The pattern, and where its panels land
folded-layers 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
pattern: "preliminary", view: "panels", samples: 200
pattern: "miura", cols: 4, rows: 3, angle: 0.35, view: "panels", samples: 200
pattern: "waterbomb", cols: 3, rows: 3, mask: 46, view: "panels", samples: 200
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.
- the folded footprint carries 3 distinct layer counts, from 28 to 36 ×5
- the miura pattern folded: 6 panels can be reached two ways and both routes place them in the same spot ×3
- the miura pattern folded: every one of its 12 panels keeps its area exactly ×3
- the waterbomb pattern folded: 13 panels can be reached two ways and both routes place them in the same spot ×2
- the waterbomb pattern folded: every one of its 30 panels keeps its area exactly ×2
- the yoshimura pattern folded: 11 panels can be reached two ways and both routes place them in the same spot ×2
- the yoshimura pattern folded: every one of its 36 panels keeps its area exactly ×2
- the miura pattern folded: the layers over the footprint add back up to the whole sheet, to within the sampling ×1
- the preliminary pattern folded: 1 panels can be reached two ways and both routes place them in the same spot ×1
- the preliminary pattern folded: every one of its 8 panels keeps its area exactly ×1
- the preliminary pattern folded: the layers over the footprint add back up to the whole sheet, to within the sampling ×1
- the twist pattern folded: 4 panels can be reached two ways and both routes place them in the same spot ×1
- the twist pattern folded: every one of its 9 panels keeps its area exactly ×1
- the twist pattern folded: the layers over the footprint add back up to the whole sheet, to within the sampling ×1
- the waterbomb pattern folded: the layers over the footprint add back up to the whole sheet, to within the sampling ×1
- the yoshimura pattern folded: the layers over the footprint add back up to the whole sheet, to within the sampling ×1
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.
Bringing the other side to the front
Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.
Error is folded too
A folded position is a composition of reflections, and a reflection in a line that is slightly off turns everything beyond it by twice as much. So an error does not stay where it was made — and whether it grows with the crease count or with its square root depends on whether it is the same error every time.
The base that tiles
The waterbomb base is the first thing most people fold and the last thing they think about. Repeat it across a sheet and it becomes a tessellation with two kinds of vertex, an assignment that has to be searched for rather than remembered, and a folded state thirty-two times smaller than the paper.
The creases a sheet gives itself
A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss.
The creases that cannot move
One vertex's foldings are always joined up. A pattern's are not, and the number of pieces they fall into is exactly two to the power of the number of creases with an interior vertex at each end — four on a square twist, six on a hexagon twist, none at all on a preliminary base. The creases a local change cannot reach are the creases that never reach the edge of the paper.
The cylinder the pattern chooses
A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.
The paper is all still there
A folded sheet is smaller than it was and none of it has gone anywhere. How much smaller it is and how many layers deep it is are not two properties of a pattern — they are one number, and their product is the sheet.
The pile, not the panel
Every technique for building a fold out of panels with depth is drawn, described and priced at one crease between two panels. A folded model has two layers nowhere except at its last fold: the printed patterns here reach eight, sixteen, thirty-two and sixty, and the length a thick panel has to find at those creases is not the published allowance but fifty-nine times it.
The property a patch does not have
A folded corrugation is described as a material — a packing ratio, a stiffness, a Poisson's ratio — and every one of those is a statement about an unbounded medium. Fold the same tiling at six sizes on the same square and the compaction climbs from 3.89 layers to 4.79 as the share of units the rim cuts falls from nine tenths to four, and it has not settled at the fine end. The number a patch gives is the material's number minus its own boundary.
The rim lies over less
A folded sheet's boundary is usually discussed as the place the theorems stop applying. It is also visible in the pile: a panel carrying a raw edge of the paper lies over fewer of the other panels than one that does not, on every printed pattern that has both kinds — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb tessellation, and never once the other way round.
Where the length sits
A pattern's folding length is a total, and a total says nothing about where the work is. Divide each printed sheet into bands by distance from its own edge and the answer separates the patterns by kind: a traditional base carries five times its share of folding in the middle 4% of the paper, a tessellation carries between 0.9 and 1.4 everywhere, and a twist unit carries none at all at its centre. On all eight, the outermost band carries less than its share.
Every generator · The flat-folding field · The patterns a reader can fold