Generator

A strip slid into a finished pattern

A generator in the designing a base library, called 55 times across 9 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

graft-strip is one function. Everything below was drawn by it when the site was last published, at settings 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.

With nothing specified

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

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

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

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

developability holds at every interior vertex — its sectors close to 360° — 4 checked ×5
Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 4 checked ×5
Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 4 checked ×5
the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 4 checked ×5
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
and the grafted pattern is still on the 10 by 10 grid both ways, which is what the rounding was for ×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
on the 10 by 10 grid the strips, slid in at their rounded widths, add exactly the rounded bill, measured over the panels the creases enclose ×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 same diagonals gathered into one row leave 5 lines against 0, because a row already spent costs nothing to spend again ×2
the the 10 by 10 grid, grafted is put past all four theorems before it is drawn ×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 grid on which the widths as asked cost nothing extra to round is exactly a grid they leave undivided ×1
a latitude of 0.02 lets every width round down as far as 0.02 before rounding up, and on every grid the expected paper falls by the whole latitude: half a spacing less 0.02 ×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
a strip 0.1 wide across the band leaves the spacing down at 0.2500 and takes the spacing across from 0.1250 to 0.0250 ×1
aiming a budget at the shorter side beats spreading it evenly by half the budget times the gap plus a quarter of the budget squared, on all 4 budgets and every gap drawn ×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 a grid exact for one family alone still pays for the other one, through the crossing as well as against the sheet ×1
and rounding up never costs less than the widths as asked, at any grid ×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 square grid a new diagonal can run on is 0.0250 both ways — the finer of the two — so a strip in one direction divides the grid in both for any crease at forty-five degrees ×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 a budget of 0.1 the same split, 80% on the strips across, is the dearest on a rectangle and the cheapest on a square, because a level sheet is the same sheet on both ×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
averaged over widths spread from 0.02 to 0.4, rounding up costs half a spacing a strip on every grid from 10 to 48, to within 0.021 ×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 45 grids, the cheapest way of spending 0.04 puts more of it on the longer family on 17, and keeps all of it on the shorter family on 7 ×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
on a rectangle, while the budget is under two spacings, dropping whichever strip sits closest above a line keeps more than 95% of the best saving, from 99% to 98%, against 49% for the rule that aims at the shorter family on the coarsest grid ×1
on a sheet that may be any rectangle, a budget of 0.1 costs most at the split that levels the two sides, 80% on the strips across, and least with all of it on the shorter side ×1
on a square sheet the closest strip keeps 92, 86, 84, 82, 80, 77, 75, 74, 75 of every hundred, against 93, 91, 94, 93, 94, 94, 95, 96, 96 for the longer family, across the same grids ×1
on a square sheet, keeping the budget on the longer family keeps 91% to 96% of the best saving at every grid, and spending it on the shorter family at most 4% ×1
on every grid of ten or finer the crossing term takes between 26.5 and 29.6 per cent of the rounding bill, give or take the small rectangle the two roundings make together ×1
on fractional widths the multiples of three beat the powers of two a spacing for a spacing: 24 at 0.328 against 16 at 0.385, 48 at 0.322 against 32 at 0.381 ×1
on widths that are fractions with denominators up to 12, every prime grid above ten costs more spacings than 24 and 48 do ×1
over 6000 designs of three strips across and two down with widths drawn from 0.02 to 0.3, the rounding bill averages 0.52, 0.47, 0.51, 0.49, 0.48, 0.50 of its bound on the grids drawn ×1
rounding up only the strips down costs a rectangle something on 37 of 37 grids and a square sheet nothing at all on 29 of them, because the margin beside the shorter side has already been paid for ×1
share of the best saving kept, from the coarsest grid drawn to the finest: closest strip 99, 99, 98, 98, 98, 98, 97, 96, 94; shorter family 49, 59, 63, 65, 66, 71, 74, 81, 84; longer family 67, 66, 67, 65, 67, 67, 70, 74, 76; at budgets of 0.32, 0.48, 0.64, 0.96, 1.28, 1.92, 2.56, 3.84, 5.12 spacings ×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 five features as asked have sides 1.42 and 1.36, and that gap of 0.06 takes 0.0009 off the level sheet on a rectangle and adds 0.0843 on a square, 94 times as much ×1
the graft moves every diagonal without turning it, so every crease keeps its direction and the pattern keeps its forty-five degrees ×1
the rounding bill is not monotone in the grid: of 17 grids from 4 to 20, a finer one costs more than the one before it 6 times ×1
the rounding bill is the rounding across times one plus the rounded strips down, plus the rounding down times one plus the strips across as asked ×1
the square sheet pays for rounding the strips down only on the grids 8, 9, 11, 14, 17, 18, 19, 21, where they overtake the strips across ×1
the thirds are exact on 6, 9, 12, 15, 18 and the quarters on 4, 8, 12, 16, 20, so the only grid drawn that holds both is 12 ×1
the with one strip across is put past all four theorems before it is drawn ×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 the budget is large enough to level the sides, the two prices meet at exactly that split ×1
with a latitude of 0.02 on each strip, used in full, the bill falls from 0.9312 to 0.7952 on a rectangle and from 1.0164 to 0.8496 on a square, where the strips down contribute 0.0000 ×1
with a latitude of 0.02, the five features cost no more than rounded up on any grid, and less than as asked first on 16 ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

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.

A grid chosen before the features

A designer fixes a box-pleating grid before knowing what will be grafted into it, so the question is what a grid costs in expectation. For widths nobody chose — spread evenly over many spacings — rounding up costs exactly half a spacing a strip on every grid, prime or composite: the sawtooth of any one design averages away, and the familiar bound on the rounding bill is twice its expected value. Divisors earn a discount only when the widths are themselves simple fractions of the sheet, and then 24 and 48 beat 16 and 32. A latitude changes the question entirely. Allowed to fall short by δ, every strip saves exactly δ on average, so the grid 1/(2δ) holds features at no expected cost — but a particular design keeps its sawtooth.

A spacing costs one plus the other side

Rounding a grafted strip up to a whole number of grid spacings is the standard remedy for a width the grid cannot hold, and with strips in one direction it costs the rounding times the cut. With strips both ways it costs more, because the crossing term is the product of the two families' totals: a spacing added across is charged once against the sheet and again against every strip running down. The same arithmetic makes the bill uneven in the grid — ten divisions beat every grid from eleven to eighteen on one design — and puts two families exact on their own grids at their least common multiple.

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 split that is worst on a rectangle

Two families of grafted strips share an allowance of shortfall, and the bill is a product of two sides whose sum the allowance fixes. So where the allowance goes is a question about a rectangle with a set perimeter: its area is largest when level and smallest when lopsided. On a sheet that may be any rectangle the split that levels the sides is the dearest there is, and it is barely dearer, because the gap enters squared. On a square sheet the same split is the cheapest, and the gap enters to first order, ninety-four times as heavily on five features. On a grid the first answer dissolves — the strip nearest a line decides, not the family — and the second does not.

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.

All the figures · The designing a base field · The patterns a reader can fold