Origami & Folding
what one sheet can do
Concept

Sensitivity — where it appears

How much an output moves for a small change in one input, holding the rest still. In tree design it prices an edge by the scale lost per unit of extra limb; on a rigid mesh it says how far a cutting error in one length pushes the sheet off closing, and so which errors matter.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

00.20.40.60.81-7-6-5-4-3-2-1error in every length, millimetres on a 150 mm sheetclosure mismatch (powers of ten)across the surface of solutionsthe direction the closure's own derivative points ina direction chosen without regard to the surfacewhich has a component along bothalong the surface of solutionsone of the twelve directions the equations do not seethe same step costs 5,130 times as much one way as the other

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.

rigid · Tolerance spending the bird's length where the arrangement prices it lowesta price is the scale lost per unit of length added to every edge of a group, re-solved at every stepstepscaledearestpricecheapestpricetight pairs00.2651head0.092tail-0.000510.2753head0.102wings0.034520.2748legs0.081wings0.012530.2781body0.072wings0.037640.2817head0.081wings0.054350.2778tail0.074head0.000560.2780head0.076tail0.0026a price holds for as long as the dearest and the cheapest group stay the same groups

A price holds until the arrangement moves

Every edge of a subject's tree has a price — the scale lost per unit of extra length — and the obvious use of a price list is to spend a fixed total of limb where it is cheapest. Done a tenth of a unit at a time, re-pricing at every step, it works and then stops: the bird's scale rises 6.3 per cent in four steps and no further. But the prices do not hold while it happens. The bird's free tail stops being free after the first tenth, and its legs nearly treble in price without being touched. The lizard's prices hold for four steps, because its arrangement keeps the same three pairs at their limit for four steps. A price is a statement about which pairs are at or near their limit, and it lasts as long as they stay there.

design · Uniaxial bases the bird rounded to a grid two waysunits in the order body, wings, head, legs, tail; the drawn tree unrounded has size 0.2651gridnearesterrorsizecheap wayerrorsize4 units1 4 2 2 316%0.26751 4 2 2 423%0.28076 units2 6 3 3 511%0.27241 6 3 3 545%0.27848 units3 8 5 4 715%0.26312 8 4 4 720%0.2782size is the scale times the sheet length one unit of the subject's own length receives; error is the worst limb's

Rounding in the cheap direction

A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.

design · Uniaxial bases

Named alongside it

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

OptimalityTrade-offTree methodUniaxial baseBox pleatingCircle packingClosureFold angleQuadrilateral meshRigid foldingTolerance

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