Origami & Folding
what one sheet can do
Series

Convergence — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. geometrypacks tocorrugation8 panels at 0.42 rad40.8% — 2.5× smallerMiura6 × 4, 15 interior vertices16.6% — 6.0× smallerfan8 sectors about one point25.0% — 4.0× smallerroll8 turns12.5% — 8.0× smallerpacked area as a fraction of deployed, computed from each geometry — not measured from any animal

    The same corrugation in four places

    A leaf, a wing, a crushed cylinder and a solar array arrive at nearly the same fold, and none of them copied any of the others. Convergence stories are cheap; this one is checkable, because the constraint that forces it can be computed rather than admired.

    part 1 · biology
  2. 0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

    Four finders, one option

    Four unrelated lineages arriving at the same corrugation is read as evidence that the corrugation is good. It is at least as much evidence that there was nothing else to arrive at: how far a folded sheet shrinks is exactly its average layer count, so a lineage choosing a packing ratio is choosing a number of layers and nothing else — and the quantity that is genuinely free turns out to be almost uncorrelated with it.

    part 2 · biology
  3. spacing rulelayers across the folded footprintratio reacheduniform12 panels, longest 1.0012.00the only one at the topgeometric-1.112 panels, longest 2.857.5038% shortgeometric-1.312 panels, longest 17.924.1565% shortalternating-212 panels, longest 2.009.0025% shortone-long12 panels, longest 3.004.6761% short12 panels · ratio = Σℓ ⁄ max ℓ · uniform is the unique maximiser, and every other rule pays for its longest panel

    The census returns one

    The rung below this one asked for a census: every pattern reaching a stated packing ratio while opening from a single input, with its crease density. The census is makeable for the corrugations and it comes back with one member. A corrugation piles its panels over a footprint as wide as its longest panel, so its ratio is the total length divided by that longest one — and that equals the panel count only when every panel is the same.

    part 3 · biology
  4. 050100150200250300350020406080100120foldspacking ratio deliveredρ 0.02 — 219 foldsρ 0.05 — 88 foldsρ 0.1 — 44 foldsρ 0.2 — 22 foldssheet of side 10 · D(k) = k(1 − k(π−2)ρ⁄S) · k* = S ⁄ 2(π−2)ρ, and the ceiling it reaches is k*⁄2

    Four materials, four optima

    The convergence argument gets its pattern and stops there. A hinge has a radius, the radius takes a fixed length of surface out of every fold, and the fold count that gets the most packing out of a sheet is inversely proportional to it — so a leaf, a wing, a gut lining and a metal array agreeing on a corrugation still disagree by an order of magnitude about how many creases to put in one.

    part 4 · biology
  5. 02040608000.20.40.60.81half-angle from shut (degrees)share of the sheet's lengthspan, every fold countclearance, 4 foldsclearance, 8 foldsclearance, 16 foldssheet 10 · the span is S·sin θ at every fold count; the clearance is (S⁄k)·cos θ and falls as the count rises

    The fold count sets the spring

    A corrugation sweeps the same span at every fold count, and the count decides only how much room the zigzag needs while it does it — which was counted as a gain with nothing pushing back. Something does push back. Every hinge is a spring, a finer corrugation has proportionally more of them, and the force to hold a given span rises exactly as the clearance falls: the product of the two is the same number whatever the count, and at each material's own best count the spring goes as one over the square of the hinge radius.

    part 5 · biology
  6. 00.20.40.60.81050100150span held, as a share of the sheetforce needed to hold it there4 folds8 folds16 foldsthe pale line under each: what it takes at flat, 4kc ⁄ Ssheet 10, hinge radius 0.05 · the corrugation springs open, and this is what has to stop it at each state

    How far open is a question about the grip

    A corrugation of hinges that rest flat is loaded when it is shut, so it opens by itself and the force in the held-state calculation is a restraint rather than a drive. Followed from shut to flat that restraint only ever falls, and by exactly π over two — so every partly open state a structure can occupy is squeezed into a band a third wide, and a grip that weakens by a third leaves the sheet nine tenths open.

    part 6 · biology
  7. -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.

    part 7 · biology
  8. 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.

    part 8 · biology

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