Origami & Folding
what one sheet can do
Generator

Every map anybody has counted

A generator in the what it costs to know library, called 12 times across 4 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.

map-count-table 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

Every map anybody has countedThe number of ways a ruled rectangle folds flat. The filled rows were computed while this page was built and each agrees with Lunnon's published value; the hollow rows are the ones this machine will not finish in a build. The table stops where it stops because exhaustive search is the only method there is.mapflat foldingsand what it took2 × 284 cells, computed here2 × 3606 cells, computed here2 × 43208 cells, computed here3 × 31,3689 cells, computed here2 × 51,9800.6 s3 × 415,55254 s4 × 4300,608not reached herethe 1 × n case is the strip, and it is the only row of this table with a fast methodnobody has a formula for any entry, and nobody has proved there is none

compute: [2,2, 2,3, 2,4, 3,3]

Every map anybody has countedThe number of ways a ruled rectangle folds flat. The filled rows were computed while this page was built and each agrees with Lunnon's published value; the hollow rows are the ones this machine will not finish in a build. The table stops where it stops because exhaustive search is the only method there is.mapflat foldingsand what it took2 × 284 cells, computed here2 × 3606 cells, computed here2 × 43208 cells, computed here3 × 31,3689 cells, computed here2 × 51,9800.6 s3 × 415,55254 s4 × 4300,608not reached herethe 1 × n case is the strip, and it is the only row of this table with a fast methodnobody has a formula for any entry, and nobody has proved there is none

view: "table", upTo: 8

Every map anybody has countedThe number of ways a ruled rectangle folds flat. The filled rows were computed while this page was built and each agrees with Lunnon's published value; the hollow rows are the ones this machine will not finish in a build. The table stops where it stops because exhaustive search is the only method there is.mapflat foldingsand what it took2 × 284 cells, computed here2 × 3606 cells, computed here2 × 43208 cells, computed here3 × 31,3689 cells, computed here2 × 51,9800.6 s3 × 415,55254 s4 × 4300,608not reached herethe 1 × n case is the strip, and it is the only row of this table with a fast methodnobody has a formula for any entry, and nobody has proved there is none

view: "objects", upTo: 8

The foldings of a strip, and how many objects they areThe published stamp-folding numbers count labelled foldings. The pale bar is that count and the dark one is the number of distinct folded objects once reading the strip from the other end and turning it over are taken as the same object.the pale bar is the published count, the dark one the objectsneither operation ever fixes a folding; doing both sometimes does, and that is why it is not a quarter2 stamps2 labelled · 1 objects · 2 fixed by doing both3 stamps6 labelled · 2 objects · 2 fixed by doing both4 stamps16 labelled · 5 objects · 4 fixed by doing both5 stamps50 labelled · 14 objects · 6 fixed by doing both6 stamps144 labelled · 38 objects · 8 fixed by doing both7 stamps462 labelled · 120 objects · 18 fixed by doing both8 stamps1392 labelled · 353 objects · 20 fixed by doing both

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

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.

The answer is bigger than the question

A twelve-square strip of stamps is twelve numbers of input and 146,376 objects of output. No algorithm writes that faster than it can be written, so 'efficient' has to be measured against the answer rather than against the question — and in folding that is the normal case.

The count counts labels

One, two, six, sixteen, fifty, a hundred and forty-four: the oldest sequence in the subject counts foldings of a strip of numbered stamps. A folded strip of blank paper has no first stamp and no top side, and neither of those operations ever leaves a folding alone — so the count of objects is 1, 2, 5, 14, 38, 120, and it is not the count over four.

The map counted from the layers

The classical map-folding counts are computed from a rule that never places a panel: work out which edge of the folded square each fold wraps around, and refuse the orderings that interleave two folds at one edge. Place the panels instead and order them by the general non-crossing rules, and the same numbers come out — 2, 6, 16, 50, 144, 8, 60, 320, 1368 — on nine sizes, by machinery that shares no line of code with the first.

What universality costs

The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.

All the figures · The what it costs to know field · The patterns a reader can fold