Origami & Folding
what one sheet can do
Generator

Five hundred and twelve rules, one object

A generator in the tessellations library, called 9 times across 4 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

rule-object 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

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

view: "funnel", cols: 3

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

cols: 3, rows: 3

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

view: "funnel", cols: 3, rows: 3

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

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.

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.

The count counts labels

One, two, six, sixteen, fifty, a hundred and forty-four: the oldest sequence in the subject counts foldings of a strip of numbered stamps. A folded strip of blank paper has no first stamp and no top side, and neither of those operations ever leaves a folding alone — so the count of objects is 1, 2, 5, 14, 38, 120, and it is not the count over four.

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.

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.

All the figures · The tessellations field · The patterns a reader can fold