Origami & Folding
what one sheet can do
Generator

Four things the model assumes

A generator in the curves and material library, called 9 times across 5 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.

what-paper-is-not 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

Four things the model assumesThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded

which: [thickness, stretch, creases, memory]

Four things the model assumesThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded

which: [stretch]

What the model assumes: no stretchThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.no stretchpaper stretches a little, which is why wet-folding works at allthe theorems are exact statements about a sheet nobody has ever folded

which: [creases]

What the model assumes: creases are linesThe idealisations every crease pattern rests on, and what each one costs when something is actually folded. None of them is a small error at the scale of a complex model, and the engineering versions of this subject are largely about the first one.creases are linesa crease has a radius; sharp folds tear and soft ones springthe theorems are exact statements about a sheet nobody has ever folded

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.

Four things that are not true

Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

One curve and one number

Folding cannot change how curved a crease is within the surface — that is what an isometry means. Everything a curved fold produces is the curvature it adds out of the surface, and one number controls all of it.

One cut short of falling apart

Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.

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.

The sixth thing that is not true

Five idealisations underlie every theorem here and each has had an essay: no thickness, no stretch, creases that are lines, perfect memory, no grain. There is a sixth, it is more basic than any of them, and it is the one nobody has ever thought to name — the paper is a disc.

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