Origami & Folding
what one sheet can do
Concept

Branch — where it appears

A point where a procedure or a mechanism can continue in more than one way. At a tie the crimp reduction has one, and at any fold angle a degree-four vertex has one, and neither is resolved by anything local.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

050100150-100100how far the vertex is drivenfold anglesectors60° 90° 120° 90°crease 1: Mcrease 2: Vcrease 3: Mcrease 4: Mcrease 2 is the odd onethree agree, one does notKawasaki holdsand it reaches flatfound by the linkage,not by the theorem

What the vertex does on the way

A four-crease vertex is a linkage on a sphere. Solving its closure gives the fold angles at every moment, and two theorems that are usually proved about the flat state turn up in the answer without being put there.

rigid · Rigid folding 10 mountains, 14 valleys3.95 wide, 2.10 deep13 mountains, 11 valleys1.98 wide, 1.79 deep

The Miura folds two ways

One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.

tessellation · Miura 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.

biology · Insect wings -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.

biology · Insect wings 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.

biology · Insect wings

Named alongside it

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

Degrees of freedomRigid foldingSpherical linkageBifurcationInsect wingsClosureFold angleLayer multiplicityMiuraQuadrilateral meshSymmetry

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