Two ways of cutting a patch out of a tessellation
patch-boundary 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: "repair", kind: "hexagonal", periods: [0.5, 0.42, 0.34, 0.28, 0.22, 0.18]
view: "share", kinds: [square, triangular, hexagonal, rhombille, elongated]
view: "repair", kind: "triangular", theta: 0.35
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.
- the share falls from 86% to 49% as the units shrink from 0.5 of the sheet to 0.18, and never reaches zero ×3
- and the panels follow: 1.73 sheet widths apart one way against closing exactly the other ×2
- assembling whole twist units on the sheet and running the outstanding pleats to the rim puts 12 pairs of creases across one another; cutting the same tessellation out of the plane puts none ×2
- the same tiling on the same square packs 3.92 layers deep at 0.5 of the sheet and 4.80 at 0.18, as the share of units the rim cuts falls from 86% to 49% ×2
- 4 of 5 tilings put creases across one another when their patches are assembled from whole units, and none of them does when the patch is clipped out of the plane ×1
- a twist unit is either whole on the paper or cut by its edge, and the share that is cut is what a patch pays for being finite ×1
- compaction is measured off the placed folded state — the average number of layers over the footprint — rather than from the construction's own area accounting ×1
- the the square grid escapes either way, because its pleats run along the sheet's own directions and meet the rim without meeting each other ×1
- the the square grid leaves every rim in one direction and crosses nowhere; the rest leave in two or three, and every one of them crosses ×1
- two creases reaching the same edge of the paper can only cross if they arrive in different directions, so the count that matters at a rim is how many directions leave it ×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 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.
Cutting a patch out of a plane
A tessellation is infinite and a sheet is not, so every picture of one is a decision about where the paper stops. Assembling whole twist units on a square and running the outstanding pleats to the rim puts 12, 18, 12 and 5 creases across other creases on four of five tilings; generating the pattern over a larger region and clipping it puts none. The panels then place exactly — and what is waiting behind the repair is a different refusal that could not be asked about before.
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 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 edge is what makes it hard
Grids, crumples, leaves, corrugations and fold-and-cut patterns all give up a consistent lettering at one step per panel with no wrong guess anywhere. The one family that does not is a tessellation clipped to a square, and what separates it from the others is not disorder, not size and not irregularity. It is having a rim.
The property a patch does not have
A folded corrugation is described as a material — a packing ratio, a stiffness, a Poisson's ratio — and every one of those is a statement about an unbounded medium. Fold the same tiling at six sizes on the same square and the compaction climbs from 3.89 layers to 4.79 as the share of units the rim cuts falls from nine tenths to four, and it has not settled at the fine end. The number a patch gives is the material's number minus its own boundary.
The vertex the list does not have
Every condition this collection checks is asked at a vertex of a crease pattern, and a crease pattern is handed to the checker as a list of points and segments. A reader is handed ink. Read the same patterns the second way and eight printed sheets gain nothing at all — while four tessellation patches gain 12, 18, 12 and 5 vertices that nobody wrote down, every one of them a place where two creases were drawn across each other.
Every generator · The tessellations field · The patterns a reader can fold