Origami & Folding
what one sheet can do
Generator

Which way up to lay the paper

A generator in the curves and material library, called 8 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.

grain-census 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

Which way up to lay the paperEach printed pattern, with the share of its crease length lying within a few degrees of the sheet's grain at its best placement and at its worst. Nothing in the subject's theorems can choose between the two, because none of them mentions a direction.patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions54% / 46%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 11%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45°

view: "patterns", band: 22.5

Which way up to lay the paperEach printed pattern, with the share of its crease length lying within a few degrees of the sheet's grain at its best placement and at its worst. Nothing in the subject's theorems can choose between the two, because none of them mentions a direction.patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions54% / 46%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 11%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45°

view: "patterns", band: 15

Which way up to lay the paperEach printed pattern, with the share of its crease length lying within a few degrees of the sheet's grain at its best placement and at its worst. Nothing in the subject's theorems can choose between the two, because none of them mentions a direction.patterncrease length near the grainbest / worstceilingThe preliminary base4 crease directions21% / 21%45°The Miura fold3 crease directions46% / 0%35°The square twist2 crease directions0% / 0%45°The hexagon twist3 crease directions30% / 0%30°The Yoshimura pattern3 crease directions29% / 0%30°Fold and cut — the triangle6 crease directions35% / 0%29°The tapered corrugation3 crease directions42% / 0%33°The waterbomb tessellation3 crease directions21% / 0%45°

view: "sweep", which: "tapered-corrugation", band: 22.5

The tapered corrugation, turned on the sheetThe share of this pattern's crease length lying near the grain, as the pattern is turned through a half turn. The peaks are the placements a folder should choose and the troughs are the ones a theorem is equally happy with.02040608010012014016018000.10.20.30.40.5how far the pattern is turned, degreesshare of crease length near the grainas printeda quarter turn

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 fifth thing that is not true

Four idealisations underlie every theorem here and each has had an essay. There is a fifth and it has never been named, because it is invisible in exactly the way the others are not: paper has a grain, no theorem in the subject mentions a direction, and so nothing in the whole apparatus can tell a folder which way up to lay the pattern down.

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