Origami & Folding
what one sheet can do
Series

Insect wings — 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

    A wing that folds into nothing

    A beetle stows a wing longer than its body under a case a fraction of that length, and the ratio is the whole engineering problem. What a fold achieves is computable from the pattern alone, and the four geometries available are not close to each other.

    part 1 · biology
  2. geometryfreedomsdrivers neededone degree-four vertexfour assignments, one motion each11Miura, 5 × 412 interior vertices, still one freedom11parallel corrugationno interior vertex to couple1112 vertices, uncoupledwhat a pattern costs when nothing constrains it1212or a sequencerone freedom is one actuator — the count is what makes a passive deployment possible at all

    No motor in the fold

    An insect's wing has muscles at its base and nothing out along its length, so the pattern has to carry the deployment by itself. The condition that makes that possible is a count: one degree of freedom means one number determines every panel, which means one thing has to pull.

    part 2 · biology
  3. 010203040506000.20.40.60.81foldsshare of the sheet lost to hinges50% of the sheet64 foldshinge radius 0.08 on a 10 unit sheet · (π − 2)ρ = 0.0913 lost per fold

    Nothing in a body folds on a line

    A crease in an organism is not a crease. It is a compliant region — a patch of thinner material that bends — and a region has a width. The width consumes surface in exact proportion to the number of folds, which puts a ceiling on how fine a pattern can usefully get.

    part 3 · biology
  4. 0102030405060010203040fold half-angle from shut (degrees)packing ratioa hinge stops here — 6°corrugation, closes in one directioncapped at 10×Miura, closes in two at oncecapped at 92×a quoted 8× is 7.2° corrugation or 20.7° Miuraa quoted 15× is 3.8° corrugation or 15.0° Miuraa quoted 30× is 1.9° corrugation or 10.5° Miuraratio = 1 ⁄ sinθ for a corrugation and 1 ⁄ sin²θ for a Miura · θ is the half-angle from shut · a hinge that stops at 6° is the dashed line

    The number is the angle

    Every packing ratio worked out so far is computed at a fold closed all the way, and a folded wing is not closed all the way. At zero thickness the ratio runs away as the fold shuts, so the size of a quoted number says how far the fold got and not what the pattern is — and the pattern contributes only an exponent, which makes the same quoted ratio mean two quite different angles depending on which geometry produced it.

    part 4 · biology
  5. the bar is how many of 200 random spring settings give two or more resting statessprings set at random fold angles, the same settings for every rowvertex 60·90·120·901982 with 1 · 198 with 2vertex 45·100·135·801964 with 1 · 196 with 2vertex 80·95·100·851964 with 1 · 126 with 2 · 62 with 3 · 8 with 4a corrugation0200 with 1a vertex's configurations are two branches through the flat state; a corrugation's are one line

    A corrugation has one resting state

    A folded wing held short of shut stores energy in its hinges, and a wing that could stay both open and folded with nothing holding it would need that energy to have two bottoms. A corrugation cannot provide them: every crease in it folds by one angle, so the energy of any set of crease springs is a parabola in that angle and has exactly one resting state, however much the springs disagree. A single degree-four vertex has two branches through the flat state, and the same springs give it two resting states on almost every setting tried.

    part 5 · biology
  6. -40-20204060801000510152025degrees of the driven crease from the flat sheetenergy in the springsthe road from one resting state to the other · sectors 60° · 90° · 120° · 90°, springs set to mixedbranch one's stateenergy 12.40branch two's stateenergy 17.65the flat sheetenergy 26.16to leave the deeperclimb 13.76to leave the shallowerclimb 8.51left of the middle is branch one, right of it branch two; they meet only at the flat sheet

    The wall is the flat sheet

    A sprung degree-four vertex usually has two resting states, one on each branch of its motion, and the branches meet in one place a sheet can pass through: the flat state. So the only road from one resting state to the other crosses the flat sheet, and on every setting of the springs tried on two vertices the flat sheet is the highest point of that road. Its energy is each spring's stiffness times its rest angle squared, summed, which does not contain the vertex's sector angles at all — the same springs put on four different vertices give a wall of exactly the same height. The geometry decides only how far below the wall each state sits, and the shallower one sits a median of five per cent below it.

    part 6 · biology
  7. the bar is the median number of resting states, over the same kind of random springsa chain shares one crease between each vertex and the next1 vertex2 states2 branch combinations · 198 of 200 settings rest on every one · 197 of 198 cross at the flat sheet2 vertices4 states4 branch combinations · 193 of 200 settings rest on every one · 200 of 200 cross at the flat sheet3 vertices8 states8 branch combinations · 181 of 200 settings rest on every one · 199 of 200 cross at the flat sheet4 vertices16 states16 branch combinations · 134 of 200 settings rest on every one · 200 of 200 cross at the flat sheetevery combination of branches holds a resting state, and every switch between them goes over the whole flat sheet

    A chain of vertices switches all at once

    One sprung degree-four vertex rests in two states, one on each branch of its motion, and switches between them only by passing through the flat sheet. Chain vertices together by sharing a crease between each and the next, and a branch can be chosen at every vertex: two, four, eight and sixteen combinations for chains of one to four. The median spring setting rests once on every combination. And every combination's curve of configurations passes through the same single point — the whole chain flat at once — and meets no other anywhere else, so every switch, even of one vertex's branch, takes the whole chain back to flat. The wall that switch climbs is every spring's flat energy added up, growing by a crease's worth for every crease, and a chain's second state sits several times further below it than a single vertex's does.

    part 7 · biology
  8. 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.

    part 8 · biology

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