Origami & Folding
what one sheet can do
Concept

Waterbomb — where it appears

A traditional base of two diagonals and two midlines collapsed into a triangle, and the tessellation made by repeating it across a grid. The tessellation has two kinds of vertex, an assignment that has to be searched for rather than remembered, and a folded state a fraction of the paper's area.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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.

tessellation · Waterbomb 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.

tessellation · Waterbomb 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.

tessellation · Waterbomb 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

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.

tessellation · Waterbomb the bar is how many rules send four panels round in a circleout of the rules that already fail the count at some vertexthe Miura fold3848 refused · every loop four panels · vertices of degree 4the tapered leaf3848 refused · every loop four panels · vertices of degree 4the Yoshimura pattern038 refused · not one closes a loop · vertices of degree 6the waterbomb tessellation120480 refused · every loop four panels · vertices of degree 4 and 6a loop of four needs the letters to alternate round one point, and only a degree-four vertex lets a repeating rule do that

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.

tessellation · Waterbomb crease density, the closest two creases come without meeting, and the two multipliedthe closest approach is between creases with no vertex in common, measured as segmentspatternm a m²closest mmproductThe Yoshimura pattern8224.52.02The waterbomb tessellation8920.01.79The square twist3136.11.13The Miura fold3625.10.91The hexagon twist4122.10.90The tapered corrugation4118.60.77parallel creases give a product of exactly one; a pattern above one carries more length than its own spacing would suggest

A paper limits spacing, not density

The density ceiling put a sheet's limit at parallel creases a crease-width apart, 1⁄w of crease a square metre, and claimed no arrangement carries more. Crossing families do: they meet at vertices, which is shared ground, and never come closer than w anywhere else. Measured over every pair of creases that do not share a vertex, density times closest spacing settles at about 1.9 for both the Miura and the waterbomb, so each can be folded nearly twice as fine as the density bound said — 262 cells a side on copier paper for the Miura rather than 139, and 141 for the waterbomb rather than 74.

material · Crease density

Named alongside it

The objects these essays reach for when they reach for this one.

TessellationMaekawa's theoremAssignmentLayer orderingVertex degreeBuried creaseCorrugationCrease assignmentCrease lengthCrease radiusDegree-fourEnumeration

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