Every vertex is a spherical linkage

A crease pattern’s interior vertex is a mechanism, and it is one this site already owns — which means it has a mobility that can be counted and measured like any other.

Four creases meet at a point. Each is a hinge. Every one of those hinge axes passes through the vertex, so no panel can translate relative to any other: the whole assembly of four panels and four hinges lives on a sphere about the vertex, and what it is, exactly, is a spherical four-bar — the object the spatial field built when it counted link lengths in angles.

The translation is worth doing slowly, because it turns a subject that looks like paper-folding into one the rest of the site can reach.

The arcs are the link lengths #

A spherical linkage’s link lengths are the arcs between consecutive axes, measured on the unit sphere. So the question is what a fold pattern’s arcs are, and the answer is on the drawing.

They are the sector angles of the flat pattern. A vertex whose four sectors are 60°, 100°, 120° and 80° is a spherical four-bar whose four arcs are 60°, 100°, 120° and 80°, and it is that at every fold — measured here at folds of 0.2, 0.5, 1.0, 1.5, 2.0 and 2.6 radians, with the worst departure of an arc from its sector being $8 \times 10^{-14}$ radians and typically $4 \times 10^{-16}$.

That is not a definition and it is not obvious. The arcs are computed from the folded configuration: the panels are placed in space by composing hinge rotations, the crease directions are read off the result, and the angle between consecutive directions is measured. That those measured angles come back as the numbers printed on the flat drawing is a statement about the mechanism — that a panel is rigid, and that the angle it holds between two of its own edges is the angle it was cut with.

It also settles something the previous essay left open. The sector angles of any flat vertex sum to a full turn, so every vertex of every crease pattern is a spherical polygon whose arcs sum to $2\pi$ — a closed spherical polygon, always, for free. Developability is that sentence and nothing more, which is why it distinguishes nothing.

The closure, written out #

It is worth writing the vertex’s closure down once, because everything above depends on it and it is three lines.

Number the creases in the cyclic order the flat pattern puts them in, and let the panel between crease $k$ and crease $k+1$ be $P_k$. Crossing crease $k+1$ from $P_k$ to $P_{k+1}$ is a rotation about that crease’s line by its fold angle. Going once round the vertex and back to where the walk started, the composed rotation has to be the identity:

$$\prod_{k} R(\hat{u}_k, \rho_k) = I.$$

Every axis passes through the vertex, so there is no translation to track and the product lives in the rotation group alone: three scalar conditions, taken as the logarithm of whatever rotation is left over. That is the same closure a spatial loop solves, with the axes concurrent — and it is why a vertex costs three constraints rather than the six a general loop would.

Two signs have to be right in that product and neither is guessable: which way round the vertex the walk goes at each step, and which end of the crease its axis is taken to point from. Getting either wrong gives a residual that is still small on a symmetric pattern, because a wrong sign is itself a symmetry there — which is why the derivative of that product is checked against a finite difference on every pattern this field draws, and why the first version of it, with the frames right and the traversal wrong, disagreed by 0.35 and converged anyway.

What the degree buys #

A spherical polygon with $d$ axes is a linkage with $d$ joints and one loop, so on the sphere it has $d - 3$ freedoms. That gives the degree of a vertex an immediate meaning.

Three creases is a spherical triangle, and a triangle does not move. A degree-three vertex is rigid, and the sheet folded at one is not a mechanism.

Four creases is a spherical four-bar: one freedom, two branches, and everything the planar four-bar has with sines where the planar case has lengths.

Five or more has two freedoms or more, and a pattern built of them is a much looser object than a Miura.

The degree-three case is worth measuring rather than asserting, because it is where a rank misleads.

At the flat state, all three of a degree-three vertex’s crease directions lie in the plane, so its three constraint rows span two dimensions and the nullity comes back as one: a freedom, at a configuration where there is none. Asked to fold to 0.05 radians the residual is $1.2 \times 10^{-3}$; at 0.2 radians $1.9 \times 10^{-2}$; at half a radian $1.2 \times 10^{-1}$. It never falls, and it grows with the fold.

That is the smallest object on this site that shows a first-order freedom which is not a motion, and it is the same phenomenon the two-bar framework makes in the plane with two bars and a pin.

The relation along a branch #

The degree-four vertex has a freedom, and following it produces a number worth having.

Drive the first crease from 0.15 radians to 2.9 and watch the second. The relation between them is thoroughly nonlinear: at $\rho_1 = 0.2$ the second crease is at 0.037 radians, at $\rho_1 = 1.0$ it is at 0.201, and at $\rho_1 = 2.6$ it is at 1.175. Nothing about that curve looks like a ratio.

Take the tangent of half of each and divide:

$$\frac{\tan(\rho_2/2)}{\tan(\rho_1/2)} = 0.184792531$$

at every point on the branch, moving by $6 \times 10^{-16}$ over the whole travel. The third crease’s ratio against the first is $-1.000000000$ and the fourth’s is $-0.184792531$.

In the half-angle tangent the mechanism is a constant ratio, which is a gearbox’s property turning up on a folded sheet of paper. Nothing was fitted; the points are solves of the vertex’s closure at twenty-two different folds, and the constancy is the measurement.

That the relation exists at all is a consequence of the vertex being flat-foldable — its opposite sectors summing to a straight angle, $60° + 120° = 100° + 80° = 180°$ here. A vertex without that property has a perfectly good branch and no such constant: driven at 0.5 radians, a vertex of 70°, 105°, 80° and 105° gives fold angles of 0.500, 0.023, $-0.488$ and $-0.023$, and the corresponding ratios wander.

This is the site’s own recurring shape — a ratio that is not a number — arriving with the sign reversed. Here a quantity that has no business being constant is, in one particular coordinate, and only for one particular family of vertices.

The two branches #

A spherical four-bar has two assembly configurations, and so does a vertex.

On the first branch one opposite pair of creases takes almost all of the fold and the other pair barely moves; on the second they change places. Driven at half a radian, the first gives fold angles of 0.500, 0.094, $-0.500$ and $-0.094$; the second gives 0.177, 0.500, 0.177 and 0.500.

These are two different mechanisms sharing one drawing. A sheet folded into either cannot reach the other without being flattened completely — which is exactly the situation a four-bar’s two assembly configurations are in, and an arm’s eight postures, and the components a solve’s paths turned out to run between.

The choice is made at the flat state, where the two branches meet, and it is made by whichever way the sheet is first pushed. That crossing is the subject of its own essay; what matters here is that the vertex has exactly two of them and that neither is reachable from the other.

Reading the mechanism off the sphere #

There is a small dividend in having the vertex as a spherical linkage, which is that everything the site knows about four-bars transfers with sines in place of lengths.

A spherical four-bar has a Grashof condition: whether a given arc can turn all the way round depends on the four arcs in the same way it does in the plane, and the answer decides whether a crease can be folded through a full turn or only through a range. On the vertex above, driving the first crease works from the flat state out to 2.9 radians and stops; the mechanism has limit positions, and they are the folds at which two of the crease directions become coplanar in the wrong way.

It has transmission angles, which decide how much of a driven crease’s motion reaches the far one — visible directly in the branch measurements, where a driven crease at 2.6 radians has moved its neighbour only to 1.17.

And it has the same two branches a planar four-bar has, arising the same way, from the two intersections of two cones instead of two circles.

None of that has to be re-derived. It is the same mechanism, and the only thing new about it is that its link lengths are printed on a flat sheet of paper as the angles between creases, which is a construction no linkage designer would have thought to use.

Why this makes the field computable #

The practical consequence is the reason the translation was worth doing.

A network of vertices is a network of spherical linkages sharing their joints. That is what makes the Jacobian cheap: at a folded state, the rows of a vertex’s block are the directions its creases take in space, three rows and one column per crease, and nothing else needs computing. It is the same statement as saying that the derivative of a loop’s closure with respect to a joint angle is that joint’s axis — the fact the spatial field’s loop solver is built on — with the axes all through one point so that the screw has no linear part.

That is why a hundred and forty-four panels can be analysed at all. Written as rigid bodies in space it would be six coordinates per panel and five constraints per hinge: 864 unknowns and 1,320 equations at twelve by twelve. Written as fold angles with spherical closures it is 264 and 363, and the same answer comes out.

What the sphere does not carry #

Two things are lost in moving to the sphere, and one of them matters.

Where the vertex is does not survive, and does not need to: a spherical linkage is about directions, so two vertices with the same sectors are the same mechanism wherever they sit on the sheet. That is what makes a tessellation of identical vertices worth building.

But the sheet is not a collection of vertices. The moment two vertices share a crease, that crease’s fold angle appears in both closures, and the assembly’s mobility stops being any vertex’s business. A network of four spherical four-bars, each with one freedom, can have one freedom or none, and which of those it has is not a property of any of them.

So the translation is exact and local, and it buys the Jacobian and nothing above it. The sphere is where a vertex lives; the assembly lives somewhere with a hundred conditions in it.

What a sheet of many vertices inherits #

The last thing worth taking from the sphere is what it says about a pattern built of many identical vertices, because it explains the shape of every deployable in this field.

If two vertices have the same four sector angles, they are the same spherical linkage, so they have the same input–output relation between their creases — the same constant of 0.184792531, if they are flat-foldable, or the same curve if they are not. A tessellation of identical vertices is therefore a network of identical mechanisms, and the compatibility conditions between neighbours are all copies of one condition rather than a hundred different ones.

That is the whole reason the Miura works. Its vertices are congruent up to reflection, so its $(n-2)^2$ conditions are a hundred copies of one statement, satisfied by construction rather than by search. The moment the vertices differ, the conditions differ too, and a designer is solving a hundred distinct equations in two hundred and forty-two unknowns with no structure to exploit.

The corresponding fact in the plane is an angulated element’s kink, which is one number chosen once for a whole ring; and the fact this site had already met is that a shared name is not a shared mechanism — two vertices drawn with the same four numbers really are one mechanism, and two drawn with different numbers are not, however alike the drawing looks.

Grashof, on paper #

The inheritance is listed above as three properties that need no re-derivation, and one of them cashes out into a statement about paper that a folder can check with a protractor — which is worth doing, because it is the clearest evidence that the translation is more than a change of vocabulary.

A spherical four-bar’s Grashof condition decides which of its links can turn all the way round: the shortest arc plus the longest, against the other two, with sines rather than lengths. The vertex’s arcs are its sector angles, so the condition is an inequality on the four angles printed on the flat pattern, evaluated before anything is folded.

What it decides, translated back, is whether a given crease is a crank or a rocker. A crank crease can be driven through its whole range; a rocker crease reaches a limit and comes back, and the limit is a fold angle short of where the folder wants to go. That is a familiar experience at the paper — a crease that will not close, that resists past a certain point and creases the panel beside it instead of folding further — and it is usually attributed to the paper. It is the sector angles.

So a folder meeting a crease that will not close has a diagnosis available from the flat drawing, and the diagnosis names the four angles at that vertex rather than the material. That is a genuinely useful thing for a spherical linkage’s classical condition to buy, and nothing about it needed a new computation: it is Grashof’s inequality with the four sector angles substituted in.

It also says something about why flat-foldable vertices are the ones the field is built on. A vertex is flat-foldable when its opposite sector angles are supplementary, and that condition interacts with Grashof’s in a way that makes the whole range available — which is what flat-foldable means physically, that every crease can be brought to a full fold. A vertex that is not flat-foldable has a limit somewhere, and the limit is what the inequality is measuring.

Which completes the translation’s account of itself. The sphere supplies the mechanism, the mechanism supplies its classical conditions, and the conditions come back as statements about angles a folder can measure on flat paper. Nothing is added and nothing is approximated; a subject about paper turns out to have been a subject about spherical linkages, and every result the spatial field established transfers with the sector angles substituted for the arcs.

The dihedral and the fold angle #

One last piece of bookkeeping, because it is a place two conventions meet and disagree.

The fold angle is what this field’s unknowns are: nought when a crease is unfolded, positive one way and negative the other. The dihedral angle of the sheet at a crease is what a geometer measures between the two panels: a straight angle when the crease is unfolded, and closing as the crease folds.

They are the same quantity with a straight angle between them, and the measurements say so: at fold angles of 11.459°, 2.124°, $-11.459$° and $-2.124$° the dihedrals come out as 177.876°, 168.541°, $-177.876$° and 168.541°, so $|\rho| + |\text{dihedral}| = 180°$ on every crease. At the far end of the branch, with fold angles of 166.158° and 113.399°, the dihedrals are 13.842° and 66.601° — a sheet nearly folded onto itself.

Nothing turns on which is used as long as one is, and this field uses the fold angle throughout because it is nought at the flat state and the flat state is where every branch starts.