Origami & Folding
what one sheet can do
Generator

A crease is not a line

A generator in the curves and material library, called 17 times across 7 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

crease-radius is one function. Everything below came out of it during this build, at arguments 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.

At its defaults

A crease is not a lineA fold carries the paper round a small radius rather than through a point, and the arc uses more of the sheet than the stack advances by. One crease loses a fraction of a millimetre. A grid with hundreds of them loses that on every line at once, which is why an ambitious tessellation is folded from thin paper and why a large grid comes out short.ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

rho: 0.12, grids: [8, 16, 24, 32, 48]

A crease is not a lineA fold carries the paper round a small radius rather than through a point, and the arc uses more of the sheet than the stack advances by. One crease loses a fraction of a millimetre. A grid with hundreds of them loses that on every line at once, which is why an ambitious tessellation is folded from thin paper and why a large grid comes out short.ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

rho: 0.12, grids: [8, 16, 24, 32, 48], side: 150

A crease is not a lineA fold carries the paper round a small radius rather than through a point, and the arc uses more of the sheet than the stack advances by. One crease loses a fraction of a millimetre. A grid with hundreds of them loses that on every line at once, which is why an ambitious tessellation is folded from thin paper and why a large grid comes out short.ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

sub: "halving-limit", thickness: 0.1, lengths: [0.297, 1, 10, 100, 1200], upTo: 13

How long a strip has to be to fold in half n timesThe length of paper single-direction folding needs, against the number of halvings, for a sheet a tenth of a millimetre thick. The vertical scale is logarithmic and the curve is still steep on it, because the requirement grows as the square of the layer count. The marked lengths are an A4 sheet, a metre, ten metres, a hundred, and the roll that was folded twelve times.24681012-202halvingsmetres of paper (powers of ten)297 mm — 6 folds, 64 layers1 m — 7 folds, 128 layers10 m — 8 folds, 256 layers100 m — 10 folds, 1024 layers1200 m — 12 folds, 4096 layerspaper 0.1 mm thick · L = (πt/6)(2ⁿ + 4)(2ⁿ − 1)the loss is the paper that goes round the closed end, and it doubles twice per fold

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

Closer than a crease is wide

One fold from a bare square leaves nine marks, seventy-five millimetres apart. Two folds leave five hundred and sixty-five, the closest pair half a millimetre apart. Three folds — using one axiom of the seven — leave half a million, and ninety-four per cent of them have another mark within a fifth of a millimetre. What bounds a folder is not what the axioms reach; it is what the paper can tell apart.

How deep is a crossing

A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

How many times can it be halved

The folklore says seven, and the folklore is a statement about one sheet of paper. What actually binds is arithmetic: every halving doubles the layers and the paper spent at the closed end grows as the square of the layer count, so the length needed for twelve folds is nearly a kilometre.

The crease has a radius

A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.

The paper had to arrive first

A model with sixty-four layers at its thickest point, folded in ordinary copier paper, is six and a half millimetres of stack. The layer count a design can reach is fixed by the substrate, not by the folder — so the elaborate tradition is downstream of a manufacturing achievement with its own dates.

The rectangle that keeps its shape

Halving a rectangle across its long side turns a proportion of r into one of 2/r, so almost every sheet comes out of the fold a different shape from the one that went in. Exactly one does not, and it is not a shape anybody chose.

What a corrugation costs

Every tessellation this repository can fold, measured the same way: how much smaller it gets, how deep the stack becomes, and how much creasing was needed to buy it. The last column is the one nobody quotes and the one a folder feels.

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