Origami & Folding
what one sheet can do
Concept

Bifurcation — where it appears

A configuration from which a mechanism can continue in more than one way. A degree-four vertex has two assemblies at any fold angle, and which one a sheet takes is not decided by the fold angle alone.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

-10010000.20.40.60.81how far the vertex is drivenstored energybranch oneMVMMbranch twoMVVVboth run downhillfrom the flat state,and end at zeroso the energy does notprefer either branch —the noise decides

Paper that folds itself

A self-folding sheet has to supply the fold and then choose what to fold into. The second half is where these things fail, and no amount of torque helps, because the two outcomes are equally downhill.

rigid · Self-folding flat-foldable at every vertex, all threeworst Kawasaki residual 0e+0 radiansone vertex moved -30 per centloop residual 2.4e-1the Miuraloop residual 5.8e-14one vertex moved 40 per centloop residual 3.6e-1every one of these is developable and flat-foldable at every interior vertex, exactly

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.

rigid · Rigid folding one crease decided, and how much of the sheet followsthe flat-folding conditions, propagated2 of 12the rigid-folding conditions, propagated12 of 12and the rigid propagation leaves 1 consistent set of fold angles

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.

rigid · Self-folding the height is the worst amplification anywhere on the sheetone line per crease; the horizontal axis is the driven crease's own fold angle3.010.32.5fold angle of the driven creasea mesh with no two vertices alike6 of 6 creases are worst near the flat sheet0 steps refused as branch changes

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.

rigid · Self-folding the bar is how many of 200 random spring settings give two or more resting statessprings set at random fold angles, the same settings for every rowvertex 60·90·120·901982 with 1 · 198 with 2vertex 45·100·135·801964 with 1 · 196 with 2vertex 80·95·100·851964 with 1 · 126 with 2 · 62 with 3 · 8 with 4a corrugation0200 with 1a vertex's configurations are two branches through the flat state; a corrugation's are one line

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.

biology · Insect wings -40-20204060801000510152025degrees of the driven crease from the flat sheetenergy in the springsthe road from one resting state to the other · sectors 60° · 90° · 120° · 90°, springs set to mixedbranch one's stateenergy 12.40branch two's stateenergy 17.65the flat sheetenergy 26.16to leave the deeperclimb 13.76to leave the shallowerclimb 8.51left of the middle is branch one, right of it branch two; they meet only at the flat sheet

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.

biology · Insect wings the bar is the median number of resting states, over the same kind of random springsa chain shares one crease between each vertex and the next1 vertex2 states2 branch combinations · 198 of 200 settings rest on every one · 197 of 198 cross at the flat sheet2 vertices4 states4 branch combinations · 193 of 200 settings rest on every one · 200 of 200 cross at the flat sheet3 vertices8 states8 branch combinations · 181 of 200 settings rest on every one · 199 of 200 cross at the flat sheet4 vertices16 states16 branch combinations · 134 of 200 settings rest on every one · 200 of 200 cross at the flat sheetevery combination of branches holds a resting state, and every switch between them goes over the whole flat sheet

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.

biology · Insect wings a 4-by-4 Miura, every crease driven in turn, at 10 angles along the motionevery consistent assignment enumerated at each crease, with both configurations found at every vertexanglecreases × states they leavethe creases that decide it0.24×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:30.44×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:30.64×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:30.84×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:31.04×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:31.24×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:31.64×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:32.04×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:32.44×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:32.84×1 8×2 4×4 8×8c:3:2 c:3:3 r:3:2 r:3:3every row is the row above it: the census is a property of the pattern, not of how far it has folded

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.

rigid · Self-folding four vertices round one panel, as a chain and as a loopa combination survives the loop only if going round it brings every fold angle back to where it startedthe faceopen chain of fourclosed loopthe same at every anglea Miura face164yesa face with no two vertices alike, seed 11161yesdriven at 0.3, 0.6, 1, 1.4, 1.8 radians · a combination counts when every crease is folded and the loop closes

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.

biology · Insect wings

Named alongside it

The objects these essays reach for when they reach for this one.

Rigid foldingQuadrilateral meshInsect wingsSelf-foldingActuationBranchBranch selectionDegrees of freedomFold angleSpherical linkagePropagationActuator

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