Origami & Folding
what one sheet can do
Concept

Actuator — where it appears

A component that supplies the torque or force driving a folded structure from one state to another. On a rigid pattern with a single degree of freedom one is enough to settle every crease, so where it is placed decides both how far an error in it travels and how many folded states the sheet is left with.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 0123400.010.020.030.040.05gearing between the two creasesradians of errormoved at the firstleft at the secondworst at a gearing of onetwo actuators disagreeing by 0.05 radians, equal stiffness · the sheet settles where the stored energy is least

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.

rigid · Self-folding the energy two actuators store in a disagreement, pair by pairenergy k₂δ² ⁄ (1 + k₂g² ⁄ k₁); the last column is the second crease's holding stiffness, k₂ + k₁⁄g², kept when k₂ is cut to a tenthgearingbalancing k₂ ⁄ k₁equal stiffnesssecond ten times stifferten times softerstiffness kept0.31610.010.915.000.09992%0.5393.440.772.560.09780%0.7841.630.621.400.09466%0.9341.150.531.030.09258%1.0001.000.500.910.09155%1.7610.320.240.310.07632%a mesh with no two vertices alike, driven at c:0:1 · energy stored in the fight, in units of δ² times the first actuator's stiffness

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.

rigid · Self-folding

Named alongside it

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

Rigid foldingSelf-foldingActuationDegrees of freedomGearingBifurcationFold angleQuadrilateral mesh

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