Generator

A strip slid into a finished pattern

A generator in the designing a base library, called 37 times across 6 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

graft-strip is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

`view: "admits", show: "diagonals"`

`view: "admits", show: "census"`

`view: "admits", show: "diagonals", grid: 8, upTo: 8`

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

a strip 0.45 wide across a cut 4 long adds 1.8000 of paper and leaves all 44 creases away from the cut at the length they were ×2
developability holds at every interior vertex — its sectors close to 360° — 4 checked ×2
diagonals with no two in a row or a column spend two lines each, so 6 of them cost 12 of the 12 a plain grid has ×2
Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 4 checked ×2
Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 4 checked ×2
over 7 strip widths the sheet grows by the width times the 4-long cut and the longest flap does not move at all ×2
the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 4 checked ×2
the same diagonals gathered into one row leave 5 lines against 0, because a row already spent costs nothing to spend again ×2
3 grafts in a row add 4.750 of paper between them, and the finished pattern carries exactly that much more than the first ×1
3 strips one way and 2 the other cross in 6 rectangles, and the excess over the separate bills is the single product of the two families' total widths ×1
a plain grid admits all 4 of its gaps each way, and creasing one square diagonally removes exactly one gap in each direction — the two its diagonal crosses ×1
and a finer grid has more lines to slide a strip into as well, so on a bare grid the two design numbers agree ×1
and strips in one direction only carry no crossing term at all, so the term exists because there are two families rather than because there are many strips ×1
and the worst of them divides the grid by 25.000000000000533 at a width 4 per cent away from a whole number of spacings ×1
at every arrangement from 1 by 1 to 3 by 0 strips, the excess over the separate bills is the product of the two families' total widths, measured rather than substituted ×1
every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
every pattern is built and put through the same cut finder a graft uses, so the counts are of lines a strip could actually be slid into ×1
every printed pattern that takes a strip both ways takes it in at least one direction only through blank margin, so none takes a strip across its creases in both directions ×1
of 7 widths, 3 leave the grid at 0.125 exactly as it was and 4 divide it ×1
of the 8 printed patterns, 4 take no graft in either direction, 3 take one in a single direction and 1 takes one both ways ×1
the a grafted pattern is put past all four theorems before it is drawn ×1
the as designed is put past all four theorems before it is drawn ×1
the cheapest feature a grid design can be given is one spacing wide, so it falls from 25 per cent of the sheet at 4 to 4.2 at 24 ×1
the with the strips slid in is put past all four theorems before it is drawn ×1
two grafts across one another cost the sum of their separate bills plus the rectangle where they cross, exactly, at 5 widths ×1

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

A design that keeps its lines clear

A strip can be slid in only along a line every crossed crease meets square, so a diagonal crease spends the lines it crosses — and the census of eight printed patterns found four taking no strip in either direction. What decides how fast a design spends them is not how many diagonals it has but which rows and columns they sit in: six diagonals on a six-by-six grid leave twelve clear lines when they share a row and none at all when no two do, from the same six creases and the same amount of paper.

A graft needs a square line

A strip can be slid into a finished crease pattern only along a line every crossed crease meets square, because only such a crease continues across the strip as itself. Tried on every line between the vertex columns and rows of eight printed patterns, four take no strip in either direction and three take one in a single direction. The eighth, the fold-and-cut triangle, appears to take strips both ways, and every line it admits runs through blank margin. No printed pattern takes a strip across its creases in both directions — and one diagonal crease in a plain grid removes exactly the row and the column it sits in.

Paying in paper

A feature added to a finished design costs exactly the paper inserted for it. Cut the crease pattern along a line, slide in a strip, and every existing crease continues across it unchanged — so the bill is the strip's width times the length of the cut, and there is no second term.

Six rectangles and one term

Two grafted strips crossing leave one rectangle both features are charged for and neither uses. Three strips crossing two leave six, and the obvious budget adds them up. It does not have to: the six rectangles sum to the product of the two families' total widths, exactly, so a design with any number of features is priced by two numbers rather than by a double sum — and a family of strips in one direction alone carries no crossing term at all, however many of them there are.

The second term

A feature grafted into a finished design costs exactly the paper slid in for it, and the bill has one term. Add a second feature across the first and it has two: the rectangle where the strips cross is paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.

The width is charged in grid

A grafted strip may be any width at all and the bill in paper is exactly width times length, with no second term — which is what the first of these essays established, and is a statement about area. The grid is a different property of the same pattern and it is not conserved: a strip whose width is not a whole number of spacings puts every vertex past the cut onto a finer grid, and a width four per cent away from a spacing divides the grid by twenty-five where a width half a spacing away divides it by two. A width nearly right costs far more than one plainly wrong.

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