Origami & Folding
what one sheet can do
Tessellations

The rule that breaks the count

The waterbomb tessellation has five hundred and twelve repeating rules for its letters and thirty-two of them fold. A hundred and twenty of the other four hundred and eighty send four panels round in a circle — the shortest circle a crease pattern can have — and every single one of those hundred and twenty has broken Maekawa's count at the very vertex the circle goes round. The theorem that closes the shortest circle, caught doing it, a hundred and twenty times.

Assumes Thirty-two rules, one object and The loop a vertex cannot close.

19 min read 8 figures Local rules, global behaviourThe pattern is the object

The waterbomb tessellation is a grid of waterbomb bases, and a repeating rule decides which way each of its creases goes: whether the horizontal crease in each row is a mountain or a valley, and which of the four half-diagonals in a cell take which letter, alternating between cells. Nine bits, so five hundred and twelve rules.

Thirty-two of them fold. That was measured by running the conditions at every vertex of a patch large enough to contain every kind of vertex, and it is the number the tessellation is drawn from: rule forty-six is the one on the printed sheet.

The waterbomb, tiledThe waterbomb base repeated across a sheet: every cell carries both its diagonals, every row of the grid is creased, and the columns are not. That last omission is what makes the corner vertices degree six rather than degree eight, and it is the pattern's whole character.two kinds of vertex, both forced16 of degree 490°, 90°, 90°, 90°9 of degree 690°, 45°, 45°, 90°, 45°, 45°40 mountain and 36 valley creases14.3 sheet-widths of foldingmountainvalleyraw edge
Waterbomb tessellation — sheet 160×160 mm — 40 mountain, 36 valley, 2290.19 mm of crease
Fig. 1 The waterbomb tessellation at rule forty-six. Its vertices come in two kinds — degree six in the middle of each cell, degree four where cells meet — and the second kind is what this essay is about.

The other four hundred and eighty are usually left as the rules that fail, which is a category and not a finding. Asking one more question of them turns it into one.

A hundred and twenty circles of four

A pattern’s letters can contradict themselves: each crease says which of its two panels lies above the other, and a circle in those statements is a proof that no order of the layers exists. The circle can in principle be as short as four panels, which is a chain going round a single interior vertex of degree four.

Run that test over all five hundred and twelve rules. A hundred and twenty of them close a circle. Every one of the hundred and twenty circles is four panels long, and every one goes round a single interior vertex of degree four. There is not one longer circle in the whole census.

Every contradiction has an even number of panels in itThe length of every circle found in the layer relation, over every population of crease patterns here. No odd length occurs, because the panels of a flat-foldable pattern two-colour; and no length of four occurs, because a circle of four goes round one vertex and the counting theorem closes it.the bar is how many circles of that many panels were found726 circles, from 6 panels to 32, over every pattern family measured here4 panels0round one vertex — Maekawa forbids it5 panels0odd — the two-colouring forbids it6 panels21129.1% of the circles measured7 panels0odd — the two-colouring forbids it8 panels21129.1% of the circles measured9 panels0odd — the two-colouring forbids it10 panels8211.3% of the circles measured11 panels0odd — the two-colouring forbids it12 panels9513.1% of the circles measured13 panels0odd — the two-colouring forbids it14 panels304.1% of the circles measured15 panels0odd — the two-colouring forbids it16 panels314.3% of the circles measured17 panels0odd — the two-colouring forbids it18 panels172.3% of the circles measured19 panels0odd — the two-colouring forbids it20 panels141.9% of the circles measured22 panels81.1% of the circles measured24 panels152.1% of the circles measured26 panels71.0% of the circles measured28 panels20.3% of the circles measured30 panels20.3% of the circles measured32 panels10.1% of the circles measuredthe empty rows are not rare cases — they are lengths that cannot occur, and each has its own reason
Fig. 2 Circle lengths measured across every family of patterns in this collection, where the four-panel row is empty. The waterbomb rule census is the one place a four turns up, and it turns up a hundred and twenty times.

And none of the hundred and twenty is among the thirty-two that fold.

It is worth being precise about how those two numbers sit together, because “none of the hundred and twenty is among the thirty-two” understates it. The thirty-two are not merely disjoint from the hundred and twenty; none of the thirty-two closes a circle of any length at all. The layer test is silent on every rule that folds and fires on a hundred and twenty of the rules that do not.

Five hundred and twelve rules, one objectThe repeating rules for a waterbomb tessellation, counted at each stage: every rule, the ones that pass the conditions on a small patch, the ones that pass on a patch containing every kind of vertex, and the number of distinct folded objects those produce. The last number is one.counting rules and counting objects are different measurementsrepeating rulesnine binary choices, one per crease of the repeating unit512pass on a small patchevery vertex of a two-by-two patch satisfies every condition56pass on a larger oneand on a patch that contains all four kinds of vertex32folded objectscounted by comparing the folded panels, not the letters1
Fig. 3 The thirty-two rules that survive, folded and compared panel by panel. Every one of them has a folded state, so none of them can have a circle in its letters — but the layer test is not reading that fact off their foldedness. It reads their crease lists, one pass each, and finds nothing.

Which is not a coincidence

The shortest chain of panels a crease pattern has is the one round a single interior vertex, and that chain can never be sent round consistently by a labelling that satisfies the conditions. The panels alternate in orientation going round a vertex — the paper turns over at every crease — so the arrows agree only if the letters alternate too, and a strict alternation has equal counts of mountain and valley.

Maekawa’s theorem says the counts differ by two. So the alternation is refused, and with it the four-panel circle.

Break the count and the circle becomes available immediately. That is what a hundred and twenty of these rules do: they put two mountains and two valleys round a degree-four vertex, which Kawasaki accepts, which the big-little-big lemma accepts, and which the folded paper does not.

Mountain, valley, mountain, valley — and the theorem that refuses itA single interior vertex lettered by the rule every beginner is taught: alternate the letters round the point. The three conditions are evaluated on it. Two of them hold; the counting one does not, and it is the same lettering that would have closed a loop among the panels.the letters alternate strictly round the pointwhich is the only lettering that could send every arc the same way round the panels4 creases · 2 mountain · 2 valleyKawasakiholdsthe sectors alternate to zerobig-little-bigholdsthe smallest sector's two creases differMaekawarefused2 mountains and 2 valleys — the difference is 0, not 2the pattern is drawn without verification, because the point of it is that it does not fold
Fig. 4 A degree-four vertex labelled by alternation, with all three conditions evaluated. Kawasaki holds because the sectors alternate to zero; the big-little-big lemma holds because the smallest sector’s two creases differ; the count is two and two.

The four-panel circles in this census are that vertex, embedded in a tessellation, a hundred and twenty times over.

Where the four panels are

The circle is short enough to point at, and pointing at it is worth more than the argument.

A waterbomb cell is a square with both diagonals drawn and a crease across the middle of each row. Four cells meet at their corners, and the vertex where they meet has four creases coming into it: the two half-diagonals from the cells on either side, and the two horizontal creases running left and right. Between those four creases sit four panels, one from each cell — a quarter of each of the four cells that share the corner.

That is the chain. Going round it, the paper turns over four times and comes back, and whether the four arrows agree depends on the four letters. Two mountains and two valleys arranged alternately, and they agree; anything else, and they do not.

The arcs the letters force, with no circle in themOne arrow per crease, drawn from the panel that must lie below to the panel that must lie above. The direction is decided by the letter and by whether the near panel has been turned over, so the whole picture is read off the crease list without placing a single layer.each arrow points from the lower panel to the higher one9 panels · 12 creases · 12 arcsno loop — the letters are consistent among themselvesthe arrows are the whole of the test — nothing here asks which panels lie over which
Fig. 5 The same kind of picture on a small pattern: panels drawn with one arrow per crease, from the panel that must lie below to the one above. A four-panel chain like the waterbomb’s corner is the smallest closed thing such a picture can contain.

A rule that assigns those four creases by alternation is not doing anything perverse. It is applying the most natural pattern there is to four creases meeting at a point, and it is the pattern a beginner is told to use. The census is, among other things, a count of how many of the five hundred and twelve possible repeating rules make that particular mistake somewhere: a hundred and twenty.

Maekawa alone, on all four hundred and eighty

The claim above names one theorem, which is only worth doing if the other two are demonstrably not involved. On this census they are not, and the check is exhaustive rather than sampled.

Every one of the four hundred and eighty failing rules fails Maekawa and nothing else. At every vertex where any rule fails, Kawasaki holds and the big-little-big lemma holds. Four hundred and eighty rules, thousands of failing vertices between them, and the refusal is the same one every time.

That is a fact about this pattern rather than about the subject — the waterbomb’s angles are fixed by its grid, so Kawasaki is satisfied at every vertex whatever the letters are, and only the counting condition is left with anything to say. But it makes the census unusually clean: the rules divide into those that satisfy a count and those that do not, and the layer contradiction lands entirely on the second group.

What the folder sees, and does not

A reader who prints one of the hundred and twenty rules and tries to fold it has a specific experience, and it is worth separating from the arithmetic.

They will not get as far as the layers. The rule fails the count at between twelve and twenty vertices of a four-by-four patch, and a vertex whose mountains and valleys are two and two simply will not close: the paper buckles, one flap has nowhere to go, and the fold stops. That is the count refusing, in the hand, immediately, and it is why nobody has ever needed the layer argument to reject these rules.

What the layer argument adds is a second, independent statement about the same sheets, made without folding anything and reaching the same verdict. Two refusals that agree are worth more than either alone — not because agreement makes them more likely to be right, but because they are computed from different things. The count reads four letters at one point. The circle reads the whole crease list and follows a chain of panels. Neither consults the other, and on a hundred and twenty patterns they land on the same vertices.

No lettering of one vertex puts its panels in a loopEvery mountain-valley labelling of a single interior vertex, at three degrees, counted twice: how many satisfy every condition the subject has, and how many of those force a directed loop among the panels round the point. The second count is zero at every degree.the bar is the letterings that pass every condition at the vertexnone of them forces a loop, because the one lettering that would is the one Maekawa forbidsdegree 48 pass · 0 loop16 letterings · 8 admissible · the alternation fails Maekawa alonedegree 630 pass · 0 loop64 letterings · 30 admissible · the alternation fails Maekawa alonechecked at equal sectors and at a skew of 0.18 radians, so the count is not a fact about a symmetry
Fig. 6 Every labelling of a single vertex at the waterbomb’s own two degrees, four and six, counted twice: how many pass every condition, and how many force a circle. None does, at either degree — which is why the hundred and twenty rules that force one have all broken a condition first.

The same on two sizes

The census was run on a three-by-three grid of cells and on a four-by-four, and the two agree in every particular: thirty-two rules pass, four hundred and eighty fail by the count alone, a hundred and twenty close a four-panel circle, none of the passing rules closes anything.

That agreement matters more than it looks. The rules that die when the patch grows are a known phenomenon here: on a two-by-two grid fifty-six rules pass, and twenty-four of those stop passing on a three-by-three, because a two-by-two patch has one corner vertex inside it and a three-by-three has four. The conditions did not get stricter; the patch stopped hiding them.

The layer census does not have that problem, and the reason is instructive. What it is finding is a fault at one vertex, and a three-by-three patch already contains every kind of vertex the tessellation has. Growing the patch adds more copies of the same vertices and therefore more copies of the same circle; it cannot add a new kind of failure.

Two vertices, two failure modes

The waterbomb has two kinds of interior vertex and only one of them can carry this failure, which sharpens what the census is measuring.

The degree-six vertex at the centre of each cell has six panels round it, so its chain is six long. A six-chain going round one vertex is closed by the same argument — the alternation is what would orient it, and Maekawa refuses an alternation at degree six as firmly as at degree four. So the centre vertices never contribute a circle either.

The degree-four vertex where four cells meet is the one the hundred and twenty circles sit on. Not because it is more fragile as a vertex, but because its chain is shorter and therefore easier to close by accident: four letters have to come out alternating, which happens for a sixteenth of all labellings of those four creases, against a sixty-fourth at degree six.

So the census’s uniform answer — a hundred and twenty circles, all of length four — is telling something about which vertices are exposed rather than about which are weak. A pattern’s shortest chains are where its letters are likeliest to trip, and a grid of cells meeting at corners has the shortest chains there are.

What a rule is, and what the census says about that

There is a reading of this that is about rules rather than about waterbombs, and it is the one worth carrying away.

A repeating rule is a promise that a local decision can be made once and applied everywhere. The waterbomb’s rule set is a complete enumeration of such promises for this pattern — every way of deciding the letters cell by cell, with the decision repeating — and the census sorts them into three classes rather than two.

Thirty-two keep the count and fold. A hundred and twenty break the count at a degree-four vertex and are refused twice: once by the count, once by the layers, and the second refusal is a consequence of the first. Three hundred and sixty break the count somewhere the layers do not notice, and are refused once.

The middle class is the interesting one because its two refusals are so far apart in cost and in kind. The count is four additions at one point. The layer circle is a proof about the whole folded object, obtained without folding anything. They are the same refusal wearing different clothes, and nothing in the way either is usually stated suggests they would be.

What the layer test catches, as a fraction

The three classes have sizes, and the sizes are worth reading as a measurement of the layer test rather than of the waterbomb.

Thirty-two rules fold and the test is silent on every one of them. That silence is neither luck nor a measurement: a rule that folds has a folded state, a folded state is an order of the layers, and an order cannot contain a circle. The test is sound — it never accuses a pattern that folds — and the soundness is a theorem rather than a result of this census.

The other direction is the measurement. Four hundred and eighty rules do not fold, and the test fires on a hundred and twenty of them, which is one failure in four. That figure is a theorem about nothing at all, and there is no reason to expect another pattern to reproduce it. It is the share of this population’s failures that fail in the one way this particular test can see.

A refutation procedure that is sound and incomplete is the ordinary kind, and stating both halves is the point of saying it. What is unusual is being able to put a number on the incompleteness, and that needs a population where every member’s true verdict is already known by other means. This one has that: the vertex conditions settle all five hundred and twelve.

Why a quarter, and whether it is the quarter it looks like

There is an arithmetic reason to expect one in four, and it is worth writing down beside the reason it is not yet established that this is it.

Take a single degree-four vertex and count its sixteen labellings. Eight satisfy Maekawa — the four ways of putting one mountain among three valleys, and the four the other way about. The remaining eight fail: six put two of each letter round the vertex, and two put all four creases in one letter. Of those eight failures exactly two alternate, and alternation is what closes the four-panel circle. So at one vertex, one count-failure in four is a circle and the other three are nothing.

The census reports a hundred and twenty circles among four hundred and eighty failures, which is also one in four, exactly.

That agreement is suggestive and it is not a derivation, because the two ratios are taken over different things. The first is over labellings of one vertex. The second is over rules, and a rule fails at between twelve and twenty vertices of a patch and is counted once however many of them are bad. For the two quarters to be the same quarter, the repeating rule would have to spread the corner vertices’ labellings evenly over the sixteen possibilities — plausible for a rule set assembled from independent bits, and not checked anywhere.

What would settle it is a census over vertices instead of over rules: label every degree-four corner in every failing rule’s patch, sort those labellings into the eight failing classes, and see whether two in eight comes back. That is the same enumeration with a different accumulator, it would cost nothing beyond what is already being computed, and it would convert a coincidence between two ratios into a fact about one of them.

Until then the modest statement is the honest one. The mechanism is exact at a vertex — breaking the count is necessary for the circle and alternation is what makes it sufficient, so most rules that break the count have no circle anywhere. Whether the population’s quarter is that vertex quarter showing through, or an accident of how nine bits happen to land, is a question this census was not accumulated to answer.

Why this census exists at all

There is a point about method here that the collection has met before in other forms.

The five hundred and twelve rules were enumerated to answer a different question: how many repeating labellings of the waterbomb tessellation fold, and whether the answer depends on how large a patch the question is asked of. That work produced a list, the list was used, and the four hundred and eighty failures were set aside as failures.

A population assembled to answer one question is the cheapest possible test bed for another, and it is a better one than a population built for the purpose, because nothing about it was chosen with the second question in mind. Had a set of patterns been constructed to demonstrate that Maekawa closes the four-panel circle, the demonstration would have been worth very little; the patterns would have been picked to show it.

How much room a pattern gives its letters to disagreeEvery pattern family here plotted by how many independent closed chains of panels it has against how often an independently drawn lettering agrees with itself. The count is Euler's relation on the panel graph and equals the number of interior vertices; it is read off the drawing before any letter is chosen.more chains is more chances for one of them to closethe printed shelftessellation patchesfold-and-cut outlinessheets folded at random00.2500.5000.7501255075100125independent closed chains of panelsshare of letterings that agree with themselvesa point at nought is nought of the draws taken, which is not a proof that no consistent lettering exists
Fig. 7 The families this collection measures letterings over, by how many independent chains their panels form. Every one of them was assembled for some other reason, which is the only thing that makes them evidence.

The patterns a checker is tested on is the standing complaint about the opposite habit: a test set drawn by the same hand as the checker, containing nothing the checker must refuse. This census is the case where an old population had something in it nobody had looked for.

What is not covered

Two limits on the claim.

The circle here is always four panels because a degree-four vertex is where it sits, and the waterbomb has degree-four vertices because its cells meet at their corners. A tessellation whose vertices are all of degree six would have no four-panel chain at all, and its shortest would be six — as the twists’ are. The census’s uniformity is a feature of this pattern’s grid.

And the hundred and twenty rules are not patterns that nearly fold. They fail the conditions at between twelve and twenty vertices apiece, so a folder would never get near the layer question. The value of the census is not that it catches something the vertex check misses — it catches strictly less — but that it shows the two failures coinciding, on a population large enough that coincidence is not an available explanation.

The refusal the letters can see, and the one only a search canSix developable quadrilateral meshes, each asked twice whether its panels can be stacked: once by reading the arcs its letters force, and once by searching every ordering. The letters agree with themselves on all six; the search refuses four of them.the bar is the nodes the ordering search visitedthe letters are consistent on every one of these, so the one-pass test says nothing about any of themmesh 37,4739 panels · 7,473 nodes · no order existsmesh 58,0079 panels · 8,007 nodes · no order existsmesh 89,3469 panels · 9,346 nodes · no order existsmesh 111,0159 panels · 1,015 nodes · an order existsmesh 141449 panels · 144 nodes · an order existsmesh 199,0629 panels · 9,062 nodes · no order existsa red bar is a pattern with no folded state, found only by visiting every ordering it might have had
Fig. 8 The other side of the same point: six patterns whose letters are all consistent and four of which have no folded state. The layer test catches strictly less than the search does, and on the waterbomb census it catches strictly less than the count does.

That is the shape of the whole finding. A theorem stated as a fact about mountains and valleys at a point turns out to be the thing standing between a crease pattern and the shortest impossibility its layers can have — and the evidence for it is a rule set that was enumerated for a different purpose entirely.

Where this goes next

counting rules and counting objects are different measurementsrepeating rulesnine binary choices, one per crease of the repeating unit512pass on a small patchevery vertex of a two-by-two patch satisfies every condition56pass on a larger oneand on a patch that contains all four kinds of vertex32folded objectscounted by comparing the folded panels, not the letters1 Thirty-two rules, one object Five hundred and twelve repeating rules for the waterbomb tessellation, fifty-six that pass on a small patch, thirty-two that pass on one containing every kind of vertex. Fold all thirty-two and compare their panels: the same panels, in the same places, with the same areas, every time. The rules are thirty-two labels on one object, and a count of them has counted the labels. the bar is the letterings that pass every condition at the vertexnone of them forces a loop, because the one lettering that would is the one Maekawa forbidsdegree 48 pass · 0 loop16 letterings · 8 admissible · the alternation fails Maekawa alonedegree 630 pass · 0 loop64 letterings · 30 admissible · the alternation fails Maekawa alonedegree 8112 pass · 0 loop256 letterings · 112 admissible · the alternation fails Maekawa alonechecked at equal sectors and at a skew of 0.18 radians, so the count is not a fact about a symmetry The loop a vertex cannot close A crease pattern's letters can contradict themselves, and the contradiction is never local. Enumerate every mountain-valley labelling of a single interior vertex at degree four, six and eight — a hundred and fifty pass every condition the subject has — and not one of them sends its panels round in a circle. The one labelling that would is refused by Maekawa, alone: Kawasaki holds on it and so does the big-little-big lemma. a 3 × 3 patch: 42 creases, 18 of them buriedthe bar is on a log scale, because the two numbers differ by four orders of magnitudepieces the 3×3 patch has262,144pieces containing a repeating rule32, one eachevery one of the 32 rules is in a piece no other rule is in Thirty-two rules, thirty-two pieces The waterbomb tessellation's surviving repeating rules fold to one object — same panels, same places, same areas. Put them in the space of letterings the patch admits and they occupy thirty-two different pieces of a quarter of a million, so no two of them can be reached from one another without unfolding the sheet.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

AssignmentEnumerationLayer orderingMaekawa's theoremNecessary conditionTessellationUnit cellWaterbomb