Paying in paper

A finished crease pattern is a solved problem. The circles were packed, the creases were derived from the packing, and every flap is exactly as long as the packing allowed. Wanting one more feature — an extra toe, a longer antenna, a horn that was not in the original sketch — looks like a reason to start again.

It is not. The pattern can be cut along a line and a strip of paper slid into the gap, and every crease that meets the line simply continues across the strip. Nothing else moves.

What a graft actually is #

The move is easier to state than to believe. Choose a straight line across the pattern. Everything on one side of it stays where it is; everything on the other side slides away by the strip’s width w; the gap left behind is filled with the continuations of whatever creases crossed the line.

The pattern in the figure is 5 cells by 4, so it carries 20 units of paper. The cut runs the full 4-unit height. A strip 0.45 wide inserted across it brings the sheet to 21.80 — an increase of exactly 1.8000, which is 0.45 × 4 and nothing more.

Forty-four creases never meet the cut. Every one of them comes out of the operation at precisely the length it went in. Five creases cross the cut, and each of those is longer by 0.45 — the strip’s width, once, not once per crossing and not scaled by anything.

The bill is one term long. That is the whole claim, and it is the reason this subject’s designs grow the way they do.

It is worth being clear about what the strip buys, because the arithmetic is about paper and the designer wants a feature. The inserted band is a region of the sheet with no flaps of its own, lying between two parts of the pattern that were previously adjacent. Whatever is folded out of that band is new — a pleat, a ridge, a short flap, a run of surface texture — and it is folded out of paper that was not competing with anything, because it did not exist a moment ago. The existing flaps did not shrink to make room for it. Nothing was taken from them, which is exactly why their crease lengths are unchanged.

Where the design came from in the first place #

To see why a one-term bill is surprising, it helps to remember what producing the pattern cost.

A flap costs a circle of paper, radius equal to the flap’s length, and two flaps whose circles overlap are asking for the same paper twice. So designing a base is packing discs into a sheet, and packing is the hard part — a problem with no general algorithm, no known optimum past a handful of equal circles, and a solution that shifts everywhere when any one radius changes.

That is the alternative a graft is being weighed against. Change the tree, re-solve the packing, re-derive the creases, and accept that the arrangement that comes back bears no particular resemblance to the one that went in. Every flap moves. The graft moves none of them.

The bill has one term #

The claim is not that the added paper is small. A strip can be as wide as the designer likes. The claim is that the cost is linear in the width and contains nothing else — no term that grows with the number of flaps, no term for the disturbance to the rest of the pattern, because there is no disturbance.

The flat line is the substantive one. Across the whole sweep, the longest crease untouched by the cut changes length by 0.0 — not “by a negligible amount”, but by no amount the arithmetic can represent. A flap’s length is what its crease lengths encode, so a graft that left flaps unchanged and a graft that left crease lengths unchanged are the same statement, and the second is the one a pattern can be measured for.

Where to put the cut is a decision, and choosing it well is an optimisation. This essay does not describe a method for making that choice and states no cost for making it; what a search of that kind costs belongs to another site in this fleet. What is established here is the price of a cut once it has been chosen.

Why every crease survives the insertion #

The graft works because of a condition on the cut, and the condition is worth stating because it is the whole of the idealisation.

Every crease that crosses the cut must be square to it. A crease running perpendicular to the cut line continues across an inserted strip as the same crease, longer by exactly w. A crease meeting the cut obliquely, at angle θ, would have to lengthen by w/sin θ — which is more than w, differs from crease to crease, and therefore changes the flaps those creases belong to. That is not a graft; it is a redesign with a strip in it.

So the admissible cuts are not arbitrary lines. They are the lines that miss every vertex and cross only perpendicular creases, and in a pattern with structure there are usually several of them and sometimes none.

The construction from a packing to a crease pattern is what makes the condition checkable rather than aesthetic. The hinges and ridges are determined by the discs, so a cut that crosses only hinges running square to it is a cut that will not disturb any disc — and the disc is what the flap costs.

A river is a graft that was planned #

The surprising part of the subject is that grafting is not a new object at all. The tree method already has one, and has had since the beginning.

A river is a strip of paper of fixed width lying between two groups of flaps, put there because the skeleton has an internal edge joining two nodes and the flaps on either side must be at least that far apart on the sheet. Its cost is its width times its length. That is the same arithmetic as a graft’s, computed at the same time as everything else, and it is the reason the rule is not really about circles at all.

So a river is a graft that was planned, and a graft is a river added late. The same strip of paper, priced the same way, arriving from opposite directions. A designer who plans a body between the legs and the arms is grafting before the packing; a designer who adds a horn to a finished pattern is grafting after it. The pattern cannot tell which happened, and the paper costs the same either way.

The identification is more than a pleasing observation, because it settles a question the graft raises on its own. Inserting a strip late looks like a licence that ought to have a catch — something that works for small features and fails for large ones, or works once and not twice. It does not, and the reason is that the object being inserted is one the tree method already handles at any width. A river ten cells wide is as legal as a river a tenth of a cell wide; the packing simply has to hold it. A graft inherits that, so the only thing a wide strip costs is paper, which is what the sweep above measures. The catch, when there is one, is never the width.

Which claim was checked, and how #

The claim is about a pattern that came out, so it is measured on that pattern and not on the recipe.

Three quantities are asserted, each against something that does not know the answer.

The paper is summed over the panels the creases enclose, by the machinery that folds a pattern flat, which knows nothing about cuts or strips. It is compared with w times the cut’s length, computed separately. The two must agree to a part in a thousand million million.

The crease lengths are collected as sorted multisets on each side of the operation, so the comparison never uses the fact that the kth crease of one pattern is the kth crease of the other. Untouched creases must match exactly; crossing creases must be longer by w exactly. A graft that renumbered the creases while conserving the lengths would pass; a graft that quietly shortened one flap to pay for another would not, which is the failure the multiset form exists to catch.

The local flat-folding conditions are re-run on the grafted pattern from scratch. A graft that broke foldability would have changed the design rather than paid for it, and the price would be a fiction.

What would have made it fail is easy to name: an oblique crease across the cut. The construction rejects such a cut rather than drawing it, so a pattern with no admissible line refuses the graft outright instead of producing a picture that quietly lies about what a strip cost.

What the picture cannot show #

The arithmetic is exact and it is narrow, and the gap between those two is where a designer gets into trouble.

It does not say the added paper is well spent. A strip buys width across the whole length of the cut, and a feature usually needs it in one place. The paper the feature does not use is still paid for, so a graft can be exactly priced and grossly inefficient at the same time.

It does not say the enlarged pattern is still a good packing. Every disc is where it was and every disc is still legal, but the sheet is larger — so the same flaps now claim a smaller share of it, and the efficiency has fallen by exactly the fraction the strip added. A design that grows by accretion drifts away from optimality with every accretion, which is a real cost that this arithmetic does not price.

And it does not say the region left between the flaps can still be filled. The construction that always works needs convex regions, and inserting a strip across a region changes its shape. A graft that turns a convex leftover into an L is a graft whose paper is correctly priced and whose pattern cannot be completed.

Finally, and least visibly: nothing above is about layer ordering. The local conditions are re-checked, but which layer goes on top is a global question, and a grafted pattern’s stacking has to be found again. The paper is conserved; the folded state is not carried across.

The paper adds and the efficiency multiplies #

The dilution named above can be put in numbers, and doing so turns “a design that grows by accretion drifts away from optimality” into something a designer can decide with.

Efficiency is the paper the flaps claim divided by the sheet’s area. A graft changes neither the numerator nor the flaps, and adds $wL$ to the denominator. So the efficiency after a graft is the efficiency before it times $A/(A+wL)$, exactly.

On the worked example that factor is $20/21.80 = 0.917$: the design gives up 8.3 per cent of its efficiency to buy the strip. On the three-graft sequence it is $30/34.75 = 0.863$, or 13.7 per cent, and the three factors are 0.952, 0.933 and 0.971 — whose product is that same 0.863.

So the bill adds and the penalty multiplies. Paper accumulates one term at a time, which is the finding this essay is named for; efficiency decays geometrically, which is the finding hidden behind it. A design can absorb one graft and cannot absorb ten.

When to graft and when to start again #

That gives the comparison the essay has been circling, and it comes out as a single number.

A redesign re-packs the whole tree and recovers whatever efficiency a fresh packing achieves. A graft keeps the packing and pays the factor above. So the graft is the right move whenever the redesign would gain less than $wL/(A+wL)$ — 8.3 per cent for the single insertion, 13.7 per cent after three.

Packings found by hand are typically within a few per cent of each other, and the method has no way to certify an optimum in any case. Against a threshold of 8 per cent, a graft nearly always wins; against 14 per cent it is a genuine question; and a design that has been grafted five or six times has diluted itself past anything a re-packing would have to beat.

That is the discipline the arithmetic recommends. Graft freely while the cumulative factor is close to one, and treat the product of the factors — not the count of grafts, and not the paper added — as the quantity that says when to go back to the packing.

It also says which grafts to prefer, which the one-term bill does not. Two strips of the same area cost the same paper and different efficiency only through the denominator they land in, so the order is irrelevant and the total is all that matters. What matters instead is $L$: a cut across a narrow part of the pattern buys the same width for less paper than a cut across a wide one, and the designer’s choice of where is a choice of how long the bill’s single term is.

Grafting on a grid, and where the strip stops being free #

The condition on the cut — every crossing crease square to the line — is a nuisance in a free packing and is automatic somewhere else.

Designing on a grid gives up the efficiency of a free packing and buys back several things, and this is one of them: a box-pleated pattern has admissible cuts nearly everywhere, in both directions, at every grid line. Grafting a feature onto a box-pleated design is close to free as a construction, whatever it costs in paper. That is a large part of why complex work drifted towards the grid, and it is a reason that is rarely given alongside the usual ones about accumulated error.

The other limit is structural. Every flap on one axis is what the method produces, and a graft preserves that — the strip runs square to the axis and the flaps stay where they are. But a feature that is not a flap along the axis is not something a strip can buy at all. The graft is exact within the family of designs the tree method makes and says nothing outside it.

Who noticed it, and when #

Grafting is generally credited to Toshiyuki Meguro and Robert Lang, arriving independently in the early 1990s as the tree method was being worked out on both sides of the Pacific. Lang’s treatment is the one most readers meet, in which grafting is presented as a way of adding surface features — scales, texture, a patterned skin — to a base whose flaps are already settled.

The independent arrival is worth noting rather than glossing. This is a subject where the two traditions were working in different languages with very little crossing between them, and simultaneous discovery is what that produces. Attribution here should be read as a description of who published, not of who thought of it first.

The idea is older than the name in the way most of this subject’s ideas are. Folders had been stretching bases — inserting extra paper into a classical base to lengthen one region without shortening another — long before there was a packing argument to say what the insertion cost. What the tree method added was not the move but the price, and a move with a price is a technique rather than a trick.

Where the ladder goes next #

The immediate continuation is the packing itself. If a strip is priced by width times length, then so is a river, and a design’s whole bill is the discs plus the strips — which is how much of the sheet the flaps can actually claim and how far short of it a real packing falls.

The other direction is where the flaps stand. A graft assumes the flaps stay put, which is true and is also a constraint: a flap in a corner of the sheet costs a quarter of what it costs in the middle, so an insertion that pushes a flap inward has a hidden second bill the strip’s width does not contain. That one is not one term long.