Generator

One node per panel is a table size

A generator in the what it costs to know library, called 48 times across 9 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.

table-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: "pleat"`

`view: "pleat", fills: [0.45, 0.62, 0.8]`

`view: "edge", cols: 6, nrows: 5`

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.

at a row height of 1.2 the sectors are 79.6°, 50.2°, 50.2°, 79.6°, 50.2°, 50.2° and 0 of them are strictly smallest, leaving 30 labellings ×5
the conditions leave 30 labellings at a vertex, and the cost holds between 0.76 and 0.91 nodes a panel ×5
the Yoshimura's row height decides its sector angles and nothing else: at 1.7320508 halves of a column the conditions leave 30 labellings a vertex and the pattern costs 57 nodes, and at 1.7320509 they leave 8 and it costs 19 ×4
the 4 families whose vertices keep eight labellings run from 0.23 to 1.00 nodes a panel ×2
√3 is 1.732050808, and the pattern everybody draws is on the expensive side of it ×1
26 patterns over 5 families: the conditions at a vertex leave 8, 8/30, 4 labellings, and the cost per panel follows that rather than the family ×1
35 patterns over 6 families: the conditions at a vertex leave 8, 4, 30 labellings, and the cost per panel follows that rather than the family ×1
38 patterns over 7 families: the conditions at a vertex leave 8, 8/30, 30, 4 labellings, and the cost per panel follows that rather than the family ×1
a crumple, deepening at 6 sizes: 8 nodes on 8 panels, 16 nodes on 16 panels, 18 nodes on 18 panels, 34 nodes on 35 panels, 38 nodes on 39 panels, 70 nodes on 71 panels ×1
it moves with the turn and the fill and with nothing else, so it is a property of the pleat rather than of the tiling it is laid over ×1
the box-pleating grid at 8 sizes: 4 nodes on 4 panels, 9 nodes on 9 panels, 16 nodes on 16 panels, 36 nodes on 36 panels, 64 nodes on 64 panels, 100 nodes on 100 panels, 144 nodes on 144 panels, 256 nodes on 256 panels ×1
the conditions leave 8 and 30 labellings at a vertex, and the cost holds between 0.89 and 0.93 nodes a panel ×1
the lemma asks for a sector strictly smaller than both its neighbours, and a Yoshimura vertex has four sectors of one size and two of another arranged so that the answer depends on which size is smaller ×1
the smallest sector at a twist corner is 46.1512° on the square, the triangular, the hexagonal and the elongated tilings alike, and at every pitch from 0.2 to 0.5 of a sheet ×1
the tapered leaf at 4 sizes: 12 nodes on 12 panels, 16 nodes on 16 panels, 20 nodes on 20 panels, 24 nodes on 24 panels ×1
the twist patches at 5 sizes: 26 nodes on 49 panels, 39 nodes on 83 panels, 39 nodes on 77 panels, 32 nodes on 62 panels, 80 nodes on 157 panels ×1
the waterbomb at 3 sizes: 28 nodes on 30 panels, 48 nodes on 52 panels, 71 nodes on 80 panels ×1
the Yoshimura, as drawn at 6 sizes: 16 nodes on 21 panels, 32 nodes on 36 panels, 47 nodes on 55 panels, 57 nodes on 65 panels, 82 nodes on 90 panels, 108 nodes on 119 panels ×1
the Yoshimura, tilted at 6 sizes: 9 nodes on 21 panels, 14 nodes on 36 panels, 17 nodes on 55 panels, 19 nodes on 65 panels, 24 nodes on 90 panels, 27 nodes on 119 panels ×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 knife edge nine decimals wide

Draw the Yoshimura with its rows 1.7320508 half-columns tall and each vertex admits thirty labellings and the pattern costs fifty-seven steps. Draw it at 1.7320509 and each admits eight and it costs nineteen. The number between them is √3, which is the proportion everybody draws — and below it the sectors are unequal and the lemma is still silent, because the small ones sit next to each other.

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.

Six creases and the same straight line

The one family here whose vertices are degree six was said to break the arithmetic that every other family obeys, on the strength of a single pattern. Built as a family — six sizes from twenty-one panels to a hundred and nineteen — the Yoshimura is exactly as linear as a grid, with no decision ever withdrawn. What degree changes is the constant, and it changes it in both directions depending on one angle.

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.

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