Origami & Folding
what one sheet can do
Generator

What a tie costs

A generator in the flat-folding library, called 20 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.

crimp-tree 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

What a tie costsThe vertex of equal sectors at each degree, with how many smaller vertices the reduction visits before it answers, against how many crimps the answer actually needs. The gold bar is the work and the green mark is the necessity.degreevertices visited per letteringcrimps needed48 of 16 fold32630 of 64 fold1038112 of 256 fold41410420 of 1024 fold2065121584 of 4096 fold12376The work grows by a factor of about 6.0 for every two creases added; the necessity grows by one.

view: "growth", degrees: [4, 6, 8, 10, 12]

What a tie costsThe vertex of equal sectors at each degree, with how many smaller vertices the reduction visits before it answers, against how many crimps the answer actually needs. The gold bar is the work and the green mark is the necessity.degreevertices visited per letteringcrimps needed48 of 16 fold32630 of 64 fold1038112 of 256 fold41410420 of 1024 fold2065121584 of 4096 fold12376The work grows by a factor of about 6.0 for every two creases added; the necessity grows by one.

view: "named"

Which vertices make the reduction chooseEvery lettering of each named vertex, sorted by whether the reduction was offered a choice. The vertices at no particular angles are decided without one; the vertices a folder actually meets are made of ties.vertexletterings that branchwidest choicethe preliminary base90°, 90°, 90°, 90°14 of 164a halved four-crease vertex60°, 60°, 120°, 120°4 of 162the waterbomb tessellation's odd vertex90°, 45°, 45°, 90°, 45°, 45°44 of 644a Yoshimura vertex60°, 60°, 60°, 60°, 60°, 60°62 of 646the preliminary base's centre45°, 45°, 45°, 45°, 45°, 45°, 45°, 45°254 of 2568a vertex at no particular angles13.8°, 68.7°, 71.1°, 94.2°, 95.1°, 17.1°0 of 641

view: "catalogue", step: 45

Every vertex a 45° grid admitsThe complete catalogue, with how many of each vertex's letterings make the reduction choose and how many of those choices decide anything. The second column is zero everywhere.sectorsletterings that branchdecided by the choice45° 45° 135° 135°4/16045° 90° 135° 90°0/16090° 90° 90° 90°14/16045° 45° 45° 45° 90° 90°44/64045° 45° 90° 45° 45° 90°44/64045° 45° 45° 45° 45° 45° 45° 45°254/2560

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 tie is not a decision

The crimp reduction decides a vertex by folding its smallest sector away, and where two sectors tie for smallest it has no forced move and must try each of them. That search is not rare — on the vertex at the centre of the first base anybody folds it happens for fourteen of the sixteen letterings — and it has never once changed the answer.

The other grid

Box pleating is drawn at forty-five degrees, and the twenty-two-and-a-half-degree grid is usually described as the same thing done finer. It is not a refinement, it is a different alphabet: five kinds of vertex become fifty-six, and the share of letterings whose decision needs a search falls from 60 per cent to 22. A finer grid is a larger vocabulary and a less ambiguous one.

Which vertices are the random ones

Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.

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