Origami & Folding
what one sheet can do
Generator

The bud chooses the fold

A generator in the folding nobody designed library, called 8 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.

bud-fit 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

The bud chooses the foldA corrugated leaf packed into a bundle, as a function of how many folds it uses. Too few and the strip is too wide to fit; too many and the stack is too thick. The smallest bundle is in between, and the bud's radius decides which counts are available at all.051015202500.511.522.5number of foldsbundle radiusthe bud, radius 1.248121624tightest at 12 foldsbud radius 1.2 · leaf area 340 · layer thickness 0.12

budRadius: 1.2, leafArea: 340, folds: [4, 8, 12, 16, 24]

The bud chooses the foldA corrugated leaf packed into a bundle, as a function of how many folds it uses. Too few and the strip is too wide to fit; too many and the stack is too thick. The smallest bundle is in between, and the bud's radius decides which counts are available at all.051015202500.511.522.5number of foldsbundle radiusthe bud, radius 1.248121624tightest at 12 foldsbud radius 1.2 · leaf area 340 · layer thickness 0.12

budRadius: 1.15, folds: [6, 10, 14, 18, 22, 30]

The bud chooses the foldA corrugated leaf packed into a bundle, as a function of how many folds it uses. Too few and the strip is too wide to fit; too many and the stack is too thick. The smallest bundle is in between, and the bud's radius decides which counts are available at all.05101520253000.511.52number of foldsbundle radiusthe bud, radius 1.1561014182230tightest at 14 foldsbud radius 1.15 · leaf area 340 · layer thickness 0.12

budRadius: 1.2, leafArea: 340, folds: [4, 8, 12, 16, 24], thickness: 0.12

The bud chooses the foldA corrugated leaf packed into a bundle, as a function of how many folds it uses. Too few and the strip is too wide to fit; too many and the stack is too thick. The smallest bundle is in between, and the bud's radius decides which counts are available at all.051015202500.511.522.5number of foldsbundle radiusthe bud, radius 1.248121624tightest at 12 foldsbud radius 1.2 · leaf area 340 · layer thickness 0.12

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.

A leaf ends its pattern

A hornbeam leaf's corrugation does not stop at the margin by being cut off: the pleats narrow until there is nothing left of them, and the margin is where the pattern reaches zero rather than where it was interrupted. The same is available to a drawn pattern and costs nothing — a corrugation tapered by a factor of eighty-two across its columns folds with a Kawasaki residual of 4×10⁻¹⁶, exactly as an untapered one does, because the column widths never enter the condition.

The bud chooses the pattern

Nothing in the geometry of a corrugation says how many folds it should have. The container does: too few folds is a strip too wide to fit, too many is a stack too thick to fit, and the window that fits at all is narrow and has a best point in the middle of it.

The container picks the member

Two large waves and eight small ones are the same metric, and geometry has nothing to say about which. Bending has nothing to say either — it rises at every step of the family, so a least-bending rule always answers the fewest waves and never anything else. What decides is the container: amplitude times wave count is constant across the family, so a ceiling on the height is a floor on the number, exactly inversely.

The plant's pattern is not a hard case

A hornbeam leaf packs into its bud by corrugating, and the pattern it uses gives up a consistent lettering at nine, twelve, fifteen, eighteen, twenty and twenty-four steps on nine, twelve, fifteen, eighteen, twenty and twenty-four panels. Nothing about the plant's problem is combinatorially difficult, and saying so is worth as much as finding a case that is.

All the figures · The folding nobody designed field · The patterns a reader can fold