Origami & Folding
what one sheet can do
Generator

Take a square out and the count goes up

A generator in the what it costs to know library, called 7 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.

polyomino-foldings 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

Take a square out and the count goes upThe number of foldings of each map, against the number of stackings its squares could possibly take. A map with a cell missing has fewer squares and fewer creases, and fewer creases means fewer things that can interfere — so removing the middle of a three-by-three multiplies the count rather than dividing it.foldings, and the stackings they are drawn frommapsquaresfoldingsa strip of five55041.7% of stackingsa plus5120every stacking worksa tee5120every stacking worksa two-by-three6608.3% of stackingsa two-by-three, one gone54033.3% of stackingsone corner gone88482.1% of stackingsthe middle gone8801619.9% of stackingsthe full square913680.4% of stackingsthe bars are on a logarithmic scale, because the counts run over three orders of magnitude

which: [2x2, 1x5, plus, tee, 2x3, 3x3], view: "shapes"

The same rule, on maps that are not rectanglesEach shape with the number of ways it folds. The count is not a function of how many squares there are, nor of the box the shape sits in: two eight-square maps in the same three-by-three box differ by nearly a factor of ten, and a five-square plus folds every way it could.how many ways each map foldsa two-by-two8of 24a strip of five50of 120a plus120= 5! — every stackinga tee120= 5! — every stackinga two-by-three60of 720the full square1368of 362880

which: [3x3, 3x3-corner, 3x3-edge, 3x3-centre], view: "shapes"

The same rule, on maps that are not rectanglesEach shape with the number of ways it folds. The count is not a function of how many squares there are, nor of the box the shape sits in: two eight-square maps in the same three-by-three box differ by nearly a factor of ten, and a five-square plus folds every way it could.how many ways each map foldsthe full square1368of 362880one corner gone848of 40320one edge cell gone2224of 40320the middle gone8016of 40320

which: [L4, 1x4, 2x2], view: "shapes"

The same rule, on maps that are not rectanglesEach shape with the number of ways it folds. The count is not a function of how many squares there are, nor of the box the shape sits in: two eight-square maps in the same three-by-three box differ by nearly a factor of ten, and a five-square plus folds every way it could.how many ways each map foldsthree in an L6= 3! — every stackinga strip of four16of 24a two-by-two8of 24

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 map that is not a rectangle

Take one square out of a three-by-three map and the number of ways it folds does not go down by an eighth. It goes up — to 848 if the square came from a corner, and to 8,016 if it came from the middle. Two maps of eight squares in the same box, differing by nearly a factor of ten, and no function of the box tells them apart.

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