Every crease of the hexagonal patch, by length
fragment is one function. Everything below was drawn by it when the site was
last published, at settings 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.
With nothing specified
view: "transition", kind: "hexagonal"
view: "zoom", kind: "hexagonal"
view: "transition", kind: "hexagonal", periods: [0.33, 0.335, 0.34, 0.345, 0.35]
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- 6 of 72 patches over a stated grid of tiling, turn and pleat width carry a crease shorter than 0.001 of the sheet ×2
- so the threshold falls in a gap of a factor of 498, and moving it anywhere inside that gap changes nothing about which creases it catches ×2
- the shortest crease anywhere in the grid is 7.9·10⁻⁶ of a sheet, and the ones that carry none have nothing shorter than 0.00118 ×2
- 12 of the 138 creases of the triangular patch are shorter than 0.001 of its side, the longest of them 7.53·10⁻⁴ and the shortest crease above them 0.0207 ×1
- 12 of the 142 creases of the hexagonal patch are shorter than 0.001 of its side, the longest of them 5.9·10⁻⁵ and the shortest crease above them 0.0294 ×1
- across a change of 0.010 in the pitch the patch goes from 142 creases and 60 interior vertices to 130 and 54, losing a ring of twists ×1
- and at 0.34 it is caught mid-loss, with 12 creases left as fragments — which is the setting the collection prints ×1
- deleting its 12 invisible creases gives 130 and 54 — exactly the counts the repair was expected to produce — and two routes to one panel that disagree by 2.3 of a sheet width ×1
- drawing the longest of them at the width of a pencil line would need the sheet enlarged 1018 times ×1
- so a crease nobody can see is not a crease that does nothing: each of these separated two panels that a fold placed differently ×1
- the hexagonal patch carries 12 creases shorter than 0.001 of its side, in 6 places, every one of them where the clip caught a pleat within a hair of a corner of the sheet ×1
- the patch as drawn has 142 creases and 60 interior vertices and its panels close to 1.2·10⁻¹⁴ of a sheet ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
A patch on a knife edge
The tessellation patch this collection prints has twelve creases nobody can see. Move the pitch of its tiling by five thousandths and they are gone — and so is a whole ring of twists. The patch sits exactly on the moment a ring of the pattern passes through the edge of the sheet, and the blemish is what that moment looks like.
The crease the drawing cannot show
Twelve creases on a printed crease pattern are eight millionths of a sheet long. They are in every count the collection takes of that patch, they pass every theorem, and no printer resolves them and no hand folds them. They are also the only thing holding the folded sheet together.
The shortest crease is not a crease
A crease pattern's density is usually quoted as total crease length over sheet area, which treats a metre of folding as a metre whether it arrives as one long line or ten thousand short ones. Reading the lengths individually instead finds twelve creases on a printed patch that are shorter than a wavelength of light.
All the figures · The flat-folding field · The patterns a reader can fold