The width is charged in grid

The first three of these essays price a graft in paper. One strip costs width times length exactly; two crossing strips cost that twice over plus the rectangle where they meet; several each way cost the same with one product for the crossings. Every one of those is an area, every one is measured by summing the panels the creases enclose, and the strip’s width appears in all of them as a free continuous parameter.

It is free in the arithmetic and it is not free in the design, because a box-pleated design sits on a grid and the grid is not an area.

Being on a grid is not a fact about how a pattern was drawn. It is a property of the finished coordinates: every vertex at a whole multiple of some spacing, and the finest such spacing is the greatest common divisor of them all. Slide in a strip whose width is not a multiple of that spacing and every vertex past the cut moves to a coordinate the old grid does not contain. What comes out has every crease continuing, every flap the same length, the bill paid exactly as promised — and no grid.

What the grid is, computed #

The spacing is Euclid’s algorithm run over the coordinates. Take every vertex’s distance from the origin along one axis, discard the zeros, and take the greatest common divisor of what is left. On an eight-by-eight grid that is 0.125, on a six-by-six grid 0.1666…, and on a pattern whose vertices do not share a common divisor at all the algorithm finds one so small it is indistinguishable from nothing — which is the correct answer and is what “not on a grid” means.

That it has to be computed rather than declared is the point. A pattern is on a grid because of where its vertices are, and an operation that moves vertices can take it off one without touching a crease.

The graft is exactly such an operation. It shifts every coordinate past the cut by the strip’s width and leaves everything else alone, which preserves every crease, every angle and every panel’s shape, and which introduces a new distance into the list the divisor is taken over.

Twenty-five for four per cent #

The factor by which the grid is divided is the part worth carrying, because it is not what a designer would expect.

A strip half a spacing wide halves the grid: every vertex past the cut moves by half a unit, so the design is now on a grid of sixteenths rather than eighths, which is a design twice as fine and no worse organised.

A strip four per cent wider than a spacing divides the grid by twenty-five. 0.13 and 0.125 have a greatest common divisor of 0.005, so the design that comes out is on a grid of two-hundredths — two hundred divisions across a sheet that had eight.

What decides the damage is the denominator of the ratio, not the distance from a whole number. A width of exactly one and a half spacings is a small change; a width of 1.04 spacings is a catastrophe; and a width a folder arrived at by eye is almost certainly the second, because a ratio arrived at by eye has no small denominator at all.

That is the inversion this essay is about. Every other cost in these essays is continuous — a slightly wider strip costs slightly more paper — and this one is not continuous anywhere. It is a function of a rational number’s denominator, which is discontinuous at every rational, and a design gets no warning from the area bill that anything has happened.

A grid is a budget of checks #

The factor matters because a design’s grid is what everything else about it is counted in, and dividing it multiplies several counts at once.

A crease pattern on a grid of $n$ has $O(n^2)$ cells to be drawn and checked. Dividing the grid by twenty-five multiplies that by six hundred and twenty-five — not because the design has more creases, which it does not, but because every position in it now has to be specified to twenty-five times the precision.

Every reference has to be folded to that precision too. A grid is produced by repeated halving where the divisions are powers of two and by an exact division construction where they are not; a grid of two-hundredths on a sheet needs a construction nobody would perform, and one arrived at by accident is one nobody has a construction for at all.

And nothing warns. The crease pattern looks the same. It folds. Every theorem it satisfied it still satisfies, because the angles have not changed. What has changed is a property nobody checks, and the check is one line of arithmetic that has never been in these essays.

The quantum under every feature #

Turn the requirement round and it puts a floor under what a feature can cost.

A strip that leaves the grid intact is at least one spacing wide. So the cheapest feature a grid design can be given is a strip one spacing wide across the whole sheet, and its area is the spacing times the sheet’s other dimension — which is $1/n$ of the sheet on an $n$ by $n$ grid.

On a four-by-four grid the smallest feature costs a quarter of the paper. On a twenty-four grid it costs a twenty-fourth. A finer grid buys cheaper features, which is a reason to design on one that has nothing to do with resolution or with how small a flap can be.

And on a bare grid the two design numbers agree: a finer grid also has more lines a strip could be slid into, so it is both cheaper to extend and has more places to be extended. The two only come apart once there are diagonals in the pattern, because what spends a clear line is a row and a column and a finer grid is drawn with proportionally more diagonals.

Where the widths come from #

A designer does not usually pick a width; they pick a feature, and the width follows from what the feature needs. That is where the awkward numbers come in.

A flap of a given length needs a strip of a given width, and the length comes from the subject — a leg is as long as a leg. A flap costs a circle of radius equal to its length, and a river between two flaps is as wide as the distance the tree says they are apart. None of those numbers arrives as a multiple of anything; they arrive as ratios in the object being folded, and rounding them changes the model.

So the designer is doing arithmetic with two incompatible sources. The subject supplies real numbers, the grid supplies a lattice, and the graft is the operation that transmits the first into the second without anybody deciding to. The rounding has to be an explicit decision because the alternative is an implicit one, and the implicit one is the twenty-five.

That is the same predicament a construction meeting the sheet it was drawn for describes at the other end of the subject: an exact relation derived on one object, applied to another, with nothing in the derivation to say which object it was about.

Why the area bill never notices #

It is worth saying plainly why three essays of exact arithmetic missed this, because the reason is structural rather than an oversight.

The paper is summed over the panels the creases enclose. A graft moves vertices and keeps creases, so every panel keeps its shape and gains or keeps its area, and the sum is exactly the old area plus the strip. Every quantity the area argument looks at is continuous in the width, and the grid is not a quantity the area argument looks at.

That is the general shape of the thing: an invariant is preserved by an operation exactly when the operation’s moving parts are the ones the invariant is blind to. The graft is blind to the grid in the same way the grid is blind to the area — a design on a grid can have any area at all, and a design of a given area can be on any grid.

So a check that a graft is sound has to ask two questions and these essays have been asking one. The first — did the paper add up — is what summing the panels answers. The second — is the result still a design of the kind that went in — is a question about a property that sum does not see.

Several strips, and one grid #

The essay before this one slides three strips one way and two the other and prices the result exactly. Every one of those five widths is subject to this, and they compound in a way the area bill does not.

The grid after several grafts is the greatest common divisor of the original spacing and every strip’s width. Five strips at widths with denominators of 3, 5, 7, 8 and 11 against a spacing of an eighth leave a grid of one part in $8 \times 3 \times 5 \times 7 \times 11$, which is 9,240 divisions of a sheet that had eight. The damage multiplies where the paper adds.

So the rounding advice is a requirement rather than a convenience once there is more than one strip. A single strip at an awkward width leaves a design that is merely finer; a family of them leaves a design that is on no grid at all in any useful sense, and the paper bill for the family is the same either way.

What a designer does about it #

The practical statement is short and it makes grafting more useful rather than less.

Choose the width in grid units, not in paper. A feature needs a certain amount of paper; round it up to the next whole spacing and the graft is free of this cost entirely. The rounding costs at most one spacing of extra paper, which on a thirty-two grid is three per cent of the sheet and on an eight grid is twelve.

And a design that wants small features wants a fine grid before it wants anything else, which is a different reason for fineness from the usual one. The usual reason is that flaps have to be thin; this one is that the increments have to be small, and the increment is the grid whatever the flaps are.

There is a third consequence and it is the one that would change practice. A design’s grid is a choice made once, at the start, and it is normally chosen for the flaps. If a design is expected to be extended, the grid should be chosen for the features that might be added later — because the quantum applies to every one of them and cannot be changed afterwards without redrawing the whole pattern. The grid is the unit of every future change, and choosing it is a decision about a design’s whole life rather than about its first version.

The rounding, priced #

The advice to round up to a whole spacing has a cost and the cost is small enough to state once.

A feature needing a width $w$ on a grid of spacing $g$ is rounded up to $g\lceil w/g \rceil$, so the worst case is one spacing of extra paper — which on an $n$ by $n$ grid is $1/n$ of the sheet’s width, times the sheet’s other dimension. Three per cent of the sheet on a thirty-two grid, twelve on an eight.

Against that, the alternative is a design on a grid twenty-five times finer, which is a pattern nobody can draw, place or fold. The exchange is not close, and the only reason it ever looks close is that the area bill is exact and visible while the grid bill is exact and invisible.

What is measured, and what is not #

The spacing is Euclid with a tolerance, relative to the sheet. An exact integer divisor was tried first and is wrong on the commonest grids in the subject: a sixth is not a whole number of millionths, so a six-by-six grid — which is on a grid by anybody’s account — had no divisor at all. The tolerance scales with the pattern rather than with the units it is drawn in.

A pattern with no common divisor is reported as having a very fine one rather than none. That is the honest output of a greatest common divisor and it should be read as a verdict rather than as a number: a grid of one part in two hundred on a sheet with eight divisions is not a grid.

The measurement is on plain grids. A real box-pleated pattern has the same property — its vertices share a spacing — and the graft does the same thing to it, but nothing here has been run on one. Whether a real pattern’s admissible lines are in the places a sensible feature wants is the clear-line question, and it binds first: a design with no clear line never reaches the question of what width to use.

And the widths are rational. An irrational width takes the design off every grid at once, which is the limiting case of the table and is not a case the divisor can report a number for.

Still open: the two quanta together #

A design has a spacing in each direction and this essay has treated them one at a time.

A strip slid in across the sheet changes the spacing along one axis and leaves the other alone, so a design can end up on a grid of eighths one way and sixteenths the other — which is still a grid, and is a rectangular grid rather than a square one. Whether that matters depends on whether the design’s diagonals are at forty-five degrees, and on a rectangular grid they are not: a diagonal of a non-square cell is at some other angle, every crease meeting it has to be re-derived, and the flat-foldability of every vertex it touches becomes a question again. So a graft in one direction only may be the more dangerous of the two, and the area arithmetic — which has no preference between the directions at all — gives no hint of it.

And the second direction’s quantum interacts with the crossing term. Rounding both widths up to whole spacings rounds both totals up, and the crossing term is the product of the two totals, so the rounding is charged twice: once in each family’s own bill and once in the product. How much that is worth on a real design is an arithmetic nobody has run, and it is the first place the two essays’ results have to be used together.

Sideways from here, the grid is not the only property of this kind. A design’s layer order, its folding sequence and its flap assignments are all facts about the finished pattern that a graft could in principle disturb, and the first of these essays’ own check tests the local theorems on the grafted pattern from scratch for exactly that reason — a graft that broke flat-foldability would have changed the design rather than paid for it. The grid is the one that slipped through because it is the one property in the list that is not local to any vertex.

The habit worth carrying is about operations that preserve what was measured. An operation proved to preserve one property has been proved to preserve one property. Three essays of exact area accounting say nothing whatever about the grid, the angles, the layer order or the folding sequence, and the reason the area survives is precisely that the operation moves the things the area does not see.