A graft needs a square line

Paying in paper found that a feature added to a finished design costs exactly the strip slid in for it: cut the pattern along a line, insert a strip, and every crease that meets the line continues across it while every crease that does not is carried along untouched. The second term added a second strip across the first and found the one extra term, the rectangle where the strips cross, equal to the product of their widths.

Both results carried a condition that neither examined. The cut has to be admissible: every crease it crosses must meet it square, so that the crease continues across the inserted strip as the same crease, longer by the strip’s width and changed in nothing else. A crease meeting the cut at an angle would have to shift sideways across the strip to reach its other half, and a shifted crease is a different pattern, with different angles at both of its ends.

The second term ends by naming the question that condition raises. On most printed patterns, it says, there is no such line in one of the two directions, and what a design has to look like for grafting to be available both ways is a condition on the crease pattern — very close, it suggests, to the condition that makes a design box-pleatable. The census below answers the first half for the patterns printed here, and the answer turns out sharper than most.

Where a cut can go #

A cut along a line that passes through a vertex would split that vertex between two sides of the strip, so the candidate lines are the ones that pass between vertices. For a vertical cut, that means the gaps between successive columns of vertex positions; for a horizontal cut, the gaps between successive rows. Every such gap is a candidate, and there is nothing else to try: two lines in the same gap cross exactly the same creases.

A candidate is admissible when every crease crossing it is square to it — for a vertical cut, every crossed crease is horizontal. The census tries every candidate in both directions on every pattern, counts the creases that cross it and how many of them are oblique, and keeps the lines with none. It also notes one thing the price does not distinguish and a designer would: whether an admissible line crosses any crease at all, or only the sheet’s own edges.

Four patterns that take no strip #

The preliminary base has two candidate lines each way, and each is crossed by two of its diagonals. The square twist has six each way, and every one is crossed by two or four of its pleats at forty-five degrees. The Yoshimura pattern has twelve vertical candidates, each crossed by five diagonals, and five horizontal ones, each crossed by twelve. The waterbomb tessellation has eight each way, and every one of those sixteen lines is crossed by eight oblique creases.

None of the four can be grafted anywhere. A feature added to any of them cannot be paid for with a strip, because there is no line along which a strip can go in; it needs a redesign, or at least an operation more invasive than a graft.

That is not a verdict on the patterns’ quality. A twist and a waterbomb are designed around diagonals, and diagonals are exactly what a graft cannot cross. A pattern built from oblique creases has committed its shape to those angles, and the strip that would lengthen one part of it would need every crossed crease to turn.

Three that take a strip one way #

The Miura fold admits six of its thirteen vertical lines and none of its four horizontal ones. The tapered corrugation admits seven of fifteen vertical lines and none of four horizontal. The hexagon twist admits two of eleven horizontal lines and none of its four vertical ones.

The Miura’s numbers come straight from its shape. Its straight creases run horizontally across the sheet, and its zigzag creases run up it. Every horizontal line crosses all seven zigzags at an angle, so none can be grafted. A vertical line through a zigzag — between the two columns of vertices one zigzag alternates between — crosses it obliquely too, and seven of the thirteen gaps are of that kind. The other six lie between one zigzag and the next, where the only creases crossing the line are the horizontal ones, and those six admit a strip.

The tapered corrugation is the same structure with one more zigzag, and its seven admissible lines are again the gaps between zigzags. A Miura is one vertex, repeated, and the census says in which direction the repetition can be interrupted by a strip: a strip between columns widens the straight panels on either side of it and leaves every vertex’s angles alone. In the other direction a strip would have to cut every zigzag, and no strip of any width can do that without shifting creases.

The hexagon twist’s two admissible lines lie close to its top and bottom edges, where the only creases crossing them — besides the sheet’s own sides — are the two pleats that happen to run vertically at the orientation it is printed in. Every vertical line on it is crossed by three or four pleats at thirty degrees from horizontal.

The one-way patterns have a second consequence that the second term makes visible. It found that for a fixed total of added width the crossing term is zero when every graft runs the same way, and largest when the width is split equally between the two directions. The Miura, the tapered corrugation and the hexagon twist can never pay the crossing term, because they can never take the second strip: the grafts they admit are all parallel, and a design that can only grow in one direction is a design whose growth is as cheap as grafting ever gets.

The pattern that seems to take strips both ways #

The fold-and-cut triangle admits two of its eight vertical lines and two of its eight horizontal ones, and on the census it is the only printed pattern marked as taking strips in both directions.

All four of its lines cross no crease. The triangle’s outline sits inside the square sheet, between about a fifth and five sixths of the way across and between a sixth and five sixths of the way up, and every one of its creases lies inside the outline. The first and last gap in each direction — between the sheet’s edge and the outline’s nearest corner — is crossed by nothing except the sheet’s two other edges. A strip inserted there widens a margin and changes no crease.

That is a graft only in the degenerate sense, and it settles the census’s answer. No printed pattern takes a strip across its creases in both directions. The arrangement the second term priced — two strips crossing inside a design — exists on this shelf only as a pair of margins. Its measurement on the triangle, two strips 0.15 wide adding an excess of 0.0225, is exactly the product of the widths as the arithmetic says it must be; the arithmetic does not care whether the strips cross creases, and a designer does.

The margin is also blank only because the triangle is printed without its outside. One straight cut builds a fold-and-cut pattern from the outline’s straight skeleton and from perpendiculars dropped to the outline’s edges, and in the full construction those perpendiculars carry on past the outline toward the edge of the sheet. A perpendicular that meets a skeleton crease is reflected into a perpendicular to the neighbouring edge, so every direction it ever takes is square to one of the triangle’s three edges — which lie at about 7, 71 and 126 degrees from horizontal. None of those directions is horizontal or vertical. Completed to the edge of its sheet, the one pattern that seemed to take strips both ways would take them in neither.

One diagonal in a grid #

The control is a plain four-by-four grid, which has only horizontal and vertical creases. It admits every one of its four vertical gaps and every one of its four horizontal gaps, and every one of those lines crosses real creases: a strip can go in anywhere, and each one extends the design rather than its margin.

Crease one of its squares along a diagonal. The diagonal crosses exactly one vertical gap — the one through its column — and exactly one horizontal gap — the one through its row. Both of those lines are now blocked, and the grid admits three of its four lines each way.

That single crease is the whole mechanism of the census. A graft line is blocked by any one oblique crease anywhere along it, however long the line and however square everything else it crosses is. The hexagon twist showed it with pleats at thirty degrees; the grid shows it with a single diagonal. Admissibility is not a matter of degree: one oblique crease on a line removes the line, and a pattern keeps a line only if every one of its oblique creases stays out of that row or column.

What box pleating keeps and gives away #

The second term’s closing remark was that the condition for grafting in both directions is very close to the condition that makes a design box-pleatable. The census and the diagonal grid say how close, and in which direction the difference runs.

A box-pleated design lives on a square grid, and its creases are horizontal, vertical and at forty-five degrees. The horizontal and vertical creases are exactly what grafting wants; the diagonals are exactly what it cannot cross. So a box-pleated design admits a strip along a grid line precisely when no diagonal crosses that line. The lattice that makes the best packing a finite question makes this a finite question too: a design has a definite number of clear lines in each direction, and it can be read off the pattern.

A design whose diagonals are confined to small patches — flaps concentrated in a few squares, with plain pleats between them — keeps many lines clear and can be grafted almost anywhere. A design whose diagonals are spread across every row and column keeps none, and is as ungraftable as a twist. Box pleating makes grafting possible; it does not make it available, and the difference is where the diagonals went.

That is also where what joins the flaps comes in. The paper between groups of flaps is paid for as a river, a strip whose width is the distance between them, and paying in paper observed that a river and a graft are the same object arriving from opposite directions. A design that routes its rivers as full-width bands of plain pleats has left clear lines where the bands run, and so has left itself room to grow; one that routes them diagonally has not.

What the census cannot show #

The census tries straight lines running the full width or height of each pattern, and those are not the only places paper can be added.

A partial strip — one that stops partway across — needs its end to be somewhere, and where a strip ends a new vertex appears. That is a different operation with different conditions, and it could extend a pattern the census counts as ungraftable, at the cost of changing more than a strip’s worth of it.

A strip along an oblique line is admissible when every crease the line crosses is square to it, which on a pattern with creases in several directions can happen along a diagonal. The census tries only the pattern’s two axis directions, because that is how a pattern is printed and how a graft has been defined here; a twist might admit a strip along one of its own pleat directions, and that is not counted. Turned by a sixth of a turn, the hexagon twist would offer a different set of lines in a direction that is no longer horizontal.

And admissibility is about the crease pattern, not the folded model. A strip that can be inserted may still cost more than paper if it lengthens a flap the design needed short, or moves a reference point another construction depended on.

The patterns the census assumes #

A candidate line passes between vertex positions, so it never runs through a vertex. A line exactly through a column of vertices would need the vertices to be split, which is not a graft.

A crease blocks a line if it crosses the line at any angle other than square. A crease that merely ends at the line’s column is not crossing it, and a crease lying along the line cannot occur, since the line avoids vertex positions.

A line that crosses only the sheet’s edges runs through margin. It is admissible, because nothing it crosses is oblique, and it is counted separately, because a strip there changes the sheet and nothing drawn on it.

The printed patterns are taken as printed, in their orientation on the sheet. A pattern turned on the page has different candidate directions, and the census counts the orientation a reader has.

How the lines were counted #

Every candidate is tried, in both directions, on every pattern, and for each the creases crossing it are found and tested for being square to it and for being creases rather than the sheet’s edges.

The count is required to agree with the cut finder the grafts themselves use on every pattern, so the census and the grafts it describes are counting the same thing. Every pattern that takes strips both ways is required to take at least one direction only through margin, which is the claim the census makes about the shelf.

The grid controls are required to come out exactly: every line admissible on the plain grid, and exactly one line removed in each direction by a single diagonal. A census in which one diagonal removed more or fewer lines would mean the test was not the square-crossing condition the argument depends on, and the figure would not draw.

Still open: a design that keeps its lines clear #

The census turns the second term’s remark into a measurable property of a design, and the property invites a design question.

For a box-pleated design, the number of grid lines in each direction that no diagonal crosses can be read off the pattern, and it measures how far the design can be extended without redesign. Two designs for the same subject, both valid, might differ greatly in that count, and a designer who expects to add features later has a reason to prefer the one with more clear lines. Nobody appears to have counted it for a real box-pleated design, and the waste census suggests where to look first: a design that already spends paper on plain pleats between its flaps may be spending it on exactly the lines a graft would need.

The other direction is the partial strip. A strip that stops at a vertex it creates is the operation a design needs when no full line is clear, and whether the vertex it creates can always be made flat-foldable — and at what cost in extra creases — is the question that would say whether an ungraftable pattern is really ungraftable or merely needs a more careful graft.

The habit worth carrying is a check on any operation that is cheap in general. Ask where it is admissible before asking what it costs. A price that is exact and simple wherever it applies can still apply almost nowhere, and the census of where is the half of the result that decides whether the price is ever paid.