Twenty lengths, four conditions
mesh-solve 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: "straightness", seeds: [11, 23, 71, 97]
view: "before-after", seed: 11
view: "seeds", seeds: [5, 11, 17, 19, 23, 27, 41, 71]
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.
- 6 of 8 meshes drawn at random were solved to machine precision, and the rest stalled ×2
- a flat-foldable quadrilateral mesh drawn at random leaves a closure mismatch of 0.066 radians, and solving all four conditions together closes them to 8.5e-14 ×1
- developability holds at every interior vertex — its sectors close to 360° — 9 checked ×1
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 9 checked ×1
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 9 checked ×1
- the 4 conditions are independent as functions of the 20 lengths, so the solutions form a surface of dimension 16 ×1
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 9 checked ×1
- the solve reaches 8.5e-14 radians in 24 steps, from 0.062 ×1
- the solved mesh's nearer crease family runs 1.21 radians off straight at its worst vertex, against 4.7e-15 for a Miura ×1
- the the mesh as drawn is put past all four theorems before it is drawn ×1
- the the mesh as solved is put past all four theorems before it is drawn ×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.
A tolerance is a direction
Cut a solved mesh a fifth of a millimetre wrong and its closure is gone. That is true of the errors it was tried with and false of errors in general: the solutions form a surface sixteen directions wide, an error along it costs five thousand times less than the same error across it, and the fifth of a millimetre is the allowance in one direction out of twenty.
Closing is not building
A quadrilateral mesh solved so that every loop closes to within a millionth of a radian is a mesh whose fold angles are consistent. It is not necessarily an object. One of the six solved here drives a panel through another at every angle of its motion — there is no part of the fold at which it could be made of solid panels — and the pair that crosses is two steps apart in the sheet, where nothing evaluated at a vertex could see it.
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.
Solved is not built
A mesh that folds because an equation holds and a mesh that folds because one crease family runs straight through every vertex are not two examples of the same thing. Cut a Miura's every dimension five per cent wrong and it still folds exactly. Cut a solved general mesh a fifth of a millimetre wrong on a 150 mm sheet and the closure is gone.
Solving every face at once
A quadrilateral mesh that folds rigidly has to close round every one of its faces, and the rung that built the general mesh could close one. Four of them at once resisted a descent that drove each free length to its own root, because closing a loop is a condition on several lengths together — and solving them jointly finds a sheet with no two vertices alike that folds, and a surface of them sixteen dimensions wide.
The Miura folds two ways
One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.
The motion has no letters to choose
A flat-folding search picks a letter for every crease and can pick badly. A rigid folding does not pick anything: the fold angles are real numbers, determined by the panels through equations that have a solution or do not. Replacing a discrete choice with a continuous solve removes every ordering question at once, and introduces a failure of its own.
Every generator · The rigid folding field · The patterns a reader can fold