Origami & Folding
what one sheet can do
Generator

What the vertex does on the way — the vertex kinematics generator

A generator in the rigid folding library, called 49 times across 11 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

vertex-kinematics is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

What the vertex does on the wayThe four fold angles of a degree-four vertex against the parameter that drives it, solved from the spherical linkage the creases make. Setting one angle sets the other three, which is the single degree of freedom. Where the vertex can reach a flat state the signs split three to one throughout the motion, which is Maekawa's theorem holding all the way and not only at the end; where it cannot, they need not, and that is the difference the two theorems are about.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

view: "wells", show: "chain-census", trials: 200

A resting state for every choice of branchesFor chains of one to four degree-four vertices sharing creases, with random springs on every crease, the median number of resting states, beside how many branch combinations the chain has and how often the road between its two deepest states crosses the flat sheet. The resting states double with every vertex added, one for each combination.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

view: "branches", sectors: [45, 100, 135, 80]

The ways a vertex can leave the flat stateFor one developable vertex of degree four, every direction in fold-angle space along which the closure still holds a little way out of the flat state. Each row is one mode, given as the ratios of the four fold angles. A mode that moves all four creases is the usual gear ratio; a mode that moves two is a simple fold along a straight crease running through the vertex.the sectors are 45°, 100°, 135°, 80° — each row is one way the vertex can start to foldcrease 1crease 2crease 3crease 4mode 10.67-0.230.670.234 of the four creases movemode 20.31-0.64-0.31-0.644 of the four creases movethe numbers are the four fold angles' ratios to one another as the vertex leaves the flat state

view: "wells", show: "chain-curves", chain: 2, trials: 200

Every configuration of a chain of 2 verticesThe energy in the springs of a chain of 2 degree-four vertices sharing creases, along every one of its 4 curves of configurations — one for each choice of branch at each vertex — against the first vertex's driven fold angle. Every curve passes through the flat sheet in the middle, and each holds a resting state of its own.-150-100-5050100150050100150the first vertex's driven crease (degrees)energy in the springsa chain of 2 vertices, one random setting of its 7 springsbranches one, onebranches two, onebranches one, twobranches two, twothe flat sheet: 27.7every curve is one choice of branch at each vertex; all of them cross at the flat sheet and nowhere else

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

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.

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.

A crease held the other way

Hold a creased sheet on the far side of flat and its memory crosses zero after τ ln(1 + φ₀⁄φₕ), a time that contains no stiffness and no fold count. Asking what happens at that moment shows that two accounts of a corrugation's energy cannot both stand: a hinge whose memory can cross zero gives one parabola and no barrier at all. The barrier belongs to a degree-four vertex, whose two branches turn out to be at right angles in the space of its fold angles. Held on the other branch its wall dips and never falls, and its two resting states trade depth at nearly the same logarithm, with lengths in place of angles. Only the exact mirror takes every rest angle through zero together, which is the fastest way to erase a crease and the least forgiving.

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.

One crease decides the sheet

Fix one crease of a flat-folding problem, propagate every condition the subject has, and three creases out of a hundred and fifty-eight follow. Fix one fold angle of a rigid one and every crease on the sheet follows, with a single consistent answer. The same experiment, two questions, opposite answers — and it is why a self-folding sheet needs one biased vertex rather than one per vertex.

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.

The vertex is geared

A rigid four-crease vertex has one degree of freedom, which says that one number decides everything and not how. The how is a fixed ratio: the tangents of the half fold angles at two creases stay in constant proportion for the whole of the motion, and the proportion is a function of the sector angles and nothing else.

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.

Two mechanisms at one point

Two creases drawn across each other cannot fold flat — Maekawa's count refuses them at every angle. They move perfectly well as rigid panels, and they move in two ways: bend along one line while the other stays flat, or the reverse. Every other developable vertex of degree four has two ways too, and in both of them all four creases move together at a fixed ratio. The crossing is the case where the two motions have nothing to do with each other.

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.

Which crease to push

Deciding one fold angle settles every other one on a quadrilateral mesh, which is what makes a self-folding sheet buildable with a single actuator. It leaves a question that sounds like an afterthought: which crease. Driving each of a mesh's twenty-four in turn gives twenty-four different answers to how far an error in it travels — and on the sheet that repeats one vertex, it gives several answers to what shape the sheet takes.

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