Origami & Folding
what one sheet can do
Concept

Branch selection — where it appears

Choosing which of the configurations available at a degree-four vertex the sheet actually takes. The two are equally low in energy, so nothing local prefers either, and a sheet folding itself has to be biased toward one or it will arrive at whichever the noise points it at.

Named by 4 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 how many folded states each crease of a four-by-four Miura leavesdriven to 0.6 radians, with every consistent assignment enumerated rather than the first eight1 state4an actuator belongs on one of these2 states82 states, so the sheet has a choice4 states44 states, so the sheet has a choice8 states88 states, so the sheet has a choicethe four that leave one are c:3:2, c:3:3, r:3:2, r:3:3 — all of them at the same corner of the sheet

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.

rigid · Self-folding 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.

ActuationBifurcationQuadrilateral meshRigid foldingSelf-foldingSymmetry breakingDegree-fourInsect wings

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