Origami & Folding
what one sheet can do
Generator

Closing and being solid are two conditions

A generator in the rigid folding library, called 8 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.

mesh-collision 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

Closing and being solid are two conditionsQuadrilateral meshes solved so that every closure holds to within a millionth of a radian, each followed through its motion and asked whether any two panels that share no crease pass through one another. Most are solid the whole way. One is not solid anywhere, and the equations that were solved cannot tell it from the others.5 of 6 solved meshes are solid at every angle sampledthe bar is the deepest interpenetration found anywhere in the motion, in panel widthsmesh 11closes to 9e-14solid at every anglemesh 17closes to 1e-121.18 — panels 3:0 and 3:2, 2 steps apartmesh 19closes to 6e-14solid at every anglemesh 23closes to 5e-14solid at every anglemesh 27closes to 2e-12solid at every anglemesh 71closes to 4e-12solid at every angle

view: "survey", steps: 20

Closing and being solid are two conditionsQuadrilateral meshes solved so that every closure holds to within a millionth of a radian, each followed through its motion and asked whether any two panels that share no crease pass through one another. Most are solid the whole way. One is not solid anywhere, and the equations that were solved cannot tell it from the others.5 of 6 solved meshes are solid at every angle sampledthe bar is the deepest interpenetration found anywhere in the motion, in panel widthsmesh 11closes to 9e-14solid at every anglemesh 17closes to 1e-121.18 — panels 3:0 and 3:2, 2 steps apartmesh 19closes to 6e-14solid at every anglemesh 23closes to 5e-14solid at every anglemesh 27closes to 2e-12solid at every anglemesh 71closes to 4e-12solid at every angle

view: "survey", seeds: [11, 17, 19, 23, 27, 71], steps: 20

Closing and being solid are two conditionsQuadrilateral meshes solved so that every closure holds to within a millionth of a radian, each followed through its motion and asked whether any two panels that share no crease pass through one another. Most are solid the whole way. One is not solid anywhere, and the equations that were solved cannot tell it from the others.5 of 6 solved meshes are solid at every angle sampledthe bar is the deepest interpenetration found anywhere in the motion, in panel widthsmesh 11closes to 9e-14solid at every anglemesh 17closes to 1e-121.18 — panels 3:0 and 3:2, 2 steps apartmesh 19closes to 6e-14solid at every anglemesh 23closes to 5e-14solid at every anglemesh 27closes to 2e-12solid at every anglemesh 71closes to 4e-12solid at every angle

view: "sweep", seeds: [11, 17], from: 0.08, to: 2.7, steps: 24

How far one panel is through anotherThe length of the segment two panels share where they interpenetrate, along the whole of a solved mesh's motion, for the mesh that collides and for one that does not. The second is flat at zero. The first never reaches zero at any fold angle, so there is no part of the motion where the object could be built out of solid panels.0.511.522.500.51the driving crease's fold angle, radiansoverlap, in panel widthsmesh 17through itself throughoutmesh 11solid throughoutboth close to within a millionth of a radian at every angle drawn

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 collision is an order

Paper passing through paper is treated here as a thing that happens during a motion and is caught by watching for it. At the flat state it is not an event at all: it is the absence of an ordering, and it can be proved rather than observed. Four of the six quadrilateral meshes this site solves for rigid folding place perfectly and admit no ordering of their nine panels — so every one of them must pass through itself, and none of them was ever driven to find out.

Closing is not building

A quadrilateral mesh solved so that every loop closes to within a millionth of a radian is a mesh whose fold angles are consistent. It is not necessarily an object. One of the six solved here drives a panel through another at every angle of its motion — there is no part of the fold at which it could be made of solid panels — and the pair that crosses is two steps apart in the sheet, where nothing evaluated at a vertex could see it.

Refused at one lettering

Four of six quadrilateral meshes here have no arrangement of their nine panels — established by searching every ordering, at the labelling each mesh arrived with. Enumerate every labelling instead and two of the four fold perfectly well at a different one. What was reported as a fact about four meshes is a fact about two meshes and two labellings.

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