Origami & Folding
what one sheet can do
Series

Waterbomb — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · tessellation
  2. 562 × 24 cells323 × 39 cells324 × 416 cells325 × 525 cellsrepeating rules that pass every condition, out of 51224 rules pass on the smallest patch and on none of the others

    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.

    part 2 · tessellation
  3. 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.

    part 3 · tessellation
  4. 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.

    part 4 · tessellation
  5. 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.

    part 5 · tessellation
  6. 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.

    part 6 · tessellation

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