The same rule, three sheets
mesh-family 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
view: "patterns", fans: [0, 0.09, 0.16, 0.24]
view: "folded", fans: [0, 0.09, 0.16, 0.24], drive: 1.25
view: "patterns", fans: [0, 0.1, 0.2]
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 straight creases of the parallel member stay parallel through the motion and the fanned members' do not — the corrugation curves by 72° at the widest fan ×3
- every member of the family folds with its panels agreeing to 2e-15 of a sheet, driven by one angle ×2
- a row moved sideways leaves Kawasaki untouched and takes the table to 0.0411, and the panels then miss by 0.0085 of a sheet ×1
- every member of the family is developable and satisfies Kawasaki at every interior vertex, by construction rather than by search ×1
- so the corrugation lies on a cone rather than on a general curved surface, and the cone's apex travels as the sheet folds ×1
- the closed form tan(ρ_column/2) = cos β · tan(ρ_row/2) agrees with a composition of four rotation matrices to 4e-16 radians, and the composition never forms a cosine ×1
- the straight creases of a folded fan member all pass through one point, to 2e-15 of a sheet-width, at every point of the motion ×1
- the table of cosines is rank one to 8.9e-16 on every member of the family, and the sheet folds with every panel where its neighbours expect it ×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.
Nothing to average over
A folded corrugation is reported with a Poisson's ratio, and both of this site's measurements of one were made on a sheet that repeats a single cell. On such a sheet every cell behaves the same way and the cell's number is the sheet's number. On a sheet with no repeating cell the cells run from −3.5 to +0.4 — some widening while others narrow — and the sheet's own figure describes none of them.
The corrugation that curves
A Miura is a flat sheet that becomes a flat slab. Open its straight creases into a fan and the same construction gives a corrugation that wraps a cone — exactly a cone, with every straight crease passing through one point to fifteen decimal places, at every moment of the fold, with the apex travelling as the sheet closes.
The family the Miura belongs to
Move one vertex of a Miura and the sheet has no rigid folded position at all — which leaves the obvious question unanswered. What else moves? A row of paper reflected in each of a fan of lines is flat-foldable for nothing at all, and whether it also folds rigidly turns out to be a condition on a table of cosines: it has to be a column of numbers times a row of numbers.
Two refusals that refuse differently
Four of the six developable quadrilateral meshes this collection solves have no ordering of their nine panels — they must pass through themselves, and a search over every ordering proves it. On all four, the letters agree with themselves perfectly. The linear proof and the exponential search are not a fast test and a slow one: they answer different questions, and neither contains the other.
Every generator · The rigid folding field · The patterns a reader can fold