A design that keeps its lines clear

A graft needs a square line establishes where a strip may be slid in and finds the answer discouraging. The cut has to be a line every crossed crease meets square, because only such a crease continues across the strip as itself; of eight printed patterns, four take no strip in either direction and three take one in a single direction. It closes by observing that for a box-pleated design the number of grid lines in each direction that no diagonal crosses can be read off the pattern, that it measures how far the design can be extended without redesign, and that nobody appears to have counted it.

The count is easy. What it is a count of turns out not to be what the closing paragraph assumed.

Why a diagonal costs anything #

A plain orthogonal grid admits a strip on every gap between its vertex lines: six each way on a six-by-six grid, twelve altogether. Every crease is horizontal or vertical, so every crease meets every candidate cut at a right angle or not at all, and a strip slid in anywhere leaves every crease continuing as itself.

A diagonal crease meets nothing square. A cut through the square it sits in crosses it at forty-five degrees, and a crease crossed obliquely does not continue across an inserted strip — it would have to bend, which means the crease pattern that comes out is not the one that went in with a strip added. So the cut is inadmissible, and the diagonal has spent that line.

The diagonal in a single square crosses one vertical gap and one horizontal one, so it spends two lines. That is the result the essay before this one reports: one diagonal crease in a plain grid removes exactly the row and the column it sits in.

Two diagonals need not cost four #

The extrapolation is that $k$ diagonals spend $2k$ lines, and the whole of this essay is that they do not have to.

Place six diagonals on a six-by-six grid so that no two share a row or a column — down the main diagonal of the grid, one per row and one per column. Every one of them spends a fresh vertical gap and a fresh horizontal one, so the twelve clear lines go at two per crease and the sixth diagonal leaves none at all. The design is finished: no strip can be slid into it anywhere, in either direction, at any width.

Place the same six diagonals along one row of squares. Each spends its own column, so the six columns go one at a time — but they all sit in the same row, and a row already spent costs nothing to spend again. Five horizontal gaps survive, so the design still admits five strips.

Same six creases, same amount of paper, same number of oblique lines in the pattern, and one design is extensible in one direction while the other is extensible in neither.

What a design spends is rows and columns, not creases. A diagonal in a row that already has one costs a column and nothing else; a diagonal in a row and a column that both already have one costs nothing whatever.

The number to put on a design #

That turns the closing paragraph’s suggestion into a slightly different quantity, and a better one.

The suggestion was to count clear lines, which is right and is the outcome. What predicts the outcome is the pair of counts

$$r = \text{distinct rows containing a diagonal}, \qquad c = \text{distinct columns containing a diagonal}$$

and the clear lines are $(n - c) + (n - r)$ on an $n$ by $n$ grid. Two designs with the same number of diagonal creases can have wildly different $r$ and $c$, and the one with the smaller pair is the one a feature can be grafted into later.

So a designer who expects to extend a design has a criterion, and it is not “use fewer diagonals”. It is gather the diagonals into as few rows and columns as the design allows — which is a statement about layout rather than about economy, and it is the opposite of the advice a cost-per-crease intuition would give.

That criterion is available at the moment a design is laid out and at no time afterwards. Designing on a grid is the decision that puts every crease at a multiple of the spacing and every diagonal inside a square; which squares those are is what the packing decides, and the packing has room in it — two flaps of the same size in the same river can usually sit a square apart one way or another without changing what the base makes. The clear count is a thing a designer can trade for at layout time and cannot recover later, which is what makes it worth computing before the pattern is drawn rather than after.

A finer grid does not help as much as it looks #

A natural response to running out of clear lines is to draw the design on a finer grid, and the arithmetic says it helps less than the count of lines suggests.

An $n$ by $n$ grid starts with $2n$ clear lines and a spread placement spends two per diagonal, so it takes $n$ diagonals to exhaust it whatever $n$ is. Doubling the grid doubles the lines and doubles the number of diagonals a design of that fineness will have, because a design drawn on a finer grid is drawn with finer features. The budget and the spending scale together, and a design that has run out of clear lines on a thirty-two grid will have run out on a sixty-four grid too.

What a finer grid does buy is the width of the strips, and that is a separate constraint with its own quantum.

Diagonals dropped at random #

The third curve in the first figure is the same diagonals placed at random cells, and it sits between the other two — nearer the spread placement than the gathered one.

That is what the arithmetic predicts and it is worth seeing rather than assuming. Six diagonals dropped into thirty-six cells of a six-by-six grid occupy, on average, a little under five distinct rows and a little under five distinct columns, because the birthday-collision rate at six items in six bins is not high. So a random design spends about nine or ten of its twelve lines on six diagonals, against twelve for the spread placement and seven for the gathered one.

A design nobody laid out with grafting in mind lands near the expensive end, and the reason is that the cheap end is a coincidence: it requires the diagonals to line up, and a design’s diagonals are placed for the shape it is making rather than for the rows they share.

Which is the honest reading of the census of printed patterns. Four of eight take no strip at all — worse than the random placement here would predict — because a real crease pattern has far more than six oblique creases and every additional one can only spend rows and columns, never recover them.

Not every diagonal is in a cell of its own #

The model above puts each diagonal inside one square of the grid, which is the box-pleated case and is not the only case.

A diagonal crease running across several squares — a ridge crease of a uniaxial base, say — sits in several rows and several columns at once, and spends all of them. A diagonal running the whole width of an $n$ by $n$ grid spends every row and every column, which is the entire budget in one crease.

So the quantity $r + c$ is really a count over the spans of the oblique creases rather than over the creases themselves, and long creases are catastrophic for it. That is the structural reason a traditional base takes no graft: its creases are long, they run corner to corner, and a handful of them covers every row and column the sheet has.

It also says where to look for an extensible design, and it is not among the elegant ones. A design whose oblique creases are short and clustered is extensible; one whose creases sweep across the sheet is not, and sweeping creases are what a compact, efficient base is made of. How much paper is wasted suggests the trade from the other side: a design that already spends paper on plain pleats between its flaps is spending it on exactly the rows a graft would need, and it is also the design most likely to have them clear.

What the clear count is for #

It is worth being explicit about what a designer would do with this number, because the earlier phrasing — a measure of how far a design can be extended without redesign — is right and is not the whole of it.

A clear line is permission to add a feature later. Paying in paper prices the feature exactly: width times the length of the cut, with every existing crease continuing unchanged and every flap the same length. That price is available only where a line is clear, so the clear count is the count of places the clean operation is available.

Where no line is clear, the alternative is redesign. A feature added to a design with no admissible cut has to be drawn in — the packing recomputed, the flaps moved, the crease pattern remade — which is not a bill in paper but a bill in work, and it is the bill grafting exists to avoid.

So the clear count is not a quality of a design in any aesthetic sense. It is a count of the cheap moves remaining, and a design at zero is a design whose every future change is a redesign. The census puts four of eight printed patterns there.

The second direction matters as much as the first, and the table above splits them for that reason. A design with five clear lines all horizontal can be widened and not lengthened — so a feature that needs paper in the other direction is a redesign even though the count is not zero. The crossing term exists only when a design is extended both ways, and a design with clear lines in one direction only never pays it, which is a saving nobody would choose.

The count a designer can carry #

Stated as something to do rather than as a measurement, the rule is three lines.

Lay the design out. Count the rows of the grid that contain an oblique crease, and the columns. Subtract each from the grid’s size and add the two: that is how many features the design can still be given cleanly, and in which direction.

Every one of those steps is available before a single crease is drawn, from the packing alone, because the packing decides which cells carry diagonals. So the count is not a diagnosis of a finished pattern; it is a number a designer can read off a layout and trade against, and it is the only number in these essays that is available that early.

What is counted, and how #

Every pattern is built and put through the graft’s own cut finder. The counts here are not derived from the $r$ and $c$ formula; they are the number of lines the graft’s own test for an admissible cut accepts, which is the test that decides where a strip may go. The formula is checked against them rather than substituted for them.

A candidate cut is a gap between adjacent vertex lines, taken at its midpoint, and it is admissible when no crease crosses it obliquely. That is the census the essay before this one ran on the printed patterns, applied here to patterns built on purpose.

And the random placement is seeded, so the middle curve is the same curve every time the figure is drawn rather than a different draw each time.

What the count does not decide #

A clear line is not a good place for a strip. The count says where a graft is admissible; it says nothing about whether the feature added there is the feature the design wants. A design with five clear lines all at one edge of the sheet is extensible in a direction that may be useless.

The model is one diagonal per square. Real box-pleated designs have squares with two diagonals, squares with none, and creases at angles other than forty-five degrees — and a crease at any angle other than a right angle to the cut spends the line just the same, so the count generalises while the arithmetic of $2k$ does not.

Nothing here is about the strip’s width. A line being clear says a strip may go in; how wide it may be without wrecking the grid is a separate constraint entirely, and the area arithmetic of the crossing term is a third.

The count says nothing about where the paper goes. A strip slid into a clear line pushes everything on one side of it outward, so a flap near the edge may end up nearer the edge still, and a flap in a corner is worth four times one in the middle. The clear count is about admissibility; what the graft does to the packing is a separate question and a harder one.

And the printed shelf is eight patterns. The claim that real designs land at the expensive end rests on those eight and on an argument about crease length, not on a survey.

Still open: the clear-line count of a real base #

The measurement here is on grids built to make a point, and the interesting number is on a design somebody folds.

Nobody has counted $r$ and $c$ for a box-pleated base. A published crease pattern for an insect or a figure has a few hundred creases on a grid of thirty-two or sixty-four, and reading off how many rows and columns carry an oblique crease is an afternoon’s work with the pattern in a file. The answer would say whether real designs are near the spread bound — every row and column spent, nothing graftable — or whether the grid’s pleated regions leave enough clear that the technique is practical at all. The census of eight printed patterns suggests the first, and those eight are mostly not box-pleated, which is exactly the family the question is about.

And two designs for one subject would settle whether it is a choice. The claim that a designer expecting to extend a design should gather the diagonals presumes that gathering them is possible without changing what the design makes. Two valid box-pleated bases for the same tree, differing in layout, would have two values of $r + c$, and the gap between them is the size of the choice. If the gap is small, the criterion is not actionable and the extensibility of a design is decided by its subject.

Sideways from here, the same question can be asked of a packing rather than of a grid. A flap costs a circle and a design is a packing of them, and the rivers between the circles are strips of paper that already exist — so a design with wide rivers has grafting room built into it without any line being clear at all, because a river can be widened where a grid line cannot be inserted. Whether river-widening and grafting are the same operation seen from two sides is a question these essays have not asked, and it would connect the clear count to a quantity the packing essays already compute.

The habit worth carrying is about resources spent by placement. When each thing spends a resource, ask whether two of them can spend the same unit — if they can, the cost is a count of units occupied rather than of things, and the two differ by exactly the amount of clustering. Here the difference between the two readings is the difference between a design that can be grown and one that is finished.