Quadrilateral mesh — where it appears
Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.
The condition that is not flat-foldability
Take away the assumption that one crease family runs straight through every vertex and ask what makes a quadrilateral mesh move. It is not flat-foldability. There is a one-parameter family of meshes, every one of them developable and flat-foldable at every vertex to machine precision, and exactly one member of it folds — the Miura. Slide a single vertex along the ray that keeps every condition exact and the sheet stops moving, first order in the displacement.
One crease decides the sheet
Fix one crease of a flat-folding problem, propagate every condition the subject has, and three creases out of a hundred and fifty-eight follow. Fix one fold angle of a rigid one and every crease on the sheet follows, with a single consistent answer. The same experiment, two questions, opposite answers — and it is why a self-folding sheet needs one biased vertex rather than one per vertex.
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.
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.
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.
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.
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.
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.
Four ways to draw a pattern
Every sentence here of the form over some crease patterns is a statement about a construction nobody declared, and it is worse than the same problem at a vertex because a pattern has a shape as well as angles. Four ways of producing a pattern that satisfies every condition disagree about how far it shrinks by a factor of twelve, about how much creasing it costs by a factor of six, and about how much of it is edge by a factor of two.
The allowance is spent at the end
A tolerance on a solved mesh was priced at one fold angle, because that is where a tolerance is priced. The surface of solutions turns out not to move as the sheet folds — the free directions at a third of a radian are the free directions at two and a half, to twelve figures — and the price of leaving it rises by a factor of thirteen along the way.
The hardest instant
Driving one crease of a quadrilateral mesh settles every other one, and an error in the driven crease arrives elsewhere multiplied. That multiplier was measured once, at one fold angle. Followed along the whole motion it is worst at the flat sheet on twenty of twenty-four creases — and on the Miura the measurement has to refuse to answer.
A collision is an order
Paper passing through paper is treated here as a thing that happens during a motion and is caught by watching for it. At the flat state it is not an event at all: it is the absence of an ordering, and it can be proved rather than observed. Four of the six quadrilateral meshes this site solves for rigid folding place perfectly and admit no ordering of their nine panels — so every one of them must pass through itself, and none of them was ever driven to find out.
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.
Refused at one lettering
Four of six quadrilateral meshes here have no arrangement of their nine panels — established by searching every ordering, at the labelling each mesh arrived with. Enumerate every labelling instead and two of the four fold perfectly well at a different one. What was reported as a fact about four meshes is a fact about two meshes and two labellings.
A search with nothing to reorder
One search on a crease pattern costs eighty steps or fifteen thousand depending on the order it takes its decisions in. The other search on the same crease pattern costs 1,188,571 steps whatever order it is given — twelve permutations of the panels, twelve identical counts. The difference between them is one line of code that neither has and one has.
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.
Only four creases decide a Miura
Driving one crease of a rigid quadrilateral mesh settles every other one — except that on the pattern everybody builds it often does not. Enumerated properly, four of a four-by-four Miura's twenty-four creases leave exactly one folded state and the other twenty leave two, four or eight. A mesh whose vertices all differ leaves one from every crease. The ambiguity is not a property of quadrilateral meshes; it belongs to the symmetry.
The deciding set does not move
A driven Miura leaves several folded states from most of its creases and exactly one from a few, and those few are where an actuator belongs. It was reported that the few change along the motion — four of twenty-four at 0.6 radians, fourteen at 0.8 — and that a five-by-five sheet had a crease leaving fifteen states where every other count was a power of two. Mapped at twenty angles from 0.1 to 3.0 radians on three sizes of sheet, neither survives. Every crease leaves the same number of states at every angle, every number is a power of two, and the same creases decide the sheet throughout. The changes were the vertex solver losing one of a vertex's two configurations on 138 of 8,640 solves, and the configurations it lost can be carried exactly from an angle where it finds both.
A loop takes choices away
A chain of four sprung degree-four vertices has sixteen combinations of branches, each a resting state, and switches between them only through the flat sheet. Close the chain into a loop round one panel and the combinations must agree when the fold angles come back round. On a face whose four vertices all differ, one combination survives; on a Miura face, four. The count is the same at every angle the face is driven to, and two surviving assignments at the same driven angle are never closer than one and a half times that angle — so they separate as the face folds and meet only when it is flat. A loop does not create the junction a region would need to switch on its own. It removes choices and leaves the switch as global as before.
Named alongside it
The objects these essays reach for when they reach for this one.
Rigid foldingFold angleBifurcationClosureMiuraNon-crossing conditionSelf-foldingSelf-intersectionActuationBranch selectionEnumerationLayer ordering