Origami & Folding
what one sheet can do
Concept

Energy minimisation — where it appears

Reading a structure's resting state as the lowest point of the energy stored in its hinges. It settles what a spring-loaded pattern does when nothing holds it, and it is what turns a question about forces into a question about the shape of a surface.

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

found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter20 interior vertices · 23 mountain and 56 valley creasesmountainvalleyraw edge

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

tessellation · Found patterns -3-2-1123050100150turn in each hinge (radians, signed by branch)energy storedrests hereand hereflat: 179the barrier isthe flat state8 folds, hinge 0.05creased to 1.6the two branches are the same sheet folded opposite ways, and the flat sheet is the only state they share

Springs that disagree do not offer a choice

A corrugation of hinges that remember different angles was expected to have more than one position in which nothing pushes. It has exactly one, at the stiffness-weighted mean of what they remember, because a sum of parabolas in one variable is a parabola. The second resting place comes from somewhere else entirely — the mirror pattern — and the flat sheet is the barrier between them, which is also why a creased sheet cannot be pulled flat at all.

biology · Convergence a corrugation remembering 0.8 rad, held at 2.4 and later opened to 0.4torques as shares of the one total both ends of the story share; the barrier as a multiple of its creased valueheld for, τremembered turncontainer suppliesopening needsbarrier0.00.80080%20%1.00×0.51.43049%51%3.19×1.01.81129%71%5.13×2.02.18311%89%7.45×3.02.3204%96%8.41×5.02.3891%99%8.92×the container's share and the opening's share always sum to the whole, whatever the holding time

Holding a fold moves the force

A creased hinge held at an angle slowly comes to remember that angle, so a leaf or a wing packed in a bud for a season is gradually holding itself and the bud has less to do. The force is not used up in the process. The torque the container must supply falls as e^(−T⁄τ), the torque later needed to open the structure rises by exactly the same amount, and the two sum to the same total at every moment of the holding. Held for three relaxation times, a corrugation creased to 0.8 radians and packed to 2.4 needs 4 per cent of the total from its container and 96 per cent from whatever opens it — and the barrier to its mirror image has grown more than eightfold. A packing that lasts buys independence from its container with a harder unfolding.

biology · Convergence

Named alongside it

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

ConvergenceCorrugationCrease radiusMembrane hingeSymmetry breakingBucklingCrumplingFound patternThe Yoshimura pattern

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