The second term

Paying in paper is a result about arithmetic being simpler than expected. Cut a finished crease pattern along a line, slide a strip of paper into the gap, and everything works: every crease that meets the line continues across the strip, every crease that does not is carried along untouched, every flap keeps its length. The sheet grows by the strip’s width times the length of the cut, and — this is the part that is worth having measured rather than assumed — there is no second term.

Add a second feature across the first and there is.

The arithmetic, and why it is not obvious #

A sheet H tall and W wide, grafted with a vertical strip w₁ wide and then a horizontal strip w₂ wide, ends up (H + w₂) tall and (W + w₁) wide. So the paper it gained is

(H + w₂)(W + w₁) − HW = w₁H + w₂W + w₁w₂

and the first two terms are exactly the two bills the single-graft result predicts. The third is not predicted by anything, and it is the rectangle where the two strips overlap.

The reason it is easy to miss is that each graft is individually correct. The first strip really does cost its width times the cut. The second strip really does cost its width times its cut — but its cut is longer than it would have been, because the first strip made the sheet wider. Each bill is right; the sum of the bills is not the total.

There is a second way of seeing why the sum of the bills is the wrong total, and it is the one that makes the result feel inevitable rather than surprising. The single-graft rule says a graft costs width × cut. The cut’s length is a property of the sheet, so the rule is not really “a graft costs w·L” — it is “a graft costs w times whatever the sheet measures across the cut at the moment the graft is made”. Applying it twice with the original sheet’s measurements is applying it to a sheet that no longer exists.

Measured rather than substituted #

The formula above is three lines of algebra and it would be easy to publish it as the finding. It is not the finding. Every number in this essay is measured off the pattern that came out.

The paper is summed over the panels the creases enclose, by the folding machinery, which knows nothing about the cut or the strip. The crease lengths are collected as sorted multisets before and after, so the comparison never uses the fact that edge k of one pattern is edge k of the other. The local theorems are re-run on the grafted pattern from scratch. And the excess is the difference between what the sheet gained and what the two bills come to, computed independently of each other.

At a strip 0.3 wide in a unit sheet: the two bills come to 0.600, the sheet gained 0.690, the excess is 0.090, and 0.3 × 0.3 is 0.090. At 0.2 the numbers are 0.400, 0.440, 0.040. At 0.8 they are 1.600, 2.240, 0.640. At unequal widths — 0.25 and 0.6 — the excess is 0.150, which is their product and not the square of either.

How much of the bill it is #

The excess is second order in the widths and the two bills are first order, so the share it takes grows with how ambitious the features are.

At a tenth of the sheet’s width it is 4.8 per cent of the total. At a fifth, 9.1 per cent. At half, 20 per cent. At 1.2 — a feature that adds more than the sheet had — 37.5 per cent.

That is the practically useful shape. A designer adding two small details to a finished base is losing a few per cent to the crossing and can reasonably ignore it. A designer adding two substantial limbs is losing a fifth of what they paid, and the paper is not anywhere either limb can use it.

Where the paper goes #

It is worth being exact about what “neither feature uses” means, because it is a design claim rather than an arithmetic one and it needs its own justification.

The rectangle is not waste in the sense of being off the sheet. It is a square of paper sitting in the middle of the pattern, fully creased, folded along with everything else. What it is not is claimed: the vertical strip was inserted to lengthen one feature and the horizontal strip to lengthen another, and the rectangle is inside both strips and therefore charged to both features while serving the purpose of neither.

A graft’s strip is inserted so that some flap can be longer, and the paper that lengthens it is the paper of the strip along the flap’s own length. The crossing rectangle is inside the first strip and inside the second, but it lies where the second cut passed through the first strip — so it is not along either flap. It is a patch of the sheet that both insertions widened and neither insertion was for.

That is the same shape of accounting as what joins the flaps, where two groups of flaps hanging from different places have to pay for the paper between them — a strip whose width is the distance between them. A river is a graft that was planned for; the crossing rectangle is a graft that nobody planned.

It is worth putting the two losses side by side, because they are usually discussed separately and they compound. A free circle packing of seven flaps claims about seventy per cent of the sheet at best; box pleating gives up a further slice for the discipline; and two substantial grafts then charge a fifth of their own cost to a rectangle nothing uses. None of those is large on its own and the product of the three is most of the paper — which is the same arithmetic the waste census reports from the packing end, arriving here one operation later.

Which theorem was checked, and how #

Three separate assertions guard the measurement, and each of them is a claim the graft could fail.

Paper conserved and paid for. The area of the grafted pattern minus the area of the original must equal the strip’s width times the length of the cut, to 10⁻¹². Both areas are summed over panels by the folding machinery, which is a completely different thing from whatever slid the strip in.

Flaps unchanged. Every crease that does not meet the cut must have exactly the same length after the graft; every crease that crosses it must be longer by the strip’s width and by nothing else. Both are checked as sorted multisets so that the identification of one crease with another is never assumed.

Still foldable. The grafted pattern is put past the four local conditions from scratch. A graft that broke flat-foldability at some vertex would be a redesign rather than a payment.

Those hold for each graft separately, and the crossing result is then the difference between two numbers that were each verified.

Where the model stops #

The two grafts here are perpendicular, and perpendicular is doing real work: a cut is admissible only when every crease crossing it is square to it, so an oblique second graft into a pattern the first one has already stretched is not a graft at all. What two grafts at some other angle would cost is not answered, and the honest reason is that the construction does not admit them.

The result also assumes each graft is a full cut, running from one edge of the sheet to the other. A partial insertion — a strip that stops halfway — is a different object with a different accounting and needs the cut to terminate somewhere, which the machinery here does not do.

And the whole account is about area. Paper costs money by area and a design’s efficiency is measured by area, so that is the right currency; but a designer also cares where the paper is, and two crossing grafts put their excess at a specific place in the middle of the sheet rather than distributing it. Where an optimal packing puts its slack is the same question asked of the layout instead of the extension. Whether that position is convenient is a question about the design and not about the arithmetic.

That last observation is worth stating as a bound rather than as a curve, and the bound is not quite the one the curve’s shape suggests. With both strips w wide on a sheet of side s, the bills come to 2ws and the excess to w², so the share is w/(2s + w). That is under a half exactly while w is under twice the sheet’s width; it is exactly a half at w = 2s, and it goes on rising toward one after that. The 37.5 per cent quoted above for w = 1.2 is that formula at 1.2/3.2, and the next entry along would have been 50 per cent.

So the crossing is a minority of the bill for any strip narrower than two sheets, which is every graft anybody would make and is not a bound in any useful sense — it is a restatement of the fact that the excess is second order. The honest summary is the one the numbers give directly: negligible for a detail, a fifth of the bill for a limb half the sheet wide, and the dominant term for a feature nobody would attempt.

Many strips, and the term that does not multiply #

The closing section calls the general case an arithmetic somebody should do rather than assume. It is short enough to do here, and the answer resolves what looks like a disagreement between two ways of counting.

Take $p$ vertical strips of widths summing to $W_1$ and $q$ horizontal strips summing to $W_2$, inserted into a sheet $H$ tall and $W$ wide in any order. The sheet finishes $(H + W_2)$ tall and $(W + W_1)$ wide, so it has gained

$$(H + W_2)(W + W_1) - HW = W_1 H + W_2 W + W_1 W_2.$$

The bills, each computed against the original sheet as the single-graft rule prescribes, come to $W_1 H$ for the vertical family and $W_2 W$ for the horizontal one — because a vertical strip does not change the sheet’s height and so does not change what any later vertical strip is charged. Parallel grafts are exactly additive. All the interaction is between the families, and the excess is $W_1 W_2$: one term, whatever $p$ and $q$ are.

Now count it the other way. Every vertical strip crosses every horizontal one, so there are $pq$ crossing rectangles on the finished sheet, and each of them is a genuine patch of paper charged twice. Their areas are $w_i w_j$ and they sum to $\left(\sum_i w_i\right)\left(\sum_j w_j\right)$, which is $W_1 W_2$ again.

Both counts are right and they are the same number. There are $pq$ rectangles and one term, because the term is the product of two totals and the rectangles are the product expanded. That is worth having explicitly, because the two descriptions sound like a discrepancy and a designer who has noticed the rectangles could easily conclude that ten features cost a hundred crossings’ worth of paper. They cost one crossing’s worth, computed on the totals.

What that changes about how to spend #

The consequence is a rule about grouping, and it is the opposite of what the rectangle count suggests.

Since the excess depends only on the two totals, how the width is divided between strips does not matter at all. Ten narrow vertical grafts costing a tenth of a sheet each cost exactly what one graft a sheet wide costs, and interact with the horizontal family by exactly the same amount. There is no saving in splitting a feature into several and no penalty for it, which is a genuinely useful thing to know when the alternative is guessing.

What does matter is the split between directions. For a fixed total of added width $T = W_1 + W_2$ on a square sheet, the excess $W_1 W_2$ is largest when the two are equal and smallest when everything goes one way — zero, in fact, if every graft is parallel to every other. So a design that can take all its extensions in one direction pays no crossing term whatsoever, and a design that needs both pays most when it needs them equally. That is a real design lever, it costs nothing to apply, and it is invisible from the single-graft rule the whole ladder started from — which is why the packing decides so much before any of this and why what a grid settles settles the directions too.

What the picture cannot show #

The three patterns in the first figure look almost identical, because a graft is designed to be invisible: no crease moves relative to its neighbours except across the cut, and no flap changes. That is the whole content of the single-graft result and it makes the two-graft result hard to draw — the excess is a rectangle in the middle of the third picture that looks exactly like the paper around it.

Nothing marks it, and nothing could: it is not a region with its own creases or its own boundary. It is an accounting entry.

The generalisation #

The statement worth taking away is not about paper. It is that a linear cost rule applied twice to a changing object is not a linear cost rule, and that the discrepancy is exactly the product of the two changes.

That is the ordinary reason a bill comes out larger than an estimate in every field where one thing is added to another and the second addition is priced against the first’s result. It has a name in accounting, in interest, in error analysis and in dimensional analysis, and in every one of them the size of the effect is second order — invisible for small changes, a third of the total for large ones.

What origami adds is that the second-order term is a place. In interest it is a number; here it is a specific square of paper, sitting at a specific point of the sheet, with creases through it, that a folder will fold and that no feature of the finished model uses. The abstraction has a location, which is unusual and is the sort of thing this subject is good for.

Who found it, and when #

Grafting is Robert Lang’s, from the tree method’s development in the 1990s, and it is described in Origami Design Secrets as the operation that makes a design extensible. Rivers — the planned version of paper between features — are from the same source and the same period.

The additivity of two grafts does not appear to have been stated either way, which is what one would expect: a designer inserting two strips measures the sheet afterwards and gets the right answer without ever forming the sum of two separate bills. The discrepancy only exists for somebody who has written the one-graft rule down as a formula and applied it twice, which is a thing a repository does and a pair of hands does not.

One further measurement is worth recording because it bounds how much of this depends on the particular pattern. The same two grafts were run on the fold-and-cut triangle, which is a completely different pattern with cuts available in both directions, at strips 0.15 wide: the sheet gained 0.3225, and the excess over two separate bills was 0.0225, which is 0.15 squared. Nothing about the excess depends on what the pattern is — it depends on the two widths and on nothing else, which is what makes it arithmetic rather than a property of a design.

Where the ladder goes next #

The obvious continuation is k grafts. Two families of parallel strips, p of them one way and q the other, give a sheet whose gain is the sum of the two families’ bills plus the product of the two families’ total widths — one crossing term, not pq of them — and whether that is right is an arithmetic somebody should do rather than assume.

The other direction is the one this essay keeps stepping around. A graft is only admissible where a cut can be made square to every crease it meets, and on most of this site’s patterns there is no such line in one of the two directions. What a design has to look like for grafting to be available in both directions is a condition on the crease pattern, and it is very close to being the condition that makes a design box-pleatable at all.