Cutting a square tessellation out of the plane, and gluing it up
rim-cost 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: "density", kinds: [square, triangular, hexagonal, elongated, rhombille], cells: [1, 2, 3]
view: "density", kinds: [triangular, hexagonal, rhombille], cells: [1, 2, 3]
view: "letters", kinds: [square, triangular, hexagonal, elongated, rhombille], cells: [1, 2, 3]
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.
- cutting the sheet out leaves every vertex where it was and every condition on it unchanged — 15 comparisons, the same count of vertices asked either way ×4
- twelve cuts across one period of the same drawing cost 25 to 33 nodes, a factor of 1.32 ×4
- the same drawing on the same rectangle costs 48 nodes cut out of the plane and 56,772 with its opposite edges joined, on 64 panels either way ×3
- 16 clipped patches over 4 tilings cost 0.52 to 0.67 nodes per panel, never rising with size ×2
- the cut version has 16 more letters to choose and asks the same 64 vertices the same conditions, so the difference is the letters and nothing else ×2
- crease count per unit area is not: it runs 104, 87, 81 on the the square grid as the cell grows, against the pattern's 69 ×1
- crease count per unit area is not: it runs 170, 145, 137 on the the triangular grid as the cell grows, against the pattern's 120 ×1
- crease length per unit area is the same at every cell size on every tiling here, because a crease the cut divides is two pieces that add back up ×1
- every one of those proofs is of something false, and what it costs grows faster than finding the lettering does ×1
- taking the cut away entirely costs 625 — where the cut falls is a detail and whether it falls is not ×1
- taking the cut away entirely costs more than 200,000 — where the cut falls is a detail and whether it falls is not ×1
- the collection's own cycle test proves the glued hexagonal cell has no consistent lettering, at 7 nodes for 12 panels, 9,619 nodes for 48 panels ×1
- the collection's own cycle test proves the glued square cell has no consistent lettering, at 3 nodes for 4 panels, 35 nodes for 16 panels, 3,455 nodes for 36 panels ×1
- the collection's own cycle test proves the glued triangular cell has no consistent lettering, at 7 nodes for 12 panels, 12,143 nodes for 48 panels ×1
- the grid, the leaf, the Miura and the crumple all sit at exactly one node per panel, so these are below the line rather than above it ×1
- what it adds is letters: 12, 24, 36 more creases to choose, being exactly the creases the cut divides ×1
- what it adds is letters: 4, 8, 10, 12, 20, 24, 30, 36 more creases to choose, being exactly the creases the cut divides ×1
- what it adds is letters: 4, 8, 10, 12, 20, 30 more creases to choose, being exactly the creases the cut divides ×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 count is not a length
Cut a rectangle out of a tessellation and it reports fifty per cent more creases than the pattern has, then twenty-five, then seventeen — converging on the truth from above and never reaching it. The crease length per unit area it reports is exact at every size, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not.
A loop that goes somewhere
Every crease says which of its two panels lies above the other, and a loop in those statements is a proof that the pattern has no flat folded state. On a sheet with no edge that sentence is false. The loops of a periodic pattern carry a lattice step each, and a loop that ends one cell to the right is not a contradiction — it is a stack of paper with no bottom layer.
A sheet with no edge
A twist tessellation repeats, so a rectangle of it is a description of the whole plane rather than a piece of paper. Joining the rectangle's opposite sides makes that explicit and produces an object every gate in this collection can read: twenty-five drawn panels become sixteen, forty crease pieces become thirty-two, sixteen vertices are all interior, and the three counts add to nothing.
A test imported without its hypothesis
The rule that a loop in a folded sheet's layer relations proves the pattern cannot fold arrives from the layer-ordering literature, where the sheet is a disc and the panels are finitely many. This collection took the rule and not the sentence that says which sheets it is about, then applied it for years to patterns whose whole interest is that they repeat.
Nothing grown was cut out of anything
A leaf's corrugation costs twelve steps on twelve panels, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four — exactly one per panel at every geometry, which is the most any pattern here costs. A tessellation patch costs half that, and the reason is that somebody cut it out of something. A leaf's creases stop at the margin because the plant stopped there.
One step per panel is a table size
Four families of crease pattern search at exactly one step per panel — a grid at nine sizes, a leaf, a Miura, six crumples — and it was read as a law about patterns that fill their own sheet. It is a number: the conditions at each of their vertices admit eight labellings. Where the conditions admit four, the cost is half. Where they admit thirty, it moves again, and the same pattern at two proportions demonstrates it with everything else held still.
Pruning on proofs alone
A search that discards a branch it cannot prove wrong is not a search. Deciding whether a periodic pattern's layer relations really contradict themselves is far dearer than the disc's one-pass test, so the cheap test is asked first — it is sufficient, so it settles almost everything — and the expensive one runs only on what the cheap one rejects. Five of nine steps on a small cell, fifty thousand of fifty-seven on a large one.
The bottom layer is at the rim
A hundred and sixty-nine panels of folded tessellation, and three of them have nothing underneath. All three touch the paper's edge, and the same is true on every tiling at every size measured. Which panel is at the bottom of a stack turns out to be a fact about where the sheet was cut rather than about the pattern, and the pattern itself has no bottom at all.
The cost of proving something false
A search closing its whole tree is the strongest result this collection can produce, and on a glued tessellation it produces one that is wrong. What it costs to reach is three steps at one period, thirty-five at four, three thousand four hundred and fifty-five at nine, and more than two hundred thousand at sixteen — growing far faster than the cost of finding the lettering it says does not exist.
The designer's grid is the dearest thing here
Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.
The edge was not what made it hard
Five families of pattern searched at one step per panel and a tessellation patch did not, and the property left standing after four alternatives were killed was having a rim. Measured under a fixed letter order the patches cost between a half and two-thirds of a step per panel, at every tiling and every size — below the line rather than above it, and the rim is why.
The lettering that was proved impossible
A search closed its whole tree on a glued square tessellation and reported that no mountain-and-valley assignment of it is consistent. Written onto ordinary patches of one, four and nine periods and handed to the four vertex theorems and a folded sheet rebuilt from scratch, the assignment it says cannot exist passes every check, on four tilings, up to fifteen hundred creases.
The most decided vertex here
Sixteen ways to letter four creases; Maekawa allows eight; the big-little-big lemma allows four. A twist polygon's corner is one of the few vertices in this collection where the second cut applies, so it keeps four labellings where a grid, a leaf, a Miura and a crumple all keep eight — and the family the collection long called difficult turns out to be the one whose conditions decide the most.
The rim is four letters a cell
Cut a rectangle out of a tessellation and it asks exactly the vertices the tessellation asks, exactly the same questions. What it adds is four free letters for every period of edge — the creases the cut divides, which become two independently answerable creases instead of one. Eight letters on a two-period square, sixteen on a four-period one, and nothing else about the two objects differs at all.
What the rim was doing
One rectangle of a twist tessellation, cut out of the plane in the ordinary way, gives up a consistent lettering in forty-eight steps. Join its opposite edges so that no crease is divided and the same drawing, at the same vertices, under the same conditions, takes fifty-six thousand seven hundred and seventy-two. The edge of the paper was never the difficulty. It was the slack.
Where you cut hardly matters
Slide the same rectangle across one whole period of the same tessellation and every position gives a different patch: different creases divided, different half-panels round the edge, panel counts from forty-nine to sixty-one. The cost of lettering them runs from twenty-five steps to thirty-three. Whether a cut is made changes the answer by three orders of magnitude; where it falls changes it by a third.
Every generator · The what it costs to know field · The patterns a reader can fold