The waterbomb, tiled
waterbomb-pattern is one function. Everything below was drawn by it when the site was
last published, at settings taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and when the generator changes, this
page changes with it.
With nothing specified
cols: 2, rows: 2, mask: 46, mm: 100, view: "pattern"
cols: 4, rows: 4, mask: 46, mm: 160, view: "pattern"
cols: 3, rows: 3, mask: 46, mm: 120, view: "pattern"
What it checked while it drew
Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.
- developability holds at every interior vertex — its sectors close to 360° — 25 checked ×6
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 25 checked ×6
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 25 checked ×6
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 25 checked ×6
- the pattern has two kinds of interior vertex, of degree 4 and 6, and both satisfy Kawasaki on the measured angles ×1
- the waterbomb tessellation is put past all four theorems before it is drawn ×1
- the Waterbomb tessellation is put past all four theorems before it is drawn ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.
A unit that folds is not a tessellation
Of the 512 repeating rules for the waterbomb tessellation, 56 pass every condition on a two-by-two patch and 32 pass on every larger one. The twenty-four that die were never foldable — the small patch simply contained one of the four kinds of vertex the pattern makes, and the failures were at the other three.
The base that tiles
The waterbomb base is the first thing most people fold and the last thing they think about. Repeat it across a sheet and it becomes a tessellation with two kinds of vertex, an assignment that has to be searched for rather than remembered, and a folded state thirty-two times smaller than the paper.
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.
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.
Where a rule can close a loop
Three corrugation families have repeating rules whose letters send four panels round in a circle, and one has none at all. The one that has none is the one whose vertices are all of degree six — and the reason is that a straight line through a point carries a single letter under any repeating rule, while a strict alternation round six creases needs the two halves of that line to differ.
Where the paper stops
Every flat-folding theorem is a statement about a full turn of paper, so a vertex at the edge of the sheet is subject to none of them. Cutting a patch out of a pattern removes conditions rather than preserving them, and a small enough patch has almost none left.
All the figures · The tessellations field · The patterns a reader can fold