Thirty-two rules, one object

The waterbomb base repeated across a sheet becomes a tessellation with two kinds of vertex and an assignment that has to be searched for rather than remembered. The search is over repeating rules: nine creases in the repeating unit, each a mountain or a valley, so five hundred and twelve rules to try.

Fifty-six of them pass every condition on a two-by-two patch and thirty-two on any larger one, and the twenty-four that die were never foldable — the small patch simply did not contain the kinds of vertex that refuse them.

Thirty-two is a satisfying number to arrive at and it invites a reading it does not support. Thirty-two rules are not thirty-two tessellations.

What was compared, and how #

A rule is a marking. What a reader folds is an object, and the two are compared by folding.

Each of the thirty-two surviving rules is built into a three-by-three patch, folded, and its panels are placed where the folding puts them. The comparison is then on the set of folded panels: for each panel, where its centroid landed and how much area it has, sorted so the order the panels were built in cannot matter.

All thirty-two produce the same set. Not the same statistics — the same set, compared centroid by centroid to four decimal places and area by area to five.

That is a stronger statement than the one the summary numbers make. Two different foldings could easily share a footprint, a layer profile and a maximum depth; sharing every panel’s position and area is a coincidence that does not happen by accident.

What the thirty-two are, then #

They are the same object described from two sides and with a symmetry group’s worth of relabelling on top.

The first half of that is measurable and clean. Sorted by how many of their creases are mountains, the thirty-two fall into two groups of sixteen — one at twenty-two mountains and one at twenty. Swapping every letter turns a folding into the same folding seen from the other side of the paper, and every theorem in this subject is blind to which side that is, so a rule and its reverse are one object with two markings. The two-colouring of the panels is the same fact in a different vocabulary: the sheet has two sides and the marking chooses which one faces out.

The second half is the sixteen. A three-by-three patch of the waterbomb tessellation has symmetries — translations of the repeating unit, reflections of the grid — and a rule that has been shifted or reflected is the same rule written down at a different starting corner. What survives the two together is one object.

Why this is not obvious from the counting #

The funnel from 512 to 56 to 32 looks like a filter tightening, and each of its steps is a genuine filter: a rule dies at each stage because some vertex of some patch refuses it. It is tempting to read the survivors as the answer set.

They are the answer set to the question that was asked, which was which repeating markings satisfy every vertex condition. The question a folder has is which tessellations can this pattern make, and the two differ by everything the marking carries that the object does not: an origin, an orientation, and a side.

The distinction has appeared on this site from the other direction. One marking, many objects shows a single crease pattern with several distinct folded states, which is the same gap opened the other way — a marking under-determining an object. Here a marking over-determines one, and both are symptoms of the fact that the marking and the object are different kinds of thing.

The four kinds of vertex, and why they do the filtering #

The waterbomb tessellation’s grid has two kinds of vertex by construction — the corners of the squares, where four creases meet, and the centres, where six do — and each of those appears in two orientations once the diagonals are drawn. That is the four kinds, and a patch that omits one of them cannot refuse the rules that only that kind refuses.

Which is exactly what the two-by-two patch does. Fifty-six rules pass on it and thirty-two on a patch large enough to contain all four, so twenty-four rules are killed by vertices the small patch does not have. The number of rules is therefore a function of what the patch contains rather than of how big it is: a four-by-four patch has the same four kinds and gives the same thirty-two.

That is the sense in which the filter is finished at thirty-two. Growing the patch adds more copies of the same four vertices, and a rule that satisfies each of them once satisfies each of them everywhere, because a repeating rule makes every copy identical.

Which theorem was checked, and how #

Every rule is enumerated, not sampled. Five hundred and twelve is the whole space of repeating markings on the unit, and the survivors are those that pass the vertex filter on the whole patch — developability, Kawasaki, Maekawa and the big-little-big lemma at every interior vertex of it.

The folding is built rather than assumed. Each survivor’s panels are placed by the folding, and the comparison uses the placed polygons; a rule whose folded state could not be constructed would be reported rather than dropped.

The comparison is on panels rather than on summaries. Footprint, maximum depth and layer profile all agree across the thirty-two, and any of them alone would be weak evidence. The set of centroids and areas is what is asserted, and the figure refuses to draw if the two rules it shows do not match.

The mountain counts must differ. If all thirty-two had the same count, the two-groups reading would be wrong and the figure would be claiming something the data does not support.

What a folder sees #

The arithmetic above is about markings, and it is worth translating into what happens at a table.

Fold the waterbomb tessellation from any of the thirty-two rules and the object in the hand is the same: a sheet drawn into a thirty-first of its area, thirty-two layers deep at its worst point, with the same panels lying in the same places. Turn it over and it is one of the other sixteen. Rotate the sheet a quarter turn before starting and it is another of the same sixteen.

So the choice a folder appears to be making when they pick a rule is not a choice about the model. It is a choice about where to start and which way up, and the two together are what the thirty-two are counting.

That has one practical consequence, and it is the reason the essay is not merely a caution about arithmetic. A designer looking for variety in a tessellation will not find it by searching the space of repeating rules — the search returns one object however long it runs. Variety comes from changing the grid, the vertex kinds or the periodicity of the marking, all of which change the object rather than its description. The twist family finds its variety exactly that way, by changing the tiling under the construction rather than the letters on it.

Where the model stops #

One patch size, one tessellation. The comparison is on three-by-three patches of the waterbomb pattern. A larger patch has the same repeating unit and the same rules; whether the same collapse happens on a different tessellation is not measured here, and there is no reason from the argument to expect it to be universal — the collapse depends on how much symmetry the pattern has.

The folded state is the flat one. Everything compared is the fully folded object; two rules that agree there could in principle differ on the way, and nothing above follows a motion. That is a real gap and it is the one a folder would notice first, because the waterbomb tessellation’s charm is what it does part-folded.

Nothing here is a decision about the whole sheet. Every rule was filtered by the conditions at its vertices, which are necessary and not sufficient — the global question is intractable and none of the thirty-two has been proved to fold as a sheet.

A repeating rule is not the only kind of assignment. The 512 are markings that repeat with the unit; a tessellation can be lettered non-periodically, and the space of those is enormously larger and is not enumerated anywhere.

And the object is one object of this pattern. The waterbomb tessellation also makes a tube, and a tube is not a flat folded state; the comparison above is between flat foldings and says nothing about the cylinder a pattern chooses.

What the picture cannot show #

Two patterns drawn side by side show two markings and cannot show that their foldings agree — the agreement is a comparison of fifty-two panels’ positions, and a picture of the folded state would show one of them rather than the identity.

Worse, a picture of the folded object is the least informative drawing in this essay. Every rule gives it, so it distinguishes nothing; the interesting content is entirely in what the markings do not determine, and absence has no picture.

The idealisation, named #

The paper has two sides and no thickness, which is what makes “the same object from the other side” a well-defined operation: reversing every letter maps a folded state to its mirror, and no measurement in the subject distinguishes them.

Real paper has a front and a back — most origami paper is coloured on one side — and for a folder the two groups of sixteen are visibly different objects: one shows colour where the other shows white. So the collapse from thirty-two to one is a collapse in the geometry and a collapse from thirty-two to two in the hand, and which of those a designer cares about depends on whether the model’s colour matters.

The generalisation #

Counting the descriptions of a thing counts the descriptions. The step everybody skips is the quotient: given a space of markings and a group of operations that do not change the object, the number of objects is the number of orbits and not the number of markings. Here the operations are the grid’s symmetries and the reversal of every letter, and thirty-two markings turn out to be one orbit.

The reason it is easy to miss is that the filtering feels like it is doing the quotient. The letterings a pattern admits fall into pieces a local change cannot cross, and a piece is a much finer object than an orbit; the two notions are easy to run together and are not the same. Going from 512 to 32 removes markings that fail a condition, and a filter that removes five-sixteenths of a space looks discriminating; but the survivors are still markings, and no condition in the subject can tell a marking from its mirror or from its own translation, so nothing in the filter could have separated them.

The practical form, for anybody counting anything in this subject: before reporting a count, ask what group acts on the things being counted. If the answer is “several”, the count is an upper bound on the number of objects and may be a very loose one.

What it would have taken to notice sooner #

The collapse is visible in the numbers that were already published, if the right two are put beside each other.

Thirty-two survivors is two to the fifth. The repeating unit has nine creases, the conditions kill four degrees of freedom, and what is left is a five-dimensional space of markings — of which one dimension is the global reversal and the other four are the ways of writing the same repeating rule down at a different origin. Nothing about that arithmetic requires folding anything.

What folding adds is certainty. A count that happens to be a power of two is suggestive and no more; two markings whose fifty-two folded panels agree centroid by centroid are the same object, and the only way to know it was to build both.

That is the habit worth carrying rather than the arithmetic: when a count comes out at a power of two, look for the group before believing the count. And when the group is found, fold two members of an orbit and compare, because a group that was guessed at is a group that might have been the wrong one.

Why the count is a power of two, without invoking a group #

The closing advice — when a count is a power of two, look for the group — is good advice, and on this pattern there is a second answer that fits the numbers better, which is worth setting out because the group story does not quite close.

A group acting on the 512 markings and collapsing them to one object would need order at least thirty-two. The translations of a three-by-three patch number nine, and nine does not divide sixteen. So whatever produces the second factor of sixteen, it is not the patch’s translations alone.

The alternative is that the filter itself is linear, and Maekawa very nearly is. At a degree-four vertex the condition $|M - V| = 2$ with $M + V = 4$ leaves $M \in \{1, 3\}$ — which is exactly the statement that $M$ is odd. Odd is a sum modulo two, and a sum modulo two is a linear equation over the two-element field. At a degree-six vertex the same reduction gives $M$ even, which admits $M \in \{0, 2, 4, 6\}$ where Maekawa wants $\{2, 4\}$, so there the parity is the linear part of a slightly stronger condition.

The waterbomb patch has four kinds of vertex and a repeating rule makes every copy of a kind identical, so the whole filter is four equations. Nine free bits less four independent linear conditions is five, and $2^5$ is thirty-two.

Which the small patch confirms by failing #

The linear reading makes a prediction that the group reading does not, and the numbers already published test it.

If the filter were purely linear, every count in the funnel would be a power of two. The two-by-two patch gives fifty-six, which is eight times seven and not a power of anything. So something non-linear is at work there — and the non-linear part is exactly the one identified above, the degree-six exclusion of all-mountain and all-valley, which a small patch encounters in a different combination than a large one does.

That is the tell. A count of 512 falling to 32 through four vertex kinds is arithmetic about linear equations; a count of 56 in between is arithmetic about something else. Reading both as evidence of a symmetry group requires a group of order sixteen that nothing in the patch supplies, and reading them as a linear filter with one awkward clause requires nothing that is not already in the conditions.

Neither reading changes the finding — the thirty-two still fold to one object, and that was established by folding rather than by counting. What changes is the moral. A power of two is evidence of a linear condition at least as often as it is evidence of a group, and on this pattern it is the first, because Maekawa is a parity check wearing an inequality’s clothes.

Who found it, and when #

The waterbomb tessellation is traditional and belongs to nobody; the rule search here is this collection’s own and the funnel from 512 to 56 to 32 was published in the essay one rung down.

The collapse to a single object was found while the rules were being counted for a different reason. That is worth recording plainly: nothing prompted the comparison except the habit of folding a thing before believing a number about it, and the number was correct — thirty-two rules do survive — while the sentence it invited was not.

Where the ladder goes next #

Two directions, both short.

The first is the partly folded state, which is where this pattern is interesting and where the comparison above says nothing. Thirty-two rules that agree at the flat state need not agree on the way there, and the waterbomb’s motion is the reason anybody folds it.

The second is the other tessellations. Every tessellation on this site is described by a repeating rule and every one of those rules has been counted; how many of those counts are counts of objects is a question with an answer per pattern, and the answer for the waterbomb turns out to be one.