Generator

Every technique is drawn for two layers

A generator in the rigid folding library, called 28 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.

stack-depth 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

`view: "where", show: "panels", n: 140`

`view: "where", show: "panel", slug: "miura-fold", n: 140`

`view: "where", show: "panel", slug: "miura-fold", n: 140, pick: 23`

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.

the cross-section is drawn at 8 layers, inside the range the canvas can separate ×5
The Miura fold: 3660 samples under the panel drawn as 95 runs of one depth ×2
The Miura fold: 9505 samples of the folded footprint drawn as 219 runs of one depth class ×2
this panel of the Miura fold lies over 4 depths, and every edge of every folded panel runs in one of 3 directions, so the steps inside a panel can only run parallel to its own sides ×2
a uniform pile is one region of the footprint, and a graded pile is deepest in several separate places — 3 on the miura fold, 4 on the tapered corrugation ×1
and the shelf holds all three kinds of pile — 3 uniform, 2 graded, 3 island ×1
every node of the straight skeleton is equidistant from each edge that defined it, so one fold serves them all — 1 checked ×1
every printed pattern is folded and its layers counted over the whole footprint, so the deepest pile is measured rather than estimated from the crease count — it runs from 7 to 60 layers across the shelf, and on 5 of the 8 patterns it stands well above that pattern's own mean rather than being its thickness ×1
on every printed pattern every panel reaches the pattern's deepest pile somewhere, so an allowance sized panel by panel to each panel's own deepest point is the allowance for the deepest pile everywhere — 8 of 8 on the preliminary base, 24 of 24 on the Miura fold, 9 of 9 on the square twist, 13 of 13 on the hexagon twist, 65 of 65 on the Yoshimura pattern, 7 of 7 on fold and cut — the triangle, 28 of 28 on the tapered corrugation, 52 of 52 on the waterbomb tessellation ×1
on every uniform pile each panel's folded image lies over a single depth, and on every graded pile each panel lies over at least three — the Miura fold 3 to 4, the tapered corrugation 4 to 4 ×1
The square twist: 2664 samples under the panel drawn as 80 runs of one depth ×1
The square twist: 9292 samples of the folded footprint drawn as 278 runs of one depth class ×1
The tapered corrugation: 3998 samples under the panel drawn as 160 runs of one depth ×1
The tapered corrugation: 5940 samples of the folded footprint drawn as 210 runs of one depth class ×1
The Yoshimura pattern: 6050 samples of the folded footprint drawn as 109 runs of one depth class ×1
The Yoshimura pattern: 9800 samples under the panel drawn as 139 runs of one depth ×1
this panel of the square twist lies over a single depth ×1
this panel of the tapered corrugation lies over 4 depths, and every edge of every folded panel runs in one of 3 directions, so the steps inside a panel can only run parallel to its own sides ×1
this panel of the Yoshimura pattern lies over a single depth ×1

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 panel is not the unit of depth

A thick-panel design gives each panel a thickness, an offset or a taper, so a graded pile could be met panel by panel only if every panel's folded image lay over one depth. On the Miura none does. Every one of its twenty-four panels lies over three or four of the four depths its pile takes, and on the tapered corrugation every panel lies over all four. The uniform piles are the opposite — every panel of the Yoshimura, the waterbomb and the preliminary base lies over exactly one depth — which is why panel-by-panel techniques look adequate on the patterns they are drawn for. On the Miura the steps between depths cross the middle of panels, and they run parallel to the panels' own sides.

The pile, not the panel

Every technique for building a fold out of panels with depth is drawn, described and priced at one crease between two panels. A folded model has two layers nowhere except at its last fold: the printed patterns here reach eight, sixteen, thirty-two and sixty, and the length a thick panel has to find at those creases is not the published allowance but fifty-nine times it.

The sheet has a thickness

Every crease pattern describes a surface with no thickness. Everything anybody builds has some, and getting it around a corner is the central problem of turning origami into hardware.

Three kinds of pile

A thick-panel technique is priced at the deepest pile a pattern has, and the depth of that pile says nothing about where it is. Mapped over the folded footprint, the printed patterns fall into three kinds. On a uniform pile the deepest count is the whole footprint — sixty layers everywhere on the Yoshimura. On an island it is a patch and the rest is shallow. On a graded pile it is a sliver — under one per cent of the tapered corrugation — while nearly nine tenths is at least half as deep, and the Miura, the pattern that gets built, is graded.

Every generator · The rigid folding field · The patterns a reader can fold