Generator

Four corrugations, every repeating rule tried

A generator in the tessellations library, called 57 times across 11 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

rule-sweep 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: "families"`

`view: "bits"`

`view: "degree", show: "families", families: [miura, leaf, yoshimura, waterbomb]`

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.

rule 0 fails the count at 15 vertices, and the angle condition and the smallest-sector lemma hold at every one of them ×3
rule 22 satisfies every condition at every one of the pattern's interior vertices ×3
every clause refuses something the clauses before it let through, and the last leaves exactly the 32 that fold ×2
every rule of the family that folds is among the 32 its clauses allow, and 6 of those do not fold — the clauses are not the whole condition ×2
of those clauses, 3 were already asked for by the vertices ×2
16 of 64 rules survive, and they are exactly the rules whose column creases change letter every time they cross a row ×1
16 of 64, 16 of 64, 26 of 64, 32 of 512 rules survive ×1
438 of the 4,096 rules alternate strictly round some vertex of degree eight, and at every such vertex the eight panels round it are forced into a ring, each above the next ×1
880 rules fail the count at grid points and nowhere else, and 112 of those close their ring of eight there ×1
960 alternate round some cell centre, and the four panels there are forced into a ring every time ×1
a rule closes a loop of panels exactly when it alternates round some vertex, and every vertex it alternates round fails the count ×1
a separate listing of all twenty-four orders of those four panels finds none that works ×1
all sixteen of the Miura's rules have a stacking, on both patches drawn ×1
among the rules that pass the vertices, those with a stacking are exactly those in which every half-diagonal differs from its reflection in the horizontal crease it meets ×1
and the waterbomb given one letter per whole diagonal folds under none of its 32 rules, while 24 of them alternate round a cell centre ×1
at a degree-four vertex with one opposite pair tied the counting theorem keeps four of eight patterns and they are a parity; at degree six with three pairs tied it keeps six of eight, which no parity picks out ×1
developability holds at every interior vertex — its sectors close to 360° — 13 checked ×1
every loop any repeating rule closes is four panels long and goes round one interior vertex of degree four ×1
every rule refused anywhere in this table is refused by the counting theorem alone — the angle condition and the smallest-sector lemma hold at every failing vertex ×1
every rule the parities let through and that does not fold is refused by one of the 2 prohibitions, and each prohibition refuses some rule the other does not ×1
for every family the counting condition read off each kind of vertex picks out exactly the rules that fold, and so does the recipe — 2 parities and 0 prohibitions for the Miura fold, 1 parity and 2 prohibitions for the Yoshimura pattern, 3 parities and 2 prohibitions for the waterbomb tessellation ×1
four of its panels are already refused, and removing any one of the four lets the rest stack ×1
from three by three on, the Yoshimura's twenty-six vertex-passing rules hold 22 with a stacking and the waterbomb's thirty-two hold 16 ×1
given one letter per whole line, the same pattern folds under none of its 256 rules, and still 102 alternate round a grid point and 192 round a cell centre ×1
Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 13 checked ×1
Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 13 checked ×1
no rule that passes every vertex alternates round any vertex, of either degree ×1
of the 144 rules that pass every vertex on a 3-by-3 patch, 144 have a folded state, found by a search that places the panels one at a time and keeps every rule a stack must keep ×1
of the 64 ways to letter a grid point with its row and column each carrying one letter, 24 give five of one letter and three of the other, 2 alternate, and none does both ×1
on 6 two-by-two patches, where every stacking can be listed, each stacking the search returns is one of the listed ones ×1
on every patch drawn, every rule that passes the vertex conditions was decided: a stacking found, or the search for one exhausted ×1
one test reads three bits of the rule and folds nothing; the other builds the folded sheet and walks its arcs looking for a circle ×1
rule 0 does not fold, which is what the two-clause recipe says of it: an even number of zigzag classes are mountains, and no course carries the letter all four zigzag classes share ×1
rule 3 folds, which is what the two-clause recipe says of it: an even number of zigzag classes are mountains, and no course carries the letter all four zigzag classes share ×1
rule 54 has no folded state on three by three cells ×1
run in order, the 1 parities and 4 prohibitions leave exactly the 22 rules with a stacking ×1
run in order, the 5 parities leave exactly the 16 rules with a stacking ×1
stated vertex by vertex every kind of degree-six vertex is needed, and at least one family's parities let fewer prohibitions do the work — 2 for 2 kinds of degree-six vertex on the Yoshimura pattern, 2 for 4 kinds of degree-six vertex on the waterbomb tessellation ×1
the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 13 checked ×1
the column widths, the row heights, the row count and the zigzag angle are all changed here, and the table of surviving rules does not move ×1
the families whose vertices are all of degree four have affine survivors and the families with degree-six vertices do not — the Miura fold affine, the tapered leaf affine, the Yoshimura pattern not affine, the waterbomb tessellation not affine ×1
the families whose vertices are all of degree four have affine survivors and the families with degree-six vertices do not — the Miura fold affine, the Yoshimura pattern not affine, the waterbomb tessellation not affine ×1
the families whose vertices are all of degree four have affine survivors and the families with degree-six vertices do not — the waterbomb tessellation not affine ×1
the families whose vertices are all of degree four have affine survivors and the families with degree-six vertices do not — the Yoshimura pattern not affine ×1
the family whose vertices are all of degree six closes none at all, because a repeating rule gives a vertex's two course-halves the same letter and they sit opposite each other ×1
the family whose vertices are all of degree six writes none, because half of six is odd and a straight line through the point has to carry one letter ×1
the four panels are a chain joined by three creases ×1
the grid with both diagonals under rule 275 is put past all four theorems before it is drawn ×1
the recipe for the waterbomb tessellation — 3 parities and 2 prohibitions — allows exactly its 32 folding rules of 512 ×1
the recipe for the Yoshimura pattern — 1 parity and 2 prohibitions — allows exactly its 26 folding rules of 64 ×1
the row letters are free: all four ways of writing them appear among the survivors, and none of them appears without the alternation ×1
the rules that write a strictly alternating lettering round some vertex and the rules that close a circle of panels are the same rules, family by family ×1
the same 144 rules pass on every patch from 3 by 3 up, and they are the same rules, mask for mask ×1
the same 16 rules survive in every geometry, the same 38 close a circle, and every refusal is the counting theorem's alone ×1
the three letters put one panel under each of its two neighbours, two panels cannot lie between the pair across whose crease they reach, and the other two creases lie on one line, where they may not interleave ×1
the waterbomb tessellation keeps a power of two of its rules and is still not affine: three of the surviving rules add up to a rule that does not fold ×1
the waterbomb, at the vertices: the recipe found allows exactly these 32 rules ×1
the waterbomb, with a stacking: the recipe found allows exactly these 16 rules ×1
the Yoshimura goes the other way: 1 parity and 2 prohibitions at the vertices, 1 parity and 4 prohibitions with the layers ×1
the Yoshimura, at the vertices: the recipe found allows exactly these 26 rules ×1
the Yoshimura, with a stacking: the recipe found allows exactly these 22 rules ×1
they agree on all 128 rules of both grid families, and neither shares a line with the other ×1
under rule 325 the eight panels round the grid point at (2, 2) are forced into a ring, each above the next ×1
with the layers counted, the waterbomb's rules are closed under adding three and need 5 parities and no prohibition at all, where at the vertices they needed 3 parities and 2 prohibitions ×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 corrugation never backtracks

As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.

A recipe needs degree four

The Miura's letters are taught as a recipe of same-and-differ clauses, and the recipe is exact: the sixteen repeating rules that fold are precisely the rules two clauses allow. The Yoshimura's twenty-six cannot have such a recipe, because twenty-six is not a power of two. The waterbomb's thirty-two is a power of two and still has none — its clauses allow sixty-four rules and half of them fail. The difference is one vertex: at degree four the counting theorem leaves a parity, and at degree six it leaves 'not all alike', which no clause of that kind can say.

Eight creases close the circle too

A repeating rule can make the letters round a vertex alternate only when half the vertex's degree is even, and every family tested so far had vertices of degree four or six. The square grid with both diagonals has eight creases at every grid point. Of its 4,096 repeating rules, 438 alternate round one, and at every such point the eight panels are forced into a ring, each above the next — 112 of them with every other vertex of the sheet passing. The prediction holds. And the same pattern described with one letter per whole line folds under no rule at all, which corrects what the rule's description was said to forbid.

Half the recipe is decoration

Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.

Half the waterbomb's rules have no stack

The recipes for the waterbomb and the Yoshimura were written for the repeating rules that pass the vertex conditions, and nobody had asked those rules for a folded state. Asked, sixteen of the waterbomb's thirty-two have none, and four of the Yoshimura's twenty-six. Four panels are enough to refuse a waterbomb rule, in five lines, by forcing two creases on one folded line to interleave. And the waterbomb's surviving rules turn out to be exactly what a recipe of its kind was proved unable to describe: five same-and-differ clauses and no prohibition. Counting the layers shortens the waterbomb's recipe and lengthens the Yoshimura's, whose refusals are not local at all.

Sixty-four rules, sixteen fold

The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.

The leaf's rules are the Miura's

A plicate leaf packs into a corrugation that is broad in the middle and narrow at both ends. Enumerate every repeating mountain-valley rule it admits and the table is the Miura fold's table, rule for rule, number for number — and it stays that table when the leaf is redrawn with even columns, a violent taper, taller rows or a steeper zigzag. The plant's geometry cannot reach its own letters.

The loop is in the rule

Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.

The plant's pattern is not a hard case

A hornbeam leaf packs into its bud by corrugating, and the pattern it uses gives up a consistent lettering at nine, twelve, fifteen, eighteen, twenty and twenty-four steps on nine, twelve, fifteen, eighteen, twenty and twenty-four panels. Nothing about the plant's problem is combinatorially difficult, and saying so is worth as much as finding a case that is.

Two sentences for the Yoshimura

No recipe made only of same-and-differ clauses picks out the Yoshimura pattern's twenty-six folding rules, because its vertices have six creases. A recipe allowed one other kind of sentence does, and it is short: an even number of the four zigzag classes are mountains, and no course carries the letter all four zigzags share. That is three clauses, one parity and a prohibition for each of the pattern's two kinds of vertex, and it allows exactly the twenty-six. The waterbomb tessellation, with four kinds of six-crease vertex, needs three parities and only two prohibitions, because its parities do half the prohibiting.

Where a rule can close a loop

Three corrugation families have repeating rules whose letters send four panels round in a circle, and one has none at all. The one that has none is the one whose vertices are all of degree six — and the reason is that a straight line through a point carries a single letter under any repeating rule, while a strict alternation round six creases needs the two halves of that line to differ.

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