Origami & Folding
what one sheet can do
Generator

Which shape of sheet the flaps want

A generator in the designing a base library, called 8 times across 3 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.

sheet-shape 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

Which shape of sheet the flaps wantHow much of the sheet the flaps can claim, as the sheet is stretched from a square to three to one at constant area. The number of flaps decides where the peak is, and for some counts it is not at the square.11.522.5360%65%70%75%sheet, longer side to shorterfraction of the sheet claimedbest at 1.8 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

n: 7, view: "sweep"

Which shape of sheet the flaps wantHow much of the sheet the flaps can claim, as the sheet is stretched from a square to three to one at constant area. The number of flaps decides where the peak is, and for some counts it is not at the square.11.522.5360%65%70%75%sheet, longer side to shorterfraction of the sheet claimedbest at 1.8 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

view: "pair", n: 6, aspects: [1, 1.2, 1.4142135623730951, 1.6, 1.8, 2, 2.5, 3], show: [1, 1.4142135623730951]

6 flaps, two sheets of the same areaThe same number of discs packed into two sheets that cost the same paper and are cut to different shapes. The discs are what a design's flaps claim, so a larger radius at equal area is a longer set of limbs from the same sheet.1.0 to 1radius 0.187566.3% of the sheet claimed1.4 to 1radius 0.198874.5% of the sheet claimed

n: 7, aspects: [1, 1.5, 2, 2.5, 3.5, 5, 7], restarts: 80, steps: 5000, view: "sweep"

Which shape of sheet the flaps wantHow much of the sheet the flaps can claim, as the sheet is stretched from a square to three to one at constant area. The number of flaps decides where the peak is, and for some counts it is not at the square.123456760%65%70%75%80%sheet, longer side to shorterfraction of the sheet claimedbest at 7.0 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

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.

One member of a family

A4 halves into A5 and keeps its shape, which is the one thing everybody knows about paper sizes. The property is not about halving and not about two: a rectangle in the ratio √n divides into n copies of itself, for every n, and every one of those rectangles can be folded out of a square one diagonal at a time.

The rectangle that keeps its shape

Halving a rectangle across its long side turns a proportion of r into one of 2/r, so almost every sheet comes out of the fold a different shape from the one that went in. Exactly one does not, and it is not a shape anybody chose.

The square is a choice

Every packing on this site has been into a square, because origami paper is sold square. Hold the area fixed and vary the shape instead and the efficiency turns out to be spiky rather than smooth — with the same peak value at every proportion that is a ratio of two factors of the flap count, and nowhere else.

All the figures · The designing a base field · The patterns a reader can fold