Origami & Folding
what one sheet can do
Generator

Rounding a design is not rounding its limbs

A generator in the designing a base library, called 9 times across 4 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.

tree-on-a-grid 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

Rounding a design is not rounding its limbsThe worst limb error against how fine the grid is, for the obvious rounding and for the best whole-number version there is. The obvious one is not monotone — a finer grid can round worse — and the best one is flat over a wide range, because the same coarse set of whole numbers goes on being the best answer.42:3:4:363:4:6:583:4:6:5125:7:11:8165:7:11:8248:11:17:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution

view: "g", g: 16

Rounding a design is not rounding its limbsThe worst limb error against how fine the grid is, for the obvious rounding and for the best whole-number version there is. The obvious one is not monotone — a finer grid can round worse — and the best one is flat over a wide range, because the same coarse set of whole numbers goes on being the best answer.42:3:4:363:4:6:583:4:6:5125:7:11:8165:7:11:8248:11:17:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution

limbs: [0.5, 1.6, 0.9, 0.8, 1.3], gs: [4, 6, 8, 12, 16]

Rounding a design is not rounding its limbsThe worst limb error against how fine the grid is, for the obvious rounding and for the best whole-number version there is. The obvious one is not monotone — a finer grid can round worse — and the best one is flat over a wide range, because the same coarse set of whole numbers goes on being the best answer.41:4:2:2:362:6:3:3:582:7:4:3:5124:12:7:6:10165:16:9:8:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution

limbs: [1, 1.37, 2.15, 1.62], gs: [4, 6, 8, 12, 16, 24], view: "pair", g: 12

The same subject in whole squares of a 12-gridA subject's limbs, and the two whole-number versions of it a grid allows. The upper bar of each pair rounds that limb on its own; the lower one is the best set of whole numbers there is. They are different sets, and the better one is not the one that rounds anything to its own nearest.limbs 1, 1.37, 2.15, 1.62 — and what a grid of 12 can say1651.37872.1512111.6298limbeachbestrounding each limb on its own: worst limb 5.9% wrongthe best whole-number version: 2.4% wrongand it uses 31 units against 35

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.

Ninety-nine in a hundred pass

A designer checks a box-pleated pattern the way every text teaches: vertex by vertex, counting mountains and valleys, watching the smallest sector. At sixteen divisions that check passes a hundred letterings in a hundred, and one of them folds. The check that separates them costs a single sweep over the crease list and is in no recipe anywhere.

Rounding in the cheap direction

A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.

Spelling a tree on a grid

Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.

The other grid

Box pleating is drawn at forty-five degrees, and the twenty-two-and-a-half-degree grid is usually described as the same thing done finer. It is not a refinement, it is a different alphabet: five kinds of vertex become fifty-six, and the share of letterings whose decision needs a search falls from 60 per cent to 22. A finer grid is a larger vocabulary and a less ambiguous one.

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