What a dashed line can say — the notation coverage generator
notation-coverage 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: "coverage", upTo: 6
view: "coverage"
view: "layers"
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 Miura strip two rows high has 2c − 3 folded states for c from 3 to 2 columns ×1
- coverage is measured over 68 seeded spacings, every assignment of each enumerated exhaustively ×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
- 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
- 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
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 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.
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.
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.
Every generator · The who found it, and when field · The patterns a reader can fold