Generator

What a dashed line can say — the notation coverage generator

A generator in the who found it, and when library, called 48 times across 11 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.

notation-coverage 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: "coverage", upTo: 6`

`view: "coverage"`

`view: "layers"`

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.

every one of the 6 folded states of the three-column patch 3 rows high is its middle column at the bottom, then its two end columns interleaved in 2 and 2 unbroken blocks, every interleaving occurring once: C(4, 2) = 6 ×9
at a shear of 20.1° the 5 by 3 patch has 16 folded states while its cells are narrower than 1 ⁄ sin 2a = 1.552 times their height, and one state when they are wider ×2
a Miura strip two rows high has 2c − 3 folded states for c from 3 to 2 columns ×1
at a shear of 20.1°, every row crease of an end column that lies inside the other end column's folded footprint welds its two panels, and every one that lies outside it does not ×1
at every height from one to 9 rows exactly one of the two end columns is welded at each of the r − 1 row creases, alternately, leaving r + 1 blocks to interleave ×1
coverage is measured over 68 seeded spacings, every assignment of each enumerated exhaustively ×1
every folded state of the 4-column strip extends to exactly one state of the 5-column strip but one, which extends to 3: 5 states become 7 ×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 probe point of panel 1 lies outside panel 3 and every probe of panel 3 outside panel 1, though the two share a strip of paper ×1
from one to nine rows the states are C(r + 1, ⌊(r + 1)⁄2⌋) — 2, 3, 6, 10, 20, 35, 70, 126, 252 — against 2, 6, 20, 70, 252, 924, 3432, 12870, 48620 interleavings of two unwelded columns ×1
from three columns on, the same states branch in the same numbers at every step, so every column adds 1, 2, 5, 9 states for one to four rows ×1
on all 12 patches the grown count equals the closed form, and on the 5 small enough to list it equals the listed count ×1
on every Miura patch listed, the signs that vary between folded states are all tied together through shared panels: the states are one choice with many answers, never several independent ones ×1
on every patch from two to twelve columns and one to four rows the grown count is 1 + (c − 2)(B − 1): 1 row adds 1 a column, 2 rows adds 2 a column, 3 rows adds 5 a column, 4 rows adds 9 a column ×1
on ten Miura patches the listing search finishes, the states grown from overlapping pairs alone are the listed states sign for sign, and their number is 1 + (c − 2)(B − 1) ×1
simple folds do not reach every flat folding, which is why the notation acquired named symbols rather than more dashed lines ×1
the field can be filled on 4 of 5, and on every one a pattern with a single folded state has no pair whose sign varies, while a pattern with more has some — the Miura, 2 by 2: 1 state, 0 of 6 signs varying; the Miura, 3 by 2: 3 states, 4 of 15 signs varying; the Miura, 3 by 3: 6 states, 9 of 36 signs varying; the Miura, 4 by 3: 11 states, 27 of 66 signs varying ×1
the field can be filled on 4 of 8, and on every one a pattern with a single folded state has no pair whose sign varies, while a pattern with more has some — the preliminary base: 1 state, 0 of 28 signs varying; the square twist: 1 state, 0 of 36 signs varying; the hexagon twist: 1 state, 0 of 66 signs varying; Fold and cut — the triangle: 2 states, 6 of 21 signs varying ×1
the field can be filled on 6 of 6, and on every one a pattern with a single folded state has no pair whose sign varies, while a pattern with more has some — the Miura, 2 by 2: 1 state, 0 of 6 signs varying; the Miura, 3 by 2: 3 states, 4 of 15 signs varying; the Miura, 4 by 2: 5 states, 12 of 28 signs varying; the Miura, 3 by 3: 6 states, 9 of 36 signs varying; the Miura, 4 by 3: 11 states, 27 of 66 signs varying; the Miura, 2 by 4: 1 state, 0 of 28 signs varying ×1
the layer order is counted in bits rather than in orderings, so it can be set beside the size of the pattern that does get recorded ×1
the share of flat foldings a sequence of simple folds can reach falls from 72% to 13% as the model grows ×1
three columns wide a patch of r rows has the central binomial coefficient C(r + 1, ⌊(r + 1)⁄2⌋) folded states — 2, 3, 6, 10, 20, 35, 70, 126, 252 for one to nine rows — and four columns wide twice that less one ×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 file has no paper

The field's interchange format is three arrays — where the vertices are, which pairs of them an edge joins, and a letter for each edge — and that is exactly the object every computation on a crease pattern starts from. A list of edges cannot say that two of them must not cross, because crossing is a property of the drawing and the list has no drawing in it. So a pattern that no paper could carry is a perfectly well-formed file, and four of this collection's own were.

A line in the columns, a binomial in the rows

A Miura strip two rows high was found to have 2c − 3 folded states for c columns, from three columns to seven, and the search that found it ran out at eight. Counted instead from the pairs of panels that actually overlap, the count runs to twenty columns and nine rows, and 2c − 3 turns out to be one row of a formula: a patch c columns by r rows has 1 + (c − 2)(B − 1) folded states, with B the central binomial coefficient of r + 1. The line in the columns has a reason that can be drawn. The binomial in the rows is measured and not explained, and it holds only while the folded columns two apart still overlap, which is the same as the pile being more than two deep.

No format has a gluing

A crease pattern file records vertices, edges, assignments, faces and layer orders. Every one of those is a feature of the paper's interior, and the boundary appears only as a kind of edge — so there is nowhere in the scheme to say that two boundary edges are the same edge, and the sheet a pattern is on cannot be written down.

One choice with eleven answers

A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.

The field is empty where it would say nothing

The interchange format for crease patterns has a field for the layer order and nothing ever fills it in. Filling it in where the folded states can be listed — four of the eight printed patterns, and Miura patches to twelve panels — finds that the preliminary base and both twists have exactly one folded state, so every one of the field's signs follows from the crease pattern and the field would record nothing a reader could not compute. The fold-and-cut triangle has two states. The Miura is different: every patch with three or more columns has several — three, six and eleven on the three-by-two, three-by-three and four-by-three — so on the pattern that gets built the field carries information from six panels up, and the field's size had been measured as log₂ of the panels' orderings, which on the preliminary base is fifteen bits for an object that has zero.

The file records no verdict

A crease pattern file records vertices, edges and letters. Every one of the square twist's two hundred and fifty-six admissible letterings makes a perfectly valid file, and two hundred and forty-eight of them describe an object that does not exist. The format has a field for the layer order — the one thing that would settle it — and nothing fills it in, so a file is a drawing rather than a claim, and the field exchanges them as though they were claims.

The first thing about layers

A folder is taught four conditions at a vertex, or is taught nothing at all, and neither one says anything about the layers — which is where most of what goes wrong actually goes wrong. There has never been a rule about layer order simple enough to teach, because the question is global and every answer to it was a search. A chain of panels whose arrows all point the same way is the first one that fits on a finger.

The half no notation records

Every notation this subject has invented writes down the crease pattern or the sequence of folds, and the crease pattern is the half that does not decide the folded object. The field's interchange format has a place for the other half and nothing fills it in — including the files published here, which carry every vertex, edge and letter of a Yoshimura and none of the three hundred bits that would say which of its layer orders the folded object is.

The reader decides the junction

Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

Two stacks shuffled in glued pairs

A Miura patch three columns wide and r rows high has a central binomial number of folded states — 2, 3, 6, 10, 20 and on to 252 at nine rows — and the count was measured without a reason. The reason is a shuffle. In every state the middle column lies at the bottom and the two end columns lie above it, interleaved like two halves of a deck. Some pairs of an end column's panels can never be parted, because the row crease joining them lies inside the other column's folded footprint, and the two columns take those welds in turn, one at every row crease. That leaves r + 1 unbreakable blocks, and the number of ways to interleave them is the binomial.

What a dashed line can say

Before the Yoshizawa–Randlett symbols a model could not be transmitted, and the subject was not cumulative. The basic notation says exactly one thing — fold this crease, this way, now — which is precisely a simple fold, and the share of flat foldings that simple folds reach collapses from 71% to 13% as a model grows.

All the figures · The who found it, and when field · The patterns a reader can fold