A boundary vertex is a strip
edge-vertex 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: "released", pattern: "square-twist"
view: "fan", bearings: [40, 100, 120], letters: "MVM", spanDeg: 180
view: "shelf"
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.
- each of the 12 creases is cut in turn and the whole pattern re-counted, so the comparison is over every cut rather than over a chosen one ×1
- each vertex is asked twice — as a ring and as the same sectors in a line — so the two counts are of one object arranged two ways ×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 pattern on the printed shelf is counted, so the comparison is over all 8 of them rather than over the ones that make the point ×1
- over 234 vertices, cutting the ring open never cost a lettering ×1
- the fan stops admitting every lettering at 50 of the 61 positions swept, and at every one of them a sector is strictly smaller than both its neighbours ×1
- the fan's creases are given in order round the vertex, so the sectors between them are the sectors of the paper ×1
- the pair differs in one letter, both members pass every condition this site checks, and one of them cannot be folded ×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 cut is a licence
What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.
A ring and a line
A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.
Most of a patch is edge
Between 34% and 91% of the vertices in the crease patterns drawn here sit on the edge of the paper rather than inside it, and on the tessellation patches — the figures that are meant to show what a repeating pattern looks like — it never falls below a third. A boundary is one unit deep whatever the unit is, so the share falls like one over the number of units across and reaches nothing at any size a page can carry.
The rim lies over less
A folded sheet's boundary is usually discussed as the place the theorems stop applying. It is also visible in the pile: a panel carrying a raw edge of the paper lies over fewer of the other panels than one that does not, on every printed pattern that has both kinds — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb tessellation, and never once the other way round.
The vertices nobody checks
Every figure on this site is gated on four conditions evaluated at every interior vertex, and the word interior has been carrying the whole sentence. On the printed patterns there are 105 vertices on the edge of the paper against 92 inside it, not one of them has ever been examined, and the condition that decides them has been available since the second phase of the collection.
Every generator · The flat-folding field · The patterns a reader can fold