Generator

Every technique is drawn for two layers

A generator in the rigid folding library, called 44 times across 7 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.

stack-depth 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

`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 while it drew, not written by hand. The count is how many separate times that claim was put to the test.

the columns overlap exactly 2 deep at a slant of 45.00 degrees ×9
the cross-section is drawn at 8 layers, inside the range the canvas can separate ×5
equal columns: 0.0% of the footprint piles deeper than 6 a row and 34.5% shallower ×2
on a 24-column leaf taper, 6 deep a row covers 41% of the footprint at the best single slant, 10 degrees, 51% with each zigzag leaning its own amount, and 66% on equal columns of the same total width ×2
the leaf taper with every zigzag leaning for four deep a row, folded rigidly with its straight creases at 60°: every panel placed from the fold angles, and no two panels disagree about a shared corner by more than 1.5e-14 of a unit ×2
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
the taper at its best single slant: 25.4% of the footprint piles deeper than 6 a row and 34.1% shallower ×2
the taper, each zigzag leaning its own amount: 2.5% of the footprint piles deeper than 6 a row and 46.4% shallower ×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
2 depth maps of a folded Miura 16 columns wide, up to 4 layers deep ×1
a corrugation passes every vertex with one slant and with a different slant on every zigzag line, and fails when its rows differ in height ×1
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
at 15.0 degrees the share at one depth climbs with every doubling of the columns, to 91 per cent on 64 ×1
at a slant of 15.0 degrees a Miura 16 columns wide folds 4 columns deep over 68 per cent of its footprint, the rest being its two ends ×1
at fifteen degrees every one of the 10 panels away from the ends lies over a single depth ×1
equal columns passes Kawasaki and Maekawa at every vertex ×1
equal columns: slants 13.3 to 13.3 degrees; a row piles 1 deep on 7.0%, 2 deep on 6.9%, 3 deep on 6.6%, 4 deep on 6.9%, 5 deep on 7.1%, 6 deep on 65.5%; mean 4.97, deepest 6 ×1
equal columns: slants 21.0 to 21.0 degrees; a row piles 1 deep on 7.5%, 2 deep on 7.4%, 3 deep on 7.1%, 4 deep on 78.0%; mean 3.55, deepest 4 ×1
equal columns: the rows fold exactly onto each other, so every depth on the folded footprint is even ×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
graded for 4 deep, the leaf taper's columns turn out of the first column's plane by up to 18.8, 24.3, 18.7, 10.6, 3.6° at straight creases of 20, 40, 80, 120, 160°, and at one slant by 0e+0° — a single slant keeps the strip straight, and grading bends it and unbends it again as it closes ×1
leaning each zigzag by half the arcsine of 2 ⁄ (4·w) puts every column at least 0.05 wider than 2 ⁄ 4 within 8% of 4 deep; the 4 narrower than 2 ⁄ 4 cannot get there at any slant and pile 5.78, 4.58, 4.58, 5.78 at best ×1
on 80 columns, every even count gives a pile 94%, 93%, 88%, 84% at one depth and every odd count leaves it mixed, 48%, 46%, 44% ×1
on equal columns 1 wide and 1 high at 15.0° pile 4 a row, 0.5 wide and 1 high at 15.0° pile 8 a row, 1 wide and 0.5 high at 15.0° pile 2 a row, 0.5 wide and 0.5 high at 15.0° pile 4 a row, 2 wide and 1 high at 45.0° pile 1 a row: the count is 2h ⁄ (w sin 2a) wherever that is whole ×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
opening from flat, with the straight creases at 170 and 150°, the steep zigzags at the narrow ends have opened 7.1 and 21.5° and the upright ones in the middle 2.6 and 7.9° — the leaf opens from its ends ×1
the 24 columns run from 0.346 to 0.999 wide; at one slant of 16.0 degrees they pile 3.78 to 10.91 a row, and leaning for 4 the zigzags run from 15.0 to 45.0 degrees ×1
the cosine table is rank one to 5.8e-15 on all three and every panel meets at five stages of the motion, to 3.0e-14; slide one vertex a tenth along its own flat-folding ray and the loop round a face misses by 0.220 radians ×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 taper at its best single slant passes Kawasaki and Maekawa at every vertex ×1
the taper at its best single slant: slants 10.0 to 10.0 degrees; a row piles 1 deep on 1.7%, 2 deep on 12.0%, 3 deep on 3.1%, 4 deep on 12.8%, 5 deep on 4.5%, 6 deep on 40.5%, 7 deep on 14.0%, 8 deep on 9.8%, 9 deep on 1.6%; mean 5.43, deepest 9 ×1
the taper at its best single slant: slants 16.0 to 16.0 degrees; a row piles 1 deep on 1.9%, 2 deep on 13.4%, 3 deep on 7.1%, 4 deep on 54.7%, 5 deep on 14.1%, 6 deep on 8.3%; mean 3.93, deepest 7 ×1
the taper at its best single slant: the rows fold exactly onto each other, so every depth on the folded footprint is even ×1
the taper, each zigzag leaning its own amount passes Kawasaki and Maekawa at every vertex ×1
the taper, each zigzag leaning its own amount: slants 15.0 to 45.0 degrees; a row piles 1 deep on 4.7%, 2 deep on 19.3%, 3 deep on 10.2%, 4 deep on 62.3%, 5 deep on 3.4%; mean 3.41, deepest 6 ×1
the taper, each zigzag leaning its own amount: slants 9.8 to 37.3 degrees; a row piles 1 deep on 4.9%, 2 deep on 16.1%, 3 deep on 4.4%, 4 deep on 14.5%, 5 deep on 6.5%, 6 deep on 51.1%, 7 deep on 2.5%; mean 4.65, deepest 8 ×1
the taper, each zigzag leaning its own amount: the rows fold exactly onto each other, so every depth on the folded footprint is even ×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
with the straight creases at 90°, the zigzag leaning 45.0° has folded to 109° and the one leaning 15.0° to 151°: the upright zigzags lead, and every one reaches flat as the straight creases do ×1

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.

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.

Parallel creases forgive every lean

A corrugation graded so its tapered columns pile evenly leans every zigzag line its own amount, and the question left was whether it still folds rigidly on the Miura's one freedom. It does, and it could not have failed to. With straight creases that are parallel, a zigzag reflecting in them keeps one angle to them all the way along, so the table of cosines that decides rigid folding has a constant in every row whatever the leans. The published rule that such rows must be copies of one another holds only when the straight creases fan. What grading costs is the shape of the motion: the zigzags close at their own rates, the strip bends up to 24 degrees out of its plane early in the fold, and the leaf opens from its ends.

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.

The slant belongs to the line

A corrugation's columns pile 2h ⁄ (w sin 2a) deep a row — twice the row height over the column's width times the sine of twice the slant — so a tapered corrugation at one slant piles its narrow columns deeper than its broad ones. The slant looked like a property of the whole pattern. It is a property of each zigzag line: at a corrugation's vertex the row creases are one straight line and the zigzag leans the same amount either side of upright, so the angles pass Kawasaki's condition whatever each zigzag's own lean. Leaning every zigzag for the columns beside it takes a 24-column leaf taper from 55 per cent of its footprint four deep a row, at the best single slant, to 62, and cuts the part piled deeper than four from 22 per cent to 3.

The slant that stacks in one depth

A folded Miura's rows lie exactly on one another, and its columns each lie one step along the folded strip from the last, so the pile at any point is the rows times the number of column images over it — on average two over the sine of twice the slant. When that number is an even whole number the pile is one depth everywhere but the ends: two columns deep at 45 degrees, four at exactly 15. When it is odd the pile stays mixed, because neighbouring columns lean opposite ways. The printed Miura at 20.05 degrees sits beside the odd count three, and its graded pile is that and nothing more; at 15 degrees the same single-freedom mechanism stacks every panel away from its ends over one depth.

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.

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