Origami & Folding
what one sheet can do
Generator

A strip with no vertex in it, rolled until it meets itself

A generator in the rigid folding library, called 18 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.

rigid-limits 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 strip with no vertex in it, rolled until it meets itselfA strip of 10 panels creased along parallel lines, seen end-on at 4 crease angles. It has no interior vertex, so every local condition the subject checks is satisfied with nothing anywhere to evaluate. The panels picked out in the last frames are the ones passing through one another, which no condition on a neighbourhood could ever have reported.20° a crease0.50 turns of papernothing touching anything34° a crease0.85 turns of papernothing touching anything36° a crease0.90 turns of paper1 pair through one another50° a crease1.25 turns of paper5 pairs through one anotherone strip of 10 panels, seen end-onit laps itself at 36.0° a crease, which is where its cross-section closesevery panel is the same length in every frame; the only thing changed is how far each crease is turned

view: "chord"

Nothing, and then a whole panelThe length of the segment two panels share where they pass through one another, on a strip of 10 panels, against the angle each crease is turned by. It is zero for as long as the strip is short of a full circle and a whole panel width the moment it is not. No local test on the sheet changes at that angle, because no local test can see two panels at once.360°/10 = 36.0°40°010203040506000.511.522.5each crease turned by, in degreesshared chord, in panel widthsat the ringed angles the cross-section closes exactly and the panels land on one another rather than throughevery condition the subject checks at a vertex holds across this whole axis, because the strip has no vertex

view: "isometry", cols: 5, rows: 4, t: 0.5, eps: [0, 0.01]

One pattern folds and its neighbour does notThe largest amount by which any edge changes length between the flat pattern and the folded position, for an exact Miura and for the same pattern with its vertices moved by 0.01 of a panel. The first is at the last bit of the arithmetic and the second is 13 orders of magnitude larger — which is to say the moved pattern has no isometric folded position of this kind at all, and the exact one does.the pattern as it isevery edge keeps its length6.7e-16the same pattern, moved by 0.01and one of them cannot1.7e-21e-181e-161e-141e-121e-101e-81e-61e-41e-21largest change in any edge length, in panel widthswhat an isometry has to do, and what it manages5 × 4 panels, at 50% folded, every edge of both comparedthe moved pattern is fitted the best single panel its own edge lengths allow before being folded at allso the gap is not a bad choice of panel — it is what is left when the best choice has been made

view: "perturb", cols: 5, rows: 4, t: 0.5, eps: [0, 0.000001, 0.00001, 0.0001, 0.001, 0.01]

Nothing is small enough to be freeHow far the exact Miura's three equations are from being satisfied, against how far the flat pattern's vertices were moved, on logarithmic axes. The exact pattern sits at the last bit of the arithmetic. Every moved pattern sits on a straight line of slope one, fitted here rather than quoted — so the failure is first order in the displacement, and there is no displacement small enough to be free.1.7e-6 at 1e-61e-61e-51e-41e-31e-21e-61e-51e-41e-31e-2how far each vertex of the flat pattern was moved, in panel widthslargest edge-length errorfitted slope0.9999over four decadesat a displacement ofexactly zero the erroris 6.7e-16, which iswhere the arithmeticstops and not wherethe geometry does5 × 4 panels at 50% folded, with one set of displacement directions scaled across the decadesa slope of one is the claim: the failure is first order, so no displacement is small enough to be free

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 mechanism that closes on itself

A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.

Panels instead of paper

Flat-foldability asks whether a pattern can reach a flat state. Rigid-foldability asks whether it can get there without any face bending on the way. The second is much stronger, and everything that gets manufactured lives inside it.

Paper through paper

Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.

The only pattern that moves

A rigid motion is not a generic property of a folded pattern. Move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind at all — and the amount by which it fails is first order in the displacement, so no move is small enough to be free.

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