The base that tiles

Almost everybody who has folded anything has folded a waterbomb base. It is the square with both diagonals and one midline, collapsed so that the corners meet — the start of the paper balloon, the frog, half the traditional repertoire’s inflatable models.

Tile it. Put a waterbomb base in every cell of a grid and let the cells share their edges, and the result is a tessellation that turns up in a stent, in a crush tube, and on any number of kitchen tables under the name magic ball.

The lines that are not there #

The pattern is a square grid with three families of crease and the third family is missing.

Both diagonals of every cell are creased, meeting at the cell’s centre.

Every horizontal grid line is creased, running the width of the sheet.

No vertical grid line is creased at all.

That last omission is the whole character of the pattern, and it is the thing most descriptions of it leave out. Put the vertical lines in and every grid corner would carry eight creases — four grid lines and four diagonals — which is a preliminary base repeated. Leave them out and the corner carries six.

Two kinds of vertex, both forced #

Tiling produces exactly two interior vertex types, and neither was chosen.

At the centre of a cell, the two diagonals cross: four creases, four sectors of ninety degrees.

At a corner of the grid, the horizontal line passes through and four diagonals arrive from the four surrounding cells: six creases, at 0°, 45°, 135°, 180°, 225° and 315°, giving sectors of 45°, 90°, 45°, 45°, 90° and 45°.

Both satisfy Kawasaki by inspection, which is the reason the pattern exists at all. At the cell centre the alternating sums are 90 + 90 and 90 + 90. At the corner they are 45 + 45 + 90 and 90 + 45 + 45. Both come to a straight angle exactly, and the generator measures the sectors off the built pattern and refuses to draw if either sum is off by half a degree.

The big-little-big lemma has nothing to forbid anywhere in the pattern, and it is worth seeing why. The lemma applies to a sector strictly smaller than both of its neighbours. At the cell centre every sector is equal. At the corner the sectors run 45, 90, 45, 45, 90, 45 around the vertex, so every 45° sector has another 45° sector on one side of it — none is strictly smallest, and the lemma is vacuous.

So the whole burden falls on Maekawa, which is unusual. In most patterns on this site the conditions share the work; here two of the four are satisfied by construction and one has nothing to say, and the entire question of which assignments fold is a question about counting mountains and valleys.

The panel count reports the missing lines #

The omission the essay is named for shows up in a count, and the count is worth deriving because it is the one place the absent creases are visible as a number.

A cell cut by both diagonals is four triangles. With every grid line creased, an $r$ by $c$ grid would have $4rc$ panels — sixty-four on a four-by-four.

The vertical lines are not creased, so the east triangle of one cell and the west triangle of its neighbour are one panel. Each row of $c$ cells therefore contributes $2c$ north-and-south triangles and $c+1$ merged east–west pieces, and

$$\text{panels} = r\,(3c + 1)$$

Four by four gives $4 \times 13 = 52$. Three by three gives $3 \times 10 = 30$. Both are the numbers the essay reports.

So the pattern has fifty-two panels where a fully creased grid would have sixty-four, and the missing twelve are exactly the $r(c-1)$ merges the absent vertical lines produce.

And the vertex count reports the two kinds #

The same arithmetic separates the two vertex types without anybody having to identify them.

Cell centres number $rc$; interior grid corners number $(r-1)(c-1)$. So

$$\text{interior vertices} = rc + (r-1)(c-1)$$

Four by four: $16 + 9 = 25$. Three by three: $9 + 4 = 13$. Again both are the reported figures.

The two terms are the two vertex kinds, degree four and degree six, and their ratio tends to one as the grid grows — so a large waterbomb tessellation is half cell centres and half grid corners, and the degree-six vertices that carry all the constraint are half of them.

That last reading is worth holding, because it says where the pattern’s difficulty is concentrated. The degree-four centres have equal sectors and admit letterings freely; the degree-six corners are where the sectors run 45, 90, 45, 45, 90, 45 and where Maekawa is doing the whole job. Half the vertices in the pattern are the ones that decide it, and that fraction does not fall with size.

It also explains the two-by-two trap the essay describes. A two-by-two patch has four centres and one corner, so its degree-six vertices are a fifth rather than a half — and a rule that fails only at corners has one chance to fail rather than nine.

The assignment is searched for #

The site’s standing rule is that a pattern’s assignment is found rather than stated, because the assignment most people would draw for the preliminary base cannot fold flat. The waterbomb needs the same treatment and gets it in a slightly different form.

A tessellation is meant to repeat, so the candidates considered are the ones that repeat: a letter for the horizontal creases depending on the row’s parity, and a letter for each of the four half-diagonals of a cell depending on the cell’s parity. That is nine bits — 512 candidate rules — and each is built and put past all four conditions at every interior vertex.

Thirty-two of the 512 survive. Not one, which is worth noticing: the pattern does not have an assignment, it has a family of them, and they are genuinely different foldings of the same lines.

The horizontal phase turns out not to matter at all — swapping every horizontal crease’s letter takes a surviving rule to another surviving rule — so the thirty-two are sixteen diagonal rules each available two ways. And the surviving rules are not the tidy ones anybody would guess. The rule this site prints has one diagonal of each cell changing assignment as it crosses the cell’s centre, which is exactly the trick that makes the Miura fold work and is exactly what nobody draws first.

There is a trap in that search and it has an essay of its own. Run it on a two-by-two patch and fifty-six rules pass rather than thirty-two, because a patch that small has only one grid-corner vertex and the extra twenty-four fail at corners it does not contain. A unit that folds is not a tessellation that folds, and the waterbomb is where this site measured the difference.

Which theorem was checked, and how #

Four things, in the order a build performs them.

The pattern is put through all four conditions at every interior vertex by the same checker every pattern here uses, and the two vertex types are then extracted by measuring how many creases meet at each interior point rather than by knowing where they are. A pattern that produced three vertex types, or one, would be refused as not being this pattern.

The sector sums are computed from the built coordinates and compared with a straight angle. This is the check that would catch a construction error in the tiling — a cell placed at the wrong offset, a diagonal drawn to the wrong corner — because such an error changes the angles and nothing else.

The rule census is run at four patch sizes and its shape is asserted: strictly more rules on the smallest patch, agreement between the larger ones, and the survivors a subset rather than a different set.

The folded state is computed by composing reflections, and its consistency is Kawasaki arriving from a direction the construction does not compute. On a four-by-four waterbomb there are twenty-five panels reachable two ways and the two routes agree to within six parts in a quadrillion.

The other tessellation unit this site prints makes a useful comparison.

The panels, and what they say about the pattern #

The pattern’s panels are all triangles, which is unusual here and worth a moment. Most of the patterns this site prints fold panels of several shapes at once, and a pattern whose panels are all one shape has an argument available to it that the mixed ones do not.

A cell of the pattern is a square cut by both diagonals into four triangles, and the horizontal grid lines are the cell edges. So the whole sheet is triangulated, and every panel is a right-angled triangle with legs in the ratio of the grid.

Triangles matter for a reason that has nothing to do with flat folding. A triangle is rigid: three edge lengths determine it completely, so a triangulated sheet of rigid panels has no way to deform within a panel and all of its motion is at the creases. That is why triangulated corrugations recur in everything anybody manufactures, and it is a property the waterbomb has and the Miura does not — a Miura’s panels are parallelograms, which are rigid as panels but which give the pattern a quite different kinematic character.

The two-colouring works here for the same reason it works everywhere: every one of the twenty-five interior vertices carries an even number of creases, four or six, and Maekawa forces that. The colouring is a check on the pattern’s construction as much as a fact about it — a tiling that had accidentally produced an odd vertex would refuse to colour, loudly, before any assignment was searched for.

What it costs and what it buys #

A four-by-four waterbomb has fifty-two panels, twenty-five interior vertices, forty mountain creases and thirty-six valley creases, and fourteen and a third sheet-widths of folding to do.

Folded flat it is thirty-two times smaller than the sheet and thirty-two layers deep at its deepest — and those are the same number, because the layer count integrated over the footprint is the area of the paper. Nothing about that is special to the waterbomb; what is specific is where it sits among the alternatives.

A Miura fold of comparable size folds six times smaller with about half the crease length. The waterbomb packs much harder and asks much more of the folder, which is the trade in one sentence: it has more creases per unit area, it stacks deeper, and it is correspondingly more work.

The crease length is the number a folder feels and it is worth quoting properly. Fourteen and a third sheet-widths means that folding a hundred-and-sixty-millimetre sheet of this pattern involves making something over two metres of crease, in seventy-six separate creases, every one of which has to be accurate. A systematic error of half a degree across that many folds is not a small matter, and the waterbomb’s reputation for being fiddly is this number rather than any subtlety in the pattern.

It also sets a floor on how fine the pattern is worth making. Doubling the grid quadruples the cell count, quadruples the layer count and doubles the crease length per unit area, while the paper consumed by the creases themselves grows with the crease count. Past some grid size — and it is a modest one for ordinary paper — the pattern stops closing because the material has run out rather than because the geometry failed. That limit is a property of the paper and the arithmetic together, and the geometry above is entirely silent about it.

The waterbomb is not the only pattern here whose interesting state is curved: the Yoshimura pattern’s folded state is also a tube, and it arrived at that shape by being what a crushed cylinder does rather than by being designed to.

What the flat state is not #

The waterbomb tessellation is the pattern on this site whose flat folded state is least interesting, and saying so is the honest limit of this essay.

Everything above is about a folded state in which every crease is fully folded and the sheet lies in a plane. That is the state the theorems are about and the state this repository can compute. It is not the state anybody uses the waterbomb tessellation for.

The pattern’s characteristic behaviour is partial: folded part of the way, it curls into a tube, and the tube’s radius is set by the pattern rather than by the paper. That is the property behind the stent-graft work and behind every crush tube built on it, and it belongs to the pattern’s kinematics rather than to its flat state. The next rung stays with the flat conditions; the tube is not something the machinery here computes, and this essay does not draw one.

What tiling a base buys, in general, is a property rather than a shape. A single waterbomb base is a shape somebody folds; a sheet of them is a material with a stiffness, a curl radius and a packing ratio, and none of those three is a property any one base has.

Who folded it first, and who noticed #

The base is traditional and old, and its date is the sort of thing this site is careful about. It is in the European and the Japanese traditions both, it appears under several names — waterbomb, balloon, fūsen — and no attribution is available or needed: it carries no authorship question, which is why this site can print it.

The tessellation is a twentieth-century object and a different matter. Repeating a base across a sheet is a modern habit, arriving with the shift from folding sequences to designed patterns, and the waterbomb tiling is one of the patterns that shows up repeatedly in that literature — as a corrugation, as a magic ball, and in the engineering work on deployable tubes.

Its most cited application is a stent graft: a tube that folds small enough to be threaded through an artery and opens to hold it. That is the deployable requirement in its most demanding form — large in use, small in transit, along a path nobody may be allowed to trust to chance — and the waterbomb answers it because its partly folded states are tubes.

The pattern printed here is generated from this repository’s own construction and its assignment was found by search, so nothing on the sheet is anybody’s design.

It is worth being exact about what that means, because a tessellation is a place where authorship questions are easy to get wrong in both directions. What is published here is a rule — a grid, a set of lines, and an assignment found by exhaustive search over a stated family — and rules of that kind are mathematics rather than models. The objects people make from waterbomb tessellations, the sculptural pieces and the shaped vessels, are designs and belong to their designers, and none of them appears anywhere here.

The distinction is the same one this site draws for the Miura fold, which is printed freely because it was published as engineering, and it is the reason the tessellation section of this site is short. Most of what is beautiful in the field is somebody’s work.

Where it sits among the alternatives is a measurement rather than an impression.

What the search did not settle #

Two things about the assignment are open and it is better to say so than to leave them looking decided.

The search covers repeating rules of one particular family: nine bits, periodic with the cell. A tessellation could have a foldable assignment with a longer period, or one with no period at all, and nothing here rules those out — the family was chosen because a tessellation is supposed to repeat, which is a design decision rather than a theorem.

And the whole search is over the local conditions. Every one of the thirty-two survivors passes all four conditions at every interior vertex, which is exactly as much as this site’s checker can establish, and not the same as folding. A large waterbomb has layer-ordering questions that no vertex condition sees, and deciding them in general is intractable.

What can be said is that the pattern is printable at true scale, and that a reader with a sheet of paper can settle the global question the way it has always been settled here — by folding it. That is not a proof either, and it is the one kind of evidence this site can hand over.

Where the ladder goes next #

The immediate continuation is the rule count: thirty-two rules survive on a large patch and fifty-six on a small one, and the twenty-four that die are the clearest statement this site has of why a patch is not a pattern.

The comparative direction is what a corrugation costs, where the waterbomb takes its place in a table beside the Miura, the Yoshimura and the twist, measured the same way. It packs hardest of the five and it is the most work, and both facts come out of the same computation.