Thirty-two rules, thirty-two pieces

Thirty-two rules, one object settled what the waterbomb tessellation’s repeating rules are. Five hundred and twelve rules; fifty-six that pass on a two-by-two patch; thirty-two that pass on one containing every kind of vertex the pattern makes. Fold all thirty-two and compare their panels centroid by centroid and area by area: the same panels, in the same places, with the same areas, every time. They differ in how many of their creases are mountains — two counts, sixteen rules each — which is which side of the paper is showing.

That answered what they are and left open what a folder can do about them, which is a different question with a different answer.

A repeating rule is a lettering of the whole patch. So each of the thirty-two is a point in the same space the sampling instrument measures — the space of letterings, cut into pieces no local change can cross. Two numbers come out of putting them there, and both are worth having.

How large the space is #

A three-by-three patch of the waterbomb tessellation has forty-two creases, thirteen interior vertices and eighteen buried creases — creases with an interior vertex at each end, whose letters were settled when the pattern was drawn and which no local change can reach.

Two to the eighteenth is 262,144. So the letterings of this patch fall into a quarter of a million mutually unreachable pieces, and a folder holding one of them is in one.

Against that, thirty-two rules. A repeating rule is a rare kind of lettering rather than a representative one — four rules for every thirty-two thousand pieces — and it is worth saying so before anything else, because the word “rule” makes them sound like the natural inhabitants of the space and they are not. Almost every lettering a hand could reach on this patch does not repeat.

Thirty-two rules, thirty-two pieces #

The second number is the essay’s.

Take each of the thirty-two rules, read off the letters on the patch’s eighteen buried creases, and compare. All thirty-two signatures are different.

So the thirty-two rules occupy thirty-two different pieces. No two of them are reachable from one another by any sequence of local changes, which means a folder holding one waterbomb tessellation cannot re-crimp it into another of the thirty-two, however patiently, without opening the sheet out.

They fold to the same object and they are mutually unreachable. Those are independent properties and both are true.

The distinctness is close to forced, which is the good news #

Thirty-two signatures all different is reported as the finding, and it is worth asking how surprising that is, because the answer changes what kind of result it is.

Take thirty-two labels drawn at random from two hundred and sixty-two thousand possibilities. The chance that no two collide is about ninety-nine point eight per cent — thirty-two things in a quarter of a million boxes essentially never share one. So distinctness by itself carries almost no information: it is what would have happened whether or not anything structural were going on.

There is also a structural reason to expect it. A rule is nine bits applied uniformly across the patch, so two rules agreeing on all eighteen buried creases would have to differ only on creases that reach the sheet’s rim — and a rule that repeats has no way to differ there while agreeing everywhere inside.

That makes the result robust rather than lucky, which is the better property. The thirty-two rules occupy thirty-two pieces not because a search happened to find them apart, but because there was no arrangement in which they could have been together. A finding that would have been surprising if it came out the other way is worth more than one that would have been surprising either way.

How fast the rules thin out #

What is genuinely informative is the ratio, and its behaviour as the patch grows is worth stating because it is not a tendency but an exponential.

The rule count is fixed. Thirty-two survive on any patch containing all four kinds of vertex, and a larger patch adds copies of vertices already present rather than new kinds — so the count stops falling at a three-by-three and stays there for ever.

The piece count is two to the number of buried creases, and the buried creases grow with the area of the patch: each interior crossing contributes its share, and crossings multiply as the patch is enlarged in both directions. A three-by-three has eighteen and a quarter of a million pieces; enlarge it and the exponent grows quadratically in the patch’s side.

So the share of letterings that are repeating rules falls as two to the minus the area. One rule per eight thousand pieces at three by three; one per a very much larger number at four by four; and past a modest size the ratio has no name worth writing.

That is the sharpest form of the essay’s first observation. A repeating rule is not a rare kind of lettering — it is an asymptotically negligible one, and the sentence the tessellation has thirty-two rules describes an object whose share of its own space is going to nothing exponentially fast in how much of the sheet is drawn.

They are unreachable, not stuck #

The distinction between being enclosed and being frozen has to be checked, or the finding is empty.

Take one rule’s lettering and try every change the patch admits: for each of the 114 pairs of creases meeting at an interior vertex, flip both and re-check the whole pattern. Sixteen of the 114 survive.

So each rule has sixteen legal changes available to it. It can be pushed. What it cannot do is reach another rule, because every one of the sixteen leaves the buried letters exactly as it found them — and the buried letters are what distinguish the thirty-two.

And what those sixteen changes lead to is worth stating: letterings that are not repeating rules. A rule’s neighbourhood in this space is made entirely of non-rules, so the first thing a folder does to a repeating tessellation destroys its repetition.

What a signature is, and why it separates them #

The comparison between rules is made on eighteen letters, and it is worth being explicit about which eighteen and why they are the right ones.

A crease is buried when both of its ends are crossings rather than points on the sheet’s rim. Flipping such a crease disturbs two vertices at once, and the smallest change the conditions permit — two creases flipped together — has only one other crease to spend, so it can repair one of the two disturbances and not both. That is the mechanism, and it is why the buried letters are constant across a piece and take every value between pieces.

So the signature of a lettering is its eighteen buried letters, read in a fixed order. Two letterings with the same signature may or may not be reachable from one another; two with different signatures certainly are not.

That is the direction the argument needs. A claim that two things are unreachable is established by finding an invariant that differs, and an invariant that differs is a proof of separation without any search at all. Nothing here walks between rules and fails — walking and failing would prove nothing — and the separation is a comparison of strings.

The check that the pieces are really there #

The signatures separate the rules; a separate measurement checks that the space really has as many pieces as the count says.

Draw independent letterings of the patch — solutions of the same constraint problem with the branch order randomised — and see how often two of them land in one piece. On a patch with eighteen buried creases the answer should be almost never, and it is: eight draws land in eight different pieces, and forty land in forty.

That is a floor rather than an estimate, and the essay treats it as one. What it rules out is the failure this kind of prediction is exposed to — a count read off a graph that has drifted away from the object it describes — and it rules it out on the same patch the rules were measured on.

Which side is showing, and where that lives #

The thirty-two split sixteen and sixteen by mountain count — twenty mountains or twenty-two on a three-by-three patch — and that split is what the earlier rung identified as which side of the paper is showing.

Turning the model over swaps every mountain for every valley, so it maps each of the sixteen to one of the other sixteen. That is a genuine correspondence and it is not a route between them: turning a folded model over is not a change a folder makes to the sheet, it is a change the folder makes to where they are standing.

So the sixteen-and-sixteen structure is a pairing across the space rather than a path through it. Each rule has a partner it is indistinguishable from and thirty other rules it cannot reach — and the partner is the same object seen from behind.

Why the patch is three by three #

Every number above is for a three-by-three patch, and the size is not arbitrary.

A unit that folds is not a tessellation established the sizes: 512 rules pass at a single unit, 56 on a two-by-two patch, and 32 on any patch containing all four kinds of vertex the pattern makes — which a three-by-three does and a two-by-two does not. Below that the count is wrong because the small patch has not met the vertices that do the refusing; above it the count is stable and the enumeration is more expensive.

The piece count is not stable with size, and it should not be. A larger patch has more interior vertices, so more buried creases, so more pieces — while the number of surviving rules stays at thirty-two. So the rules get rarer as the patch grows, which is the expected direction and is a reason to quote the ratio at a stated size rather than as a property of the tessellation.

What a folder finds #

The result has a table-level statement and it is testable in about ten minutes with a printed pattern.

Fold a waterbomb tessellation. Look at it. Now try to change which way one of the central diagonals goes without letting the sheet open — push, crimp, reverse a crease at a time. What happens is that the change either refuses immediately or propagates into a neighbouring cell and refuses there, and the pattern does not settle into a different repeating one.

The sixteen legal changes are the reason the sheet feels as though it might be about to go: there are things to push and they move. The eighteen buried creases are the reason nothing comes of it.

That is a considerably more specific claim than “the tessellation is stiff”, and it is a claim about the letters rather than about the paper. It survives with the paper replaced by anything that folds.

Sixteen changes that lead nowhere #

The sixteen legal changes each rule admits are worth looking at rather than counting, because what they lead to says what a folder is actually holding.

Every one of the sixteen flips two creases meeting at one interior vertex. Every one leaves the patch satisfying all four conditions at all thirteen of its vertices. And every one produces a lettering that is not a repeating rule: the two flipped creases now disagree with their counterparts in the neighbouring cells, so whatever regularity the patch had is gone at that point and nowhere else.

So the neighbourhood of a repeating rule, in this space, is a shell of near-rules — patterns that repeat everywhere except at one vertex. Which is the picture a folder has of a tessellation that has gone slightly wrong, and it is exactly right: a tessellation is one flip away from being a tessellation with a defect, and thirty-one flips away from being a different tessellation, and the thirty-one are not available.

Against the twists #

The twists have the same shape of result at a size a reader can hold, and putting the three together says what the buried-crease count is measuring.

A square twist has twelve creases, four buried, sixteen pieces — and its four buried creases are the four sides of its central polygon, so its sixteen pieces are the sixteen ways of lettering that polygon. That is the direction the middle turns, decided when the pattern was drawn.

A hexagon twist has eighteen creases, six buried, sixty-four pieces, on the same principle.

The waterbomb patch has forty-two creases, eighteen buried and 262,144 pieces, and its buried creases are not the sides of one polygon — they are scattered through the interior, one per pair of adjacent crossings. So its pieces are not indexed by a single feature of the folded object the way a twist’s are, and there is no equivalent of “which way the middle turns” to name them by.

That is the difference between a unit and a tessellation, stated in letters. A twist’s decisions are concentrated in one place and have a name; a tessellation’s are spread over the whole sheet and do not.

What a rule is, as a lettering #

The word “rule” hides what the object is, and unpacking it is what lets it be placed in a space of letterings at all.

A repeating rule for the waterbomb tessellation is nine bits: one deciding which way the horizontal creases alternate, and eight deciding the four half-diagonals at each of the two kinds of cell. Five hundred and twelve rules, which is 2⁹.

Applying a rule to a patch writes a letter on every crease of it — forty-two letters on a three-by-three — so a rule is a function from nine bits to forty-two letters, and its image is a single point of the space this essay measures.

That is why the thirty-two can be located at all. They are not a different kind of object from the other letterings; they are ordinary points that happen to be in the image of a very small map. Nine bits into forty-two letters is an enormous compression, which is what makes a repeating tessellation a thing a person can describe in a sentence — and it is also why the thirty-two are so sparse in the space they live in.

What this does not say #

Three limits.

The pieces are pieces of the locally admissible set. Every lettering counted satisfies the four conditions at every interior vertex; whether it folds globally is NP-hard and is not decided. The thirty-two rules are known to fold because the earlier rung folded them; the other quarter of a million pieces contain candidates.

Thirty-two distinct signatures is a lower bound on nothing. It is exact: the signatures were computed for all thirty-two rules and compared pairwise, so this is a complete statement about the rules rather than a sample.

And “unreachable” is under the move this site uses — two creases meeting at an interior vertex, flipped together. A folder who cuts the sheet has other options, and a cut is precisely the operation that joins pieces: one cut anywhere on this patch would take the piece count down by at least a factor of four.

The shape of the whole finding #

Three facts about the same thirty-two objects, found by three different measurements, and they fit together into something worth stating once.

They are a count of markings, five hundred and twelve reduced to thirty-two by the vertex conditions on a patch large enough to have all four kinds of vertex.

They are one folded object, established by folding all thirty-two and comparing panels.

They are thirty-two places, established by reading their buried letters, and no local change goes between any two of them.

Put together: the waterbomb tessellation has one folded object, thirty-two ways of writing it down, and no route between any two of those ways. Which is a small and complete demonstration that in this subject, being the same object and being reachable from one another are unrelated properties.