Where the branches meet

Every folding of a crease pattern passes through one configuration: the flat sheet. It is where the paper starts, and it is the only state two different foldings of the same pattern have in common — which makes it a crossing rather than an ordinary configuration.

It is also the one configuration at which the field’s instrument is wrong about all of them.

Why the rank falls there #

At the flat state every crease lies in the plane. The rows a vertex contributes to the constraint Jacobian are the directions of its creases in space, three components each — and if every direction has a zero third component, those rows span two dimensions rather than three.

So the rank falls by exactly one per interior vertex, and the nullity rises by the same.

A three-by-three Miura sheet has twelve creases and four interior vertices. Folded, its rank is 11 and its mobility is 1. Flat, its rank is 8 — exactly $2 \times 4$ — and its nullity is four. At four by four the flat nullity is 6 against a folded 1; at five by five, 8 against 1; at six by six, 10 against 1.

None of that is a numerical accident or a shortcoming of the routine. It is what the flat state is.

The vertex, in the coordinate where it is simple #

Before the sheet, the vertex — because the branches are visible there in a way they are not on a patch.

Drive one crease of a flat-foldable degree-four vertex from a fold of 0.15 radians out to 2.9, and record another crease’s angle. The curve is not a line: at 0.2 the second crease is at 0.037, at 1.0 it is at 0.201, at 2.6 it is at 1.175.

Take the tangent of half of each and the ratio is $0.184792531$ at every point, moving by $6 \times 10^{-16}$ across the whole travel. In that coordinate the branch is a straight line through the origin, and the two branches are two lines through the origin with different slopes.

That is the clearest possible picture of what a bifurcation is. Two straight lines crossing at a point: away from the point each is a one-dimensional mechanism, and at the point the pair of them spans a plane that neither of them is in. A rank taken at the crossing reports the plane.

Why the tangent of the half angle is the coordinate that does this is a property of the vertex being flat-foldable — its opposite sectors summing to a straight angle — and it is the subject of the vertex’s own essay. What it buys here is that the branch structure can be drawn rather than inferred.

What the extra directions are #

Three of the three-by-three sheet’s four apparent freedoms are not motions of the folded mechanism. They are the tangent directions of other branches.

A degree-four vertex has two folding modes. On the first, one opposite pair of creases takes almost all of the fold and the other pair barely moves; on the second the pairs change places. Both pass through the flat state and cross there and nowhere else.

At the crossing, the tangent directions of both branches are available to the constraint matrix at once, and the matrix cannot distinguish “a direction some branch leaves in” from “a direction this mechanism can move in”. The nullity counts them all.

That is the general shape of a bifurcation and it is not peculiar to paper. It is the same object this site met when a solve’s branches turned out to be components of a configuration space, and again when an arm’s eight postures turned out to be separated by configurations at which the Jacobian drops rank. The difference here is that the crossing is not somewhere out in the workspace. It is the state the mechanism is manufactured in.

The reading that cannot be made #

The consequence is worth stating as sharply as the measurement allows, because it is the reason this rung exists.

Take a Miura pattern, which folds. Take the same grid with its interior vertices displaced by a tenth of a panel, which folds to no angle at all. Measure both at the flat state.

Twelve creases, four interior vertices, rank 8, nullity 4 — for both.

Identical rank, identical nullity, identical everything the first-order analysis can produce, on a mechanism with one freedom and on a structure with none. The flat state’s first-order behaviour does not know the difference, and the difference is entirely second order.

So the honest instrument for “does this pattern fold” is not a rank at the flat state. It is to push the pattern off the flat state and see whether anything closes.

Pushing it off #

Which means the flat state cannot be used as a starting point either, and that has a practical consequence for every fold in this field.

A Newton step taken at the flat state goes into a subspace that leaves the sheet flat. The solve converges instantly to the configuration it started in and reports success — a residual of nought, achieved by not moving. A rigidly foldable pattern has to be pushed off its flat state by hand, and which way it is pushed decides which branch it lands on.

The push used throughout this field is the pattern’s own mountain-and-valley assignment written as signs, at a tenth of a radian. Seeded that way the Miura sheet converges in six or seven iterations to the Miura mode.

Seeded with every crease of a vertex at the same magnitude it converges somewhere else — to a configuration in which some creases come to rest at exactly nought and the sheet folds along the rest. That is a genuine rigid folding, with mobility one like the other, and it is not the Miura mode. It is a fold along a straight line that happens to be made of two collinear creases, which every flat-foldable degree-four vertex has.

A solver lands on whichever branch it was aimed at, exactly as a four-bar’s solve lands on whichever assembly configuration its seed was near. What is different is that here the seed is not a refinement of a known answer; it is the choice of mechanism.

Counting the branches #

How many branches leave the flat state is a question with a partial answer and it is worth being precise about which part.

For a single degree-four vertex the answer is two, and it is established by enumeration rather than by search: every assignment of a sign to every crease at two magnitudes is handed to the solver — thirty-two solves for four creases — and the distinct answers are collected. Two proper modes come back, plus the degenerate line-folds. That is an enumeration, so a branch cannot be missed by a seed landing badly.

For a pattern the count is larger and this field does not claim it. Four vertices with two modes each is at most sixteen combinations, most of which are incompatible; which of them survive is a question about the assembly and is the crease-pattern subject’s rather than this one’s. What is measured here is the dimension of the tangent cone at the flat state — four on a three-by-three sheet — and the mobility on each branch that a solve reaches, which is one.

The gap between those two numbers is the whole content of the rung: four tangent directions, and every branch through them one-dimensional.

What a nullity of four means, exactly #

It is worth being careful about the four, because “four freedoms at the flat state” invites a picture that is wrong.

It does not mean the flat sheet can move in four independent ways. It cannot move in any way at all while staying flat, and the four directions are not motions.

What it means is that the tangent cone to the configuration set at that point lies inside a four-dimensional subspace. The configuration set near the flat state is a union of branches, each a curve; their tangents together span at most four dimensions; and the null space is the smallest linear thing containing all of them. A tangent cone is not a tangent space, and at a crossing point the second is strictly larger than the first is useful.

The same distinction has a name everywhere it appears. In the algebra field it is the difference between a variety and its tangent space at a singular point; here it is the difference between what a sheet can do and what its linearisation permits. Both are the reason a rank is an upper bound and not an answer.

Why the flat state is not a defect #

There is a temptation to treat the flat state as a degeneracy to be avoided, and it is worth resisting for a reason that is about mechanisms rather than about arithmetic.

The flat state is where a folded structure is made, stored and shipped. It is the configuration a deployable spends most of its life in and the one it has to be able to leave reliably. A mechanism whose only branch point is at its own manufacturing state is not a mechanism with a defect; it is a mechanism whose design problem includes deciding which branch it will leave along, and providing something that decides it — a crease already pressed in, a stop, a spring.

This site has met that shape of problem before in the plane. A four-bar has two assembly configurations and cannot pass between them without being taken apart; a designer picks one and builds it. The difference is that a four-bar is built in the configuration it will work in, and a folded sheet is built in the configuration where the choice is still open.

How the flat state was measured at all #

A configuration at which the Jacobian is rank-deficient is a configuration a solver will not converge to, so it is worth saying how the flat state’s numbers were obtained.

They were not obtained by solving. The flat state is known exactly — every fold angle is nought — so the Jacobian is written down there directly and its rank taken, with no iteration anywhere in it. That is the one configuration in this field where the mechanism does not have to be solved for before it can be drawn, and it is the reason the measurement is trustworthy: there is no seed, no convergence and no tolerance in the configuration, only in the rank.

The rank decision itself is not close. On the three-by-three sheet the eight retained singular values run over about one order of magnitude and the four discarded ones are at $10^{-16}$ of the largest; the gap is effectively infinite. So the rank of 8 is not a threshold effect, and the claim that it equals $2V$ exactly on every size in the table is a claim about the geometry rather than about the arithmetic.

There is one thing the flat measurement cannot be asked, and it is worth naming because it is the natural next question. The nullity says how many directions there are; it says nothing about how many branches use them, and a four-dimensional cone could be spanned by four branches, or by ten, or by two branches and a two-dimensional family. Deciding that needs the branches to be found, which is a solve from a seed, which is the machinery above.

The same crossing in the plane #

It is worth one paragraph on what the corresponding thing looks like in a mechanism with no folding in it, because the flat state can otherwise read as a peculiarity of paper.

A lazy tong fully closed — every scissor at zero opening, every bar on top of its neighbour — is the same kind of point. The connection lines all coincide, the constraint rows lose independence, and the assembly can leave that configuration in more than one way: opening to the left of the line and opening to the right are different mechanisms sharing one state. Nothing about the tong’s arithmetic anywhere else in its range shows it, because everywhere else the tong has exactly the four freedoms it is counted with.

What is different about a crease pattern is only that the branch point is where the mechanism is made, so it cannot be designed around by keeping the mechanism away from it. Every folded structure starts flat and has to leave, and which way it leaves is a decision somebody has to take.

The mountain-and-valley marks are the branch #

The flat state being a branch point explains a convention every origami diagram carries and no crease pattern computation can do without, and it is worth naming because it turns an annotation into a piece of the mathematics.

A bare crease pattern — lines on paper, no other marking — is at the branch point and stays there. It does not determine a folded shape, because the branches leaving the flat state are several and nothing in the lines says which. Adding mountain and valley marks to each crease is exactly the choice of branch: a sign per crease, chosen once, and the folded state follows.

So the marks are not a convenience for the folder. They are the missing data, and a diagram without them specifies an ambiguous object in precisely the way a four-bar’s lengths specify an ambiguous mechanism until the assembly is chosen. The site’s own vocabulary covers it: a branch is a component, and the marks name the component.

The physical version of the same fact is a failure mode of real deployable structures, and it is the reason they are not simply flat sheets with hinges. A sheet at its flat state can be pushed onto either branch by whatever pushes it, so a structure deployed by an actuator at the flat state may open into the wrong shape — not a shape with an error in it, a different shape, reached by a legitimate folding of the same pattern.

The remedy in hardware is the same as the remedy in the solver, which is the pleasing part. A solver seeded at the flat state goes nowhere; seeded with the pattern’s own mountain-and-valley assignment written as a small fold, it converges onto the intended branch. A structure biased with a small pre-fold at every crease — a scored line, a moulded hinge, a memory in the material — leaves the branch point in the intended direction for the same reason. The seed and the pre-fold are the same object, one in a computation and one in a part.

Which gives the branch point its proper standing in the field. It is not a degeneracy to be avoided, since it is where the sheet is made and stored; it is a place where the mechanism has to be told which way to go, and both the analysis and the hardware have to carry that instruction. A pattern that does not carry it is under-specified, and the under-specification is invisible in every first-order quantity the field computes.

What it does to the rest of the field #

Two consequences, and both have been used already.

Every rank in this field is taken at a folded state, never at the flat one. The scaling table’s rank of 263 at twelve by twelve is measured after Newton has converged to a fold of 0.4 radians, and the same sheet measured flat would give 242 and a nullity of 22. The tables say which, because the two numbers answer different questions.

And the flat state is where the second-order machinery earns its place. A first-order flex may be a motion or nothing, and the flat state of a crease pattern is the configuration where that distinction stops being an edge case and becomes the ordinary situation: every pattern has one, every pattern’s nullity is too large there, and on a pattern that does not fold all of the nullity is spurious.

The degree-three vertex is the smallest instance and it is worth keeping in mind as the shape of the whole thing. Three creases, three constraints, rank 2 at the flat state, nullity 1 — and no fold exists at any angle, with a residual that starts at $1.2 \times 10^{-3}$ and grows. One apparent freedom, no actual ones, and nothing in the linear algebra to say so.