Origami & Folding
what one sheet can do
Generator

Shrinking a shape that turns back on itself

A generator in the designing a base library, called 10 times across 2 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.

split-events 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

Shrinking a shape that turns back on itselfAn outline with a reflex corner, drawn with the shrunk copies of itself that define its straight skeleton, and the track that corner takes. Every other corner moves inward; this one moves out, and where it reaches an edge on the far side the shrinking region breaks into two.the outline, shrunk — and the one corner that moves outwardwhat the shrink finds6 edges, 1 of them meeting at a corner that turns back3 skeleton nodes8 arcs traced by the cornersthe last of them forms at 0.150 of a sheet

view: "wavefront", shape: "ell", steps: 5

Shrinking a shape that turns back on itselfAn outline with a reflex corner, drawn with the shrunk copies of itself that define its straight skeleton, and the track that corner takes. Every other corner moves inward; this one moves out, and where it reaches an edge on the far side the shrinking region breaks into two.the outline, shrunk — and the one corner that moves outwardwhat the shrink finds6 edges, 1 of them meeting at a corner that turns back3 skeleton nodes8 arcs traced by the cornersthe last of them forms at 0.150 of a sheet

view: "census", shapes: [triangle, pentagon, house, dart, ell, star]

Six outlines and the skeletons they shrink toEvery outline this site folds and cuts, convex and not. The bar is how many arcs its straight skeleton has; the note says how many corners it has, how many nodes the shrink produces, and whether a mountain-and-valley assignment satisfying every vertex condition exists.the bar is the arcs of the outline's straight skeletonthe triangle33 edges · 1 nodes · 30 letteringsthe pentagon55 edges · 1 nodes · 420 letteringsthe house75 edges · 3 nodes · 1 letteringthe dart75 edges · 3 nodes · no lettering foldsthe ell86 edges · 3 nodes · 1 letteringthe star1010 edges · 1 nodes · 420 letteringsa corner that turns back adds arcs and, on one of these, takes away every lettering that folds

view: "wavefront", shape: "star", steps: 4

Shrinking a shape that turns back on itselfAn outline with a reflex corner, drawn with the shrunk copies of itself that define its straight skeleton, and the track that corner takes. Every other corner moves inward; this one moves out, and where it reaches an edge on the far side the shrinking region breaks into two.the outline, shrunk — and the one corner that moves outwardwhat the shrink finds10 edges, 5 of them meeting at a corner that turns back1 skeleton nodes10 arcs traced by the cornersthe last of them forms at 0.122 of a sheet

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.

One cut for a star

The fold-and-cut construction here could reach a triangle, a pentagon and a house, and refused everything that turned back on itself, because shrinking an outline with a reflex corner needs an event the shrink did not implement. With split events it reaches a five-pointed star — ten creases through one point, four hundred and twenty letterings that fold, and every edge of the outline landing on one line to a part in 10^16.

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

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