Origami & Folding
what one sheet can do
Generator

Two models, and the sheet that has carried both

A generator in the curves and material library, called 6 times across 1 essay. 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.

overlay-union 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

Two models, and the sheet that has carried bothThe two patterns and their union. The union's creases are exactly the creases of the two patterns; what it has that neither of them had is the vertices where one pattern's creases cross the other's, marked in magenta where they fail.two models, and the sheet that has carried both4 crossings, of which 0 cannot fold flatThe preliminary baseThe square twistthe sheet after bothno crease has moved and none has been added; what is new is where they cross

view: "pair", a: "preliminary-base", b: "hexagon-twist"

Two models, and the sheet that has carried bothThe two patterns and their union. The union's creases are exactly the creases of the two patterns; what it has that neither of them had is the vertices where one pattern's creases cross the other's, marked in magenta where they fail.two models, and the sheet that has carried both12 crossings, of which 10 cannot fold flatThe preliminary baseThe hexagon twistthe sheet after bothno crease has moved and none has been added; what is new is where they cross

view: "crossing"

A crossing folds flat only at a right angleWhere a crease of one pattern crosses a crease of another, the vertex they make has opposite sectors equal. Kawasaki asks the alternate pairs to be supplementary, and two pairs of equal angles are supplementary only when all four are right angles.a crossing of two creases folds flat only at a right angle35°90°145°90°180°0so a sheet can carry a second model only where every new crease meets every old one square

view: "census"

Two models in one sheetFor pairs of the printed patterns: how many vertices their union has that neither pattern had, and how many of those satisfy Kawasaki. A pattern overlaid with itself produces none of either, which is what makes the rest evidence.two models in one sheet: what the crossings are, and what they docrossingsthat fold flatThe Yoshimura pattern and The waterbomb tessellation1200The Yoshimura pattern and The tapered corrugation760The hexagon twist and The Yoshimura pattern5832The tapered corrugation and The waterbomb tessellation570The hexagon twist and The waterbomb tessellation504The square twist and The Yoshimura pattern400The preliminary base and The Yoshimura pattern362The square twist and The waterbomb tessellation3628the ones that do fold are the crossings that happen to be square

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 sheet remembers

Perfect memory is the fourth idealisation, and the least examined of the four. It is usually read as a complaint that paper will not lie flat again; the large half is the opposite. A sheet folded once is no longer blank, so folding a second model into it is folding the union of two patterns — and a union folds flat only where every new crease meets every old one at a right angle.

All the figures · The curves and material field · The patterns a reader can fold