One term, and no second one.

The second term adds a strip across the first and finds a second term after all — the rectangle where the two strips overlap, paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.

That essay ends by saying the obvious continuation is $k$ grafts, that two families of parallel strips ought to give a sheet whose gain is the sum of the two families’ bills plus the product of their total widths — one crossing term, not $pq$ of them — and that whether it is right is an arithmetic somebody should do rather than assume.

This is that arithmetic, done by measuring rather than by substituting.

Why it is not obvious #

The rectangles are real and there are six of them.

Three vertical strips and two horizontal ones overlap in six places, each of them a rectangle of a vertical strip’s width by a horizontal strip’s width. Each is paper that two features are both charged for and neither uses, in exactly the sense the second of these essays established for the single case. So a budget that says “add up the crossings” is describing the object correctly.

What it gets wrong is the arithmetic. Let the vertical strips have widths $w_1, \dots, w_p$ and the horizontal ones $u_1, \dots, u_q$. The rectangles have areas $w_i u_j$ for every pair, and their sum is

$$\sum_{i}\sum_{j} w_i u_j = \left(\sum_i w_i\right)\left(\sum_j u_j\right)$$

which is a product of two sums rather than a sum of $pq$ products. The double sum factorises, and a designer who wants the number does not have to enumerate the crossings; they need two totals and one multiplication.

That is not a deep identity. It is the distributive law, and it is easy to see once written. What makes it worth a measurement is that the identity is being claimed about paper, and the paper’s behaviour is what the first two of these essays were about establishing.

The measurement #

Nothing below substitutes into the formula.

Each strip goes in on a line the pattern admits at the time — which is not the same as a line the original pattern admits, because a graft moves every coordinate past its own cut, so the second vertical strip’s cut is found on a pattern one strip wider. Each graft is then checked to have added exactly its own width times its own cut, by the same test the first of these essays used: the paper is summed over the panels the creases enclose, by a method that knows nothing about the cut.

The last two columns are the same number at every arrangement, including one the table does not have room for: three strips of 0.08, 0.14 and 0.20 crossing two of 0.30 and 0.06, where the six rectangles have six different areas, the sheet gains 0.9312 against 0.7800 charged separately, and the excess of 0.1512 is $0.42 \times 0.36$ to the last digit.

The unequal case is the one that does the work. A check run only on strips of equal width would pass on a formula with a $p q w^2$ in it just as happily as on the product of the totals, and the two are the same number when every width is $w$. Unequal widths and unequal counts are what tell them apart.

The row with a zero in it #

The last row of the table is three strips in one direction and none in the other, and its excess is zero.

That is the sentence the whole result turns on, and it is easy to read past. Three strips crossing nothing produce no rectangles, so they cost exactly their separate bills — and the separate bills, for three parallel strips, are simply additive. The crossing term exists because there are two families, not because there are many strips.

So the cost structure has a shape a designer can hold. Adding features along one axis is linear: each one costs its width times the sheet’s height, and adding a tenth costs no more than the first did. Adding features along both axes is linear plus one product, and the product depends on the totals rather than on how the totals are divided up.

Which means a design with many small features in both directions is charged the same crossing term as a design with two large ones of the same total width. Twelve strips of a tenth crossing twelve strips of a tenth cost the same rectangle — $1.2 \times 1.2$ — as one strip of 1.2 crossing one of 1.2. How the paper is spent does not matter; only how much.

The share rises, and it is not the count that raises it #

Read the last column of the table as a share of everything spent and it runs 5.7, 7.4, 10.7, 12.6, 17.1 per cent, rising steadily as strips are added. That looks like a penalty for having many features, and it is not.

The share is $WU/(W+U+WU)$, which depends only on the totals. Strips of 0.12 give totals of 0.12, 0.24, 0.24, 0.36, 0.48 in the first family across those five rows, and the share follows the totals exactly. A design that added its features as four strips of 0.03 rather than one of 0.12 would sit on the first row, not the fourth.

So the cost of many features is the cost of much paper, and nothing else. A designer told that crossing grafts are expensive will naturally try to use fewer of them, and that is the wrong economy: the saving is in the total width, which is to say in how large the features are, and splitting a feature into several changes nothing at all.

Halving every strip halves both totals and so quarters the excess, at every arrangement, with the number of rectangles unchanged. That is the cleanest statement of the identity available: the column that halves is the width and the column that does not move is the count.

What this does to the corner #

The single-graft picture is worth re-reading in the light of the totals.

The second term reports the crossing rectangle as nine per cent of the bill at a strip a fifth wide and nearly a third at four fifths. Those are shares of a two-strip bill, and with families they behave differently.

For $p$ strips of total width $W$ crossing $q$ strips of total width $U$, on a unit sheet, the gain is $W + U + WU$ and the excess is $WU$. So the excess as a share of everything spent is

$$\frac{WU}{W + U + WU}$$

which depends only on the two totals — and for equal totals $W = U = t$ it is $t/(2+t)$, rising from nothing at a bare sheet towards a third as the totals approach the sheet’s own size. At totals of a fifth each it is nine per cent; at four fifths each it is twenty-nine.

Those are exactly the numbers the two-strip case gave, which is the identity working: a two-strip sheet with widths $t$ and $t$ and a twenty-strip sheet with totals $t$ and $t$ are charged identically. So the second of these essays’ figures were never about two strips; they were about two totals, and nobody had a reason to know that until the family case was computed.

Where the strips actually go #

The arithmetic holds wherever the strips can go, and where they can go is these essays’ other half.

A four-by-four grid has three interior gaps each way, so it takes at most three strips in each direction before the cuts start landing on lines a previous strip has already used. The four-by-three arrangement in the table is therefore already at the edge of what the base admits, and the cut finder is choosing the nearest admissible line to each target fraction rather than an arbitrary one.

That matters for reading the table honestly. The arrangements are not arbitrary counts of strips; they are counts the pattern can actually accommodate, and a pattern with fewer admissible lines caps the design before the arithmetic has anything to say. The formula prices what a design can do and the census says how much of it a design can do at all, and the second is the binding constraint on every printed pattern measured.

Where the sheet’s own shape comes in #

One thing the family arithmetic does change is how quickly the sheet stops being the sheet.

A single graft of a fifth widens a square by a fifth and leaves something close to a square. Four grafts of a fifth in one direction and none in the other widen it by four fifths in that direction alone, and what comes out is a rectangle nearly two to one. So a design gathering its features on one axis is not only paying a linear bill — it is changing the proportion of the paper it wants, and a construction assumes its sheet in ways that do not survive that.

That is a cost with no term in the formula. The arithmetic here is exact and it is an arithmetic about area, and area is one of several things a designer is spending. How much of the sheet the flaps can claim is another; where a flap stands is a third, and it is the one grafting most nearly touches, because a strip pushes everything on one side of it outward.

A budget a designer can carry #

Put the three terms together and the whole of grafting’s economics is four numbers and one multiplication.

Take the sheet’s own width and height. Add up the widths of every strip to be slid in one way, and every strip the other. The paper required is the product of the two sums with the two sheet dimensions — nothing else enters, and in particular neither the number of features nor where any of them goes.

Where a strip goes is free. The cut finder in the measurement above places each strip at an admissible line near a target fraction, and moving those targets changes the resulting pattern completely and changes the bill by nothing. That is worth saying because it is the one part of this a designer would not guess: a feature grafted near the edge of a base costs the same as one grafted through the middle, even though a flap in a corner is worth four times one in the middle once it is there.

So grafting has a curious shape as a design operation. It is completely insensitive to position and completely sensitive to total width, where almost everything else in this field is the other way round. Circle packing prices a flap by where it sits and hardly at all by what it is for; grafting prices a feature by how wide it is and not at all by where it goes.

What the measurement does not show #

The strips go in on lines the pattern admits, and a real design may not have enough of them. A graft needs a square line finds four of eight printed patterns taking no strip in either direction, and the arithmetic here is run on a grid, which admits every gap. A design with three admissible lines cannot take four strips at any price.

Every strip is a full-width strip. A strip that stops part-way across is a different operation — it creates a vertex where it stops — and nothing here bears on it.

The widths are free and they are not. A strip whose width is not a whole multiple of the grid’s spacing leaves a design with no grid, which is a price the area arithmetic has no term for at all.

And the grid is not a design. An orthogonal grid is the simplest pattern that admits grafts everywhere, which is what makes it the right object for isolating the arithmetic and the wrong object for saying what a design costs. A box-pleated base has flaps, rivers and a packing, and every one of them is a constraint on where a strip may go — designing on a grid is what that constraint looks like from the inside, and it is the reason the census matters more than the formula in practice.

One graft, read again #

With the family arithmetic in hand, the first of these essays’ result reads differently.

Paying in paper says a feature costs exactly the paper inserted for it, with no second term, and the emphasis in that sentence has always been on no second term — the surprise being that a design operation could be priced so simply. What the families show is that the simplicity is the distributive law and not a fact about a single strip.

Write the whole thing at once. A sheet $W_0$ by $H_0$ receiving total widths $W$ and $U$ becomes $(W_0 + W)(H_0 + U)$, so the gain is

$$W_0 U + H_0 W + W U$$

and every result in these essays is a reading of those three terms. The first of these essays sets $U = 0$ and finds one term. The second sets $p = q = 1$ and finds the third term. This one lets $p$ and $q$ be anything and finds that the third term did not notice.

There was never a sequence of increasingly complicated bills. There is one product of two binomials, and the essays here have been expanding it a term at a time — which is worth saying plainly because the next essay is about a cost that is genuinely not in it.

Still open: the third direction #

Two families is not the general case, and the general case is a question about the sheet rather than about the arithmetic.

Strips at forty-five degrees are a third family, and a box-pleated design has diagonals through most of its squares, so a diagonal strip would be the natural way to widen a design along its own diagonals. Whether a diagonal cut can be square to every crease it meets is the question the census asks of horizontal and vertical lines and has never been asked of a diagonal one — and if such cuts exist, the crossing term becomes three products rather than one, with a geometry in it: two strips at forty-five degrees cross in a parallelogram whose area is not the product of the widths.

And the totals are the budget, so the design question is what they buy. The arithmetic says $W$ and $U$ are the only numbers that matter and says nothing about how a designer should split them. A design whose features are all in one direction pays no crossing term and distorts the sheet; a design that balances them pays the term and keeps its proportion. The cheapest split is not the balanced one and it is not the extreme one, and finding it is a one-variable minimisation over a criterion nobody has written down — which would have to price the proportion of the sheet as well as its area.

The habit worth carrying is about costs that come in pairs. A term that appears when two things interact is not a term per pair of things. The second of these essays found one rectangle and called it the second term, which invited a count of rectangles; the count is right and the sum is not a count, and the difference between them is the difference between a budget that grows with a product of counts and one that grows with a product of totals.