Rigid folding — where it appears
Named by 24 essays across 4 fields — each of them below, with the objects they name alongside it.
From a shell to a solar array
The Miura fold was published in 1970 and flew on a satellite in 1995. The gap is not ignorance — the pattern was known, understood and available the whole time — and the same twenty-five year lag appears between every folding result and the hardware that uses it.
No motor in the fold
An insect's wing has muscles at its base and nothing out along its length, so the pattern has to carry the deployment by itself. The condition that makes that possible is a count: one degree of freedom means one number determines every panel, which means one thing has to pull.
Paper through paper
Every test the subject has for rigid folding is a statement about a neighbourhood, and a neighbourhood cannot see the far side of the sheet. So a pattern can satisfy all of them while driving one panel straight through another, and the sharpest witness has no interior vertex in it at all.
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.
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.
Which crease to push
Deciding one fold angle settles every other one on a quadrilateral mesh, which is what makes a self-folding sheet buildable with a single actuator. It leaves a question that sounds like an afterthought: which crease. Driving each of a mesh's twenty-four in turn gives twenty-four different answers to how far an error in it travels — and on the sheet that repeats one vertex, it gives several answers to what shape the sheet takes.
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.
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.
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.
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.
A mechanism that closes on itself
A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.
Thickness round a closed loop
Real panels have thickness, and every technique for accommodating it works by shifting a hinge off the ideal crease by a small amount. On a flat sheet the shifts accumulate outward and end at the edge. On a closed sheet they accumulate round a loop and have to come back to where they started, which is a condition none of the techniques was designed to satisfy.
A corrugation has one resting state
A folded wing held short of shut stores energy in its hinges, and a wing that could stay both open and folded with nothing holding it would need that energy to have two bottoms. A corrugation cannot provide them: every crease in it folds by one angle, so the energy of any set of crease springs is a parabola in that angle and has exactly one resting state, however much the springs disagree. A single degree-four vertex has two branches through the flat state, and the same springs give it two resting states on almost every setting tried.
The wall is the flat sheet
A sprung degree-four vertex usually has two resting states, one on each branch of its motion, and the branches meet in one place a sheet can pass through: the flat state. So the only road from one resting state to the other crosses the flat sheet, and on every setting of the springs tried on two vertices the flat sheet is the highest point of that road. Its energy is each spring's stiffness times its rest angle squared, summed, which does not contain the vertex's sector angles at all — the same springs put on four different vertices give a wall of exactly the same height. The geometry decides only how far below the wall each state sits, and the shallower one sits a median of five per cent below it.
A chain of vertices switches all at once
One sprung degree-four vertex rests in two states, one on each branch of its motion, and switches between them only by passing through the flat sheet. Chain vertices together by sharing a crease between each and the next, and a branch can be chosen at every vertex: two, four, eight and sixteen combinations for chains of one to four. The median spring setting rests once on every combination. And every combination's curve of configurations passes through the same single point — the whole chain flat at once — and meets no other anywhere else, so every switch, even of one vertex's branch, takes the whole chain back to flat. The wall that switch climbs is every spring's flat energy added up, growing by a crease's worth for every crease, and a chain's second state sits several times further below it than a single vertex's does.
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.
Two drivers and one freedom
Two actuators on a sheet with one degree of freedom are two commands for one number, and if they disagree by a hundredth of a radian the sheet cannot satisfy both. Where it settles is decided by the gearing between the two creases: a strongly geared pair absorbs the disagreement and leaves a quarter of it standing, while a weakly geared pair keeps ninety per cent. The loosest coupling is the expensive one, which is the opposite of what coupling usually means.
A gearing reflects stiffness squared
Two actuators on a sheet with one freedom disagree, and the sheet settles where their stored energy is least. With unequal stiffnesses the answer depends on them only through k₂g² ⁄ k₁ — the second actuator, seen from the first crease, is a spring of stiffness k₂g², the gearing entering squared as a gear train reflects any stiffness. That settles which actuator to make compliant. On a rigid mesh's loosest pair a second actuator ten times stiffer than the first stores fifty times the fighting energy of one ten times softer, and softening it gives up only 8 per cent of how firmly that crease is held, because the first actuator already holds it ten times over through the gearing. On the tightest pair softening saves four times the energy and gives up 68 per cent of the hold. Compliance is cheap exactly where the fight is expensive.
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.
Quadrilateral meshDegrees of freedomBifurcationFold angleSelf-foldingActuationInsect wingsMiuraBranchClosureActuatorBranch selection