Origami & Folding
what one sheet can do
Generator

A flap in a corner costs a quarter

A generator in the designing a base library, called 9 times across 3 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

edge-flap 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

A flap in a corner costs a quarterThree flaps of the same length on one sheet: one in the middle, one on an edge, one in a corner. Each consumes every point of paper within its own length of it, but only the paper that is actually there — so the same flap costs a whole disc, half of one, or a quarter, and the boundary is the cheapest place to stand.1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it

view: "circle", L: 0.28, n: 7

A flap in a corner costs a quarterThree flaps of the same length on one sheet: one in the middle, one on an edge, one in a corner. Each consumes every point of paper within its own length of it, but only the paper that is actually there — so the same flap costs a whole disc, half of one, or a quarter, and the boundary is the cheapest place to stand.1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it

view: "circle", L: 0.28

A flap in a corner costs a quarterThree flaps of the same length on one sheet: one in the middle, one on an edge, one in a corner. Each consumes every point of paper within its own length of it, but only the paper that is actually there — so the same flap costs a whole disc, half of one, or a quarter, and the boundary is the cheapest place to stand.1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it

view: "packing", n: 4

The boundary is paper tooOne set of flaps packed under two rules. On the left each flap's disc has to lie wholly inside the sheet; on the right a flap may stand on the edge or in a corner, where it only pays for the part of its disc that is on paper. The second rule fits the same number of longer flaps onto the same sheet.whole discs, insideflaps of 0.2500edges and corners allowedflaps of 0.5000the same 4 flaps, packed twice: whole discs inside the sheet, and discs allowed to stand on the edgeevery flap is 2.00 times longer under the looser rule, and it is the same paper

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.

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 hole is cheap paper

A flap claims every point of paper within its own length of its tip, but only the paper that is actually there — so an edge is half price and a corner a quarter. A hole in the middle of the sheet is more edge, and holding the area fixed, a square with a hole in it is cheaper paper than a solid square at every flap length measured.

Spending the cheap paper

A hole makes a sheet cheaper by the square inch, because a flap against its rim claims only half a disc. Whether a design can spend that was left open, because no packing search here could express a region that is not paper. It can now, and the mechanism turns out not to be the discount at all: across a hole, two flaps may overlap, and two tips three fifths of a sheet apart carry flaps of 0.500 rather than 0.300.

The corner is worth four times the middle

A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.

Every generator · The designing a base field · The patterns a reader can fold