A unit that folds is not a tessellation

A tessellation is a unit repeated, so it is tempting to check the unit and stop. The unit is small, the conditions are local, and every vertex of the pattern is supposedly a copy of every other.

That last clause is where it goes wrong, and the waterbomb tessellation shows exactly how, with a number attached.

The family being searched #

The waterbomb tessellation is a square grid with both diagonals in every cell, every horizontal grid line creased and none of the vertical ones. Its geometry is fixed; what is not fixed is which creases are mountains.

The assignments considered here are the ones a tessellation ought to have — the ones that repeat. A horizontal crease’s letter depends on the parity of its row. Each of the four half-diagonals of a cell has a letter depending on the parity of the cell. That is nine bits, so 512 candidate rules, and each one can be built at any size and checked exhaustively.

This is the site’s standing habit applied at one level up. A pattern’s assignment is found rather than remembered, because the assignment most people would draw for the preliminary base cannot fold; a tessellation’s assignment rule is found the same way, for the same reason.

What the small patch cannot see #

A two-by-two patch has five interior vertices: four cell centres and one grid corner.

That single corner is the problem. The corner vertex is not one thing repeated. Its six creases are two halves of a horizontal grid line, whose letter depends on the parity of its row, and four half-diagonals arriving from the four surrounding cells, whose letters depend on the parities of those cells. So the corner comes in four varieties, according to whether its row index and its column index are odd or even.

A two-by-two grid has interior corners only at row one and column one. It contains one of the four varieties and says nothing whatever about the other three.

A three-by-three grid has corners at rows one and two and columns one and two, which is all four. That is why the count falls from fifty-six to thirty-two between those two sizes and then stops falling: every larger patch contains the same four varieties and nothing new.

Where the twenty-four die #

Taking one of the doomed rules and looking at its four corners settles it completely.

On a three-by-three patch its corner vertices come out at two mountains and four valleys, four mountains and two valleys, two and four again — and, at the fourth, zero mountains and six valleys.

Maekawa requires the two counts to differ by exactly two. Three of the four corners manage it and the fourth does not, by the widest possible margin: every crease at that vertex is the same letter, so the difference is six. The vertex is developable, its sectors satisfy Kawasaki, the big-little-big lemma has nothing to object to, and the pattern is dead anyway.

That is a satisfying failure to find, because it is the one the waterbomb’s structure predicts. Two of the four conditions hold by construction everywhere in this pattern and a third is vacuous, so the whole question of which assignments fold is a question about counting mountains and valleys — and the rules that die, die on the count.

It is also a failure with no ambiguity in it, which is worth noting because most near-misses in this subject are numerical. A vertex whose alternating sums come to 179.4° and 180.6° is a judgement call about tolerance; a vertex with six creases of the same letter is not. The rule is not marginal, not sensitive to how the angles were computed, and not something a slightly different implementation would decide differently. It is simply wrong, and it passed a two-by-two patch.

Forty-three per cent of the rules that pass on a two-by-two patch are wrong in that way. That is not a proportion anybody would guess, and it could not have been argued for without counting.

Four corners, four bits, and none of them independent #

The two counts are worth reading as powers of two, because the arithmetic says how much the small patch actually knew.

Nine bits give 512 rules and thirty-two survive, so the full set of conditions removes exactly $\log_2(512/32) = 4$ bits — one, pleasingly, for each variety of corner vertex.

The two-by-two patch leaves fifty-six, which removes $\log_2(512/56) = 3.19$ bits. So the small patch already supplies four-fifths of the evidence and still passes forty-three per cent of the rules it accepts in error.

That is the shape of the finding rather than a restatement of it. The last fifth of the constraint is not a refinement; it is the difference between a census that is right and one that is wrong about nearly half of what it reports.

And the corners overlap heavily #

The counts also say something the corner argument on its own does not.

If the three unseen corner varieties each imposed an independent halving, fifty-six would fall to seven. It falls to thirty-two — a factor of 1.75 rather than eight — so the four corner conditions are very far from independent, and most of what the three unseen ones demand is already demanded by the one the small patch contains, together with the cell centres.

That redundancy is why the count stops. A pattern whose corners genuinely constrained one another independently would keep losing rules as it grew, and there would be no size at which a census could be trusted. Here the four varieties exhaust the pattern’s vocabulary, three of the four are largely implied by the rest, and the residue is twenty-four rules.

The right lesson is therefore not that bigger patches are safer. It is that a patch must contain every kind of vertex the pattern makes, and that the smallest patch which does is three by three — after which nothing is learned by growing it at all.

Counting rules is not counting assignments #

This site already counts how many mountain-and-valley assignments of a fixed pattern survive the local conditions, and it is worth separating that question from this one, because the two are easy to conflate and behave differently.

How many assignments fold asks about a fixed pattern and lets every crease choose independently. The number of candidates is two to the power of the crease count, it grows explosively with the pattern, and what the local conditions do to it is throw away a shrinking fraction while the absolute number of survivors grows.

The census here asks something else. The candidates are rules, there are 512 of them at every size, and the pattern’s growth does not add candidates — it adds tests. So the count of survivors can only fall, and what is being measured is not the strength of a filter but whether the sample was representative.

The distinction matters because the two answers point in opposite directions. The assignment count says the local conditions are a weak filter that leaves more and more survivors as a pattern grows. The rule count says a periodic pattern has very few genuinely different foldings — thirty-two, from 512 — and that a small patch overstates even that.

Both are true of the same pattern at the same time. They are counting different things.

Two different ways a tiling refuses #

This site now has two cases where a tessellation is more than its unit, and they fail in different places. Putting them side by side is the point of this rung.

In the twist census, the unit is foldable for every regular polygon and only three of them tile the plane at all. The refusal is geometric and happens before any assignment is considered: a pentagon does not tile, so there is no pentagonal twist tessellation to assign letters to.

In the waterbomb, the tiling is never in doubt — a square grid tiles — and the refusal is about the assignment. The unit folds, the tiling exists, and a rule that gives every cell the same treatment still produces a vertex somewhere that cannot fold.

So a tessellation can fail its unit twice over: the unit’s shape may not tile, and the unit’s assignment may not survive being tiled. Only the second is invisible on a small patch, which is why it is the one worth counting.

The other pattern here whose assignment had to be searched for is the hexagonal twist, whose unit folds and whose tiling is what the plane permits, and whose crease count is too large to enumerate whole — so a stated search budget stands where an exhaustive count would.

Which theorem was checked, and how #

The claim being tested is a shape rather than a number, and it is asserted as one.

The rule census must fall from the smallest patch to the next: strictly more rules on two-by-two than on three-by-three, or the figure refuses to draw. It must then stop falling: the three larger counts must agree. And the survivors must be a subset of the earlier set rather than a different set of the same size, which is the condition that would catch a search whose results were varying for some reason other than the patch.

Then one dying rule is taken individually and made to demonstrate the mechanism: it is shown passing on the two-by-two, failing on the three-by-three, and the two patches’ interior-vertex counts are compared to establish that the failure is at a vertex the small patch lacks.

The whole census is exhaustive rather than sampled. There are 512 rules, each pattern has a few dozen vertices, and checking every rule at every size costs a fraction of a second — which means the counts are exact and the words “fifty-six” and “thirty-two” are not estimates.

There is a bar above the one the census applies, and none of the survivors has cleared it.

What this does not establish #

Three limits, and the first is the one most likely to be over-read.

The surviving thirty-two pass the local conditions and that is all. Every one of them satisfies developability, Kawasaki, Maekawa and the big-little-big lemma at every interior vertex of every patch tried, and that is not the same as folding flat. The global question is NP-hard and no census of vertices reaches it.

Only periodic rules of one family were searched. The nine-bit family was chosen because a tessellation is meant to repeat with its cell. A rule with a longer period, or an aperiodic one, is outside the search and nothing here says whether such a thing folds. The number 512 is a property of the question asked, not of the pattern.

Nothing here says the count is stable forever. Three sizes agreeing is evidence and not proof. The argument that it should be stable is the parity one — a three-by-three patch already contains all four varieties of corner vertex and all the neighbour arrangements those varieties have — and that argument is what makes the agreement expected rather than lucky. It is worth having both: the reasoning says where the count should settle, and the census says where it does.

It is worth keeping the two limitations apart, since growing the patch answers only one of them.

The test worth carrying #

The practical form of all this is a rule of thumb for anybody checking a repeating pattern, and it is short.

Grow the patch until the count of survivors stops falling, and until the survivors are a subset of the previous set. If the count is still falling, the patch is too small. If the survivors are not a subset, something is varying that should not be.

The second half matters as much as the first and is easier to forget. A census that returned thirty-two rules at every size, but a different thirty-two each time, would be reporting a bug rather than a pattern — and the counts alone would look perfectly convincing.

The underlying reason the test works is the boundary. A vertex on the edge of the patch carries no conditions at all, so a small patch is not a small pattern; it is a much less constrained one, and it hides failures rather than scaling them down. The rule census is that fact made countable.

Whether the rule family was the right family #

The nine-bit family is a choice and it deserves defending, because the number 512 is entirely a consequence of it.

The choice made here is that a rule may depend on parities: the parity of a horizontal crease’s row, and the parity of a cell’s position on the checkerboard. That is the smallest interesting family — anything less is a rule that treats every cell identically, and a rule that treats every cell identically cannot produce the alternating structure a flat-foldable pattern needs.

Two alternatives were available and each would answer a different question. A family with period three rather than two would have many more members and would contain everything found here, since a period-two rule is also a period-six rule; searching it would establish whether the pattern has foldings that need a longer repeat. And an unrestricted search over individual creases would answer the assignment question rather than the rule question, and would be intractable at any interesting size — a four-by-four waterbomb has seventy-six creases, which is two to the seventy-six candidates.

So the family is the smallest one capable of producing an answer, and the counts belong to it. What the census establishes is not “the waterbomb has thirty-two foldings” but “the waterbomb has thirty-two foldings that repeat with its cell”, which is the useful statement for a tessellation and is a smaller claim.

The distinction is easy to lose in the reporting and worth keeping, because it is the same distinction the essay is about at one level up. A search restricted to a small family finds what that family contains; a check run on a small patch finds what that patch contains. Both are honest as long as the restriction is stated with the number.

One tessellation on this site is immune to the whole problem, and saying why sharpens the rule.

Who has counted this, and when #

Enumerating mountain-and-valley assignments is standard in the computational treatment of flat-foldability and the counting problem for a single vertex is classical. What is less commonly done is enumerating rules rather than assignments, and the reason is that it only makes sense for a periodic pattern — which is to say, for tessellations, which are a comparatively recent object of study.

The habit of checking a patch large enough to contain every vertex neighbourhood, rather than large enough to look convincing, belongs to the same literature and to crystallography before it, where the equivalent statement — that a unit cell must contain every distinct environment — is old and completely standard.

What is this repository’s is the specific census: 512 rules, four patch sizes, the counts 56 and 32, and the demonstration that the twenty-four that drop out die on Maekawa at a corner the small patch does not contain.

What the count would be worth knowing exactly #

There is one number in this essay that is exact and one that is not, and the difference is worth marking.

The counts — 512, 56, 32 — are exact. Every rule in the family is built and checked, nothing is sampled, and the arithmetic behind them would be the same on any machine.

What is not exact is the claim that thirty-two is where it stops. Three patch sizes agreeing is strong evidence and the parity argument explains why they should agree, and neither is a proof that a patch of forty by forty would not remove one more. Establishing that would need an argument about the pattern rather than a census of it: something of the form every vertex neighbourhood a waterbomb produces already appears in a three-by-three patch, which is a statement about the tiling and is very likely true and is not demonstrated here.

That is the shape of most claims on this site and it is worth being explicit about it once. The computation says what happened; the reasoning says what should happen; and where the two agree the result is worth publishing with both stated, so that a reader can see which part would have to give if the answer were wrong.

Where the ladder goes next #

Sideways, to what a corrugation costs, where the waterbomb is measured against the other tessellations this repository can fold on the one quantity a deployable is bought with.

And back to the conditions, because the same lesson has a second form.

The tapered corrugation folds as one row and fails as four, for exactly the reason twenty-four rules pass on a two-by-two patch: the vertex that objects was not in the picture. That case is the cleaner statement of the two because nothing is being counted — a single pattern, drawn three sizes, passing and then failing.

The waterbomb census is the harder-won statement because it puts a number on how much a small patch flatters, and because the number is one nobody would have guessed.