Origami & Folding
what one sheet can do
Generator

The four letterings a small map refuses

A generator in the what it costs to know library, called 6 times across 1 essay. 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.

map-windows 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

The four letterings a small map refusesThe four letterings of a three-by-three map, out of the 256 that pass every vertex, whose forced arcs close a loop, drawn as crease patterns with mountains and valleys. They are the whole of what the cheap refusal ever catches on a map of this size. Each is unchanged by a quarter turn: they are two mirror-image pinwheels, each lettered both ways.the four letterings of a three-by-three map the loop test refusesdash-dot: mountain; dashed: valley; each passes all four vertices and still has no layer orderone in sixty-four of the letterings a three-by-three map allows

view: "refused"

The four letterings a small map refusesThe four letterings of a three-by-three map, out of the 256 that pass every vertex, whose forced arcs close a loop, drawn as crease patterns with mountains and valleys. They are the whole of what the cheap refusal ever catches on a map of this size. Each is unchanged by a quarter turn: they are two mirror-image pinwheels, each lettered both ways.the four letterings of a three-by-three map the loop test refusesdash-dot: mountain; dashed: valley; each passes all four vertices and still has no layer orderone in sixty-four of the letterings a three-by-three map allows

view: "forced"

One crease at every vertex is forcedA 4-by-3 map with its creases drawn and, in the second colour, one crease at each interior vertex that the other three decide. The rest can be lettered freely, so the map has exactly two to the 11 letterings that pass every vertex.the creases a map's letters leave free, and the ones they forcesecond colour: the crease each vertex forces; grey: creases free to take either letter4 by 3: 17 creases, 6 interior verticesevery vertex needs three creases of one letterand one of the other, so its fourth is forced11 free creases: 2048 letterings

view: "exact"

Every refusal on a small map is a small refusalFor maps from two by two to four by four: creases, interior vertices, letterings that pass every vertex, how many the loop test refuses, how many of those contain a refused three-by-three window and how many do not, the share refused, and the share one would expect if each window refused one lettering in sixty-four independently.every lettering of every map to four by four, read whole and window by windowwindow estimate: one minus 63⁄64 to the power of the number of three-by-three windowsmapcreasesverticesletteringsrefusedin a windowin noneshare refusedwindow estimate2 by 24180000.00%0.00%4 by 21031280000.00%0.00%6 by 216520480000.00%0.00%3 by 31242564401.56%1.56%4 by 31762048646403.13%3.10%5 by 32281638476076004.64%4.61%4 by 42493276821122048646.45%6.11%no map two cells across is ever refused; up to five by three every refusal is a refused window; four by four is the first tohave refusals no window explains

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.

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 map refuses in small pieces

The one-pass refusal — a circle in the arcs a map's letters force — fires on four of a three-by-three map's 256 letterings and on none below it, and the question was how it grows. It grows by windows. Every lettering it refuses on a four-by-three or five-by-three map contains one of those four refused three-by-three patterns somewhere inside it, so the share refused is almost exactly one minus 63⁄64 to the power of the map's three-by-three windows. Four by four is the first map with a refusal no window explains, 64 of its 2,112. The rise is smooth, not sharp: a quarter at six by six, near half at eight, seven tenths at ten.

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