29 Pencil realizations: hinges concurrent within a common panel

In dimension three, this chapter and section 30 strengthen KT Theorem 5.6 in the hinge-coplanar model of section 23.5. A pencil realization gives each body, besides its panel, a point of that panel, through which every hinge at the body passes (definition 29.2). The hinges at a body then form a pencil of lines through a point in a plane. We show that, over every infinite field, every spanning multigraph \(G\) has a pencil realization of rank \(6(|V(G)| - 1) - \operatorname {def}(\tilde G)\), the deficiency rank of theorem 23.66. If \(G\) is simple, the realization can be chosen with the points of adjacent bodies projectively distinct. If moreover \(G\) has a nondegenerate pencil realization (definition 29.23), it has one of that rank. These claims make up theorem 29.34. Besides section 23.5, the reader needs the deficiency and rigid subgraphs of section 20.

Section 29.1 and section 29.2 give the definitions. Section 29.4 places the two independent hinges of Katoh and Tanigawa’s base case (KT Lemma 5.3) through one point of their common panel. Section 29.6 gives an added edge a hinge through the points of both its ends. Such a hinge exists only when the point of each end lies in the panel of the other (lemma 29.12). These conditions can fail at the edges that Katoh and Tanigawa add to a realization of a minimal spanning subgraph. So we induct over every multigraph, not only over the minimal ones (theorem 29.15). Section 29.7 states this induction and proves its cases at a loop, on at most two bodies and at a cut edge. Section 29.8 adds the distinct points and the nondegenerate realizations to the induction statement. Katoh and Tanigawa’s induction statement likewise asks for nonparallel realizations (KT Theorem 5.5). The cases at a loop, on at most two bodies and at a cut edge hold for the strengthened statement too, so it follows from the two remaining cases (theorem 29.30). Section 29.9 proves these two cases, where \(G\) has a proper rigid subgraph and where it is two-edge-connected without one. At a simple two-edge-connected multigraph both follow from two statements (definition 29.31), which section 30 proves by lifting pictures of \(G\) in the plane.

Two parts of the chapter are not used in the proof of theorem 29.34. Section 29.3 shows that the polarity of lemma 25.9 takes a pencil realization to one with the panel normals and the points exchanged. This chapter uses the polarity nowhere else. Section 29.5 realizes the cycles on three and four bodies as pencil realizations.

29.1 Extensors through a point

Hinge-coplanarity is a containment condition: a screw element lies in the panel with normal \(n\) when it is the extensor of points of the hyperplane \(n^{\perp }\) (definition 23.24). We dualize this to concurrency. A screw element passing through a point replaces containment in the hyperplane by containment of the point in the element’s span: at \(k = 2\) this says a line — the extensor of two points — contains the point.

For \(k \ge 2\) the zero element passes through every point (take every \(p_i = q\)), so we treat the nonzero-ness of \(C\) as a separate hypothesis wherever the notion is used below, where \(k = 2\). For \(k = 1\) the extensor of \(q\) is \(q\), and the zero element passes through no nonzero point.

29.2 Pencil realizations

A pencil realization asks that, at every body, the incident hinges lie in the body’s panel and pass through one point of that panel. This is the classical picture of a pencil of lines through a point in a plane. In the molecular reading (section 25.1) the panel is the plane of an atom’s bonds and the point is the atom’s centre.

The incidence \(q_v \cdot n_v = 0\) is forced at any body with a nonzero hinge: the hinge’s spanning points lie in \(n_v^{\perp }\) and contain \(q_v\) in their span, so \(q_v \in n_v^{\perp }\). We require it at every body, isolated ones included, so that the normal and the point play symmetric roles under the polarity below (section 29.3). Two hinges in one panel always pass through a common point of it (lemma 29.8). So at a body of degree at most two, asking the hinges to pass through one point adds nothing to hinge-coplanarity. The requirement first has content at a body with three or more hinges.

29.3 The polarity and self-duality

Work at grade \(k = 2\) (\(d = 3\)) for the rest of this subsection, the setting of lemma 25.9. There the polarity sends the extensor of two homogeneous points \(v_0, v_1 \in K^4\) to the panel-support extensor \(C(v_0,v_1)\) (definition 23.1), over every field \(K\): the join-of-two-points \(\leftrightarrow \) meet-of-two-panels identity. The two transport lemmas below also hold over every field \(K\). They show that the polarity turns passing through the point \(q\) into lying in the panel with normal \(q\), and lying in that panel into passing through \(q\).

We record this symmetry for its own sake: the proof of theorem 29.34 does not use it. It is one reason why definition 29.2 allows the points of a link’s two endpoints to coincide: requiring them to be projectively distinct would break the symmetry. section 29.8 gives a second reason, and imposes distinctness as a separate condition.

29.4 The coincident-panel pencil pair

This subsection supplies the pencil condition at the base case of theorem 29.15: every loopless multigraph on at most two vertices (lemma 29.21). Katoh–Tanigawa’s two-vertex base realization of a parallel edge pair (lemma 23.34, KT Lemma 5.3, p. 669) places both hinges in a single shared panel \(n^{\perp }\) with two linearly independent supporting extensors (lemma 23.31), which fill the whole screw space \(K^6\) and make the two bodies rigid. On the pencil condition the same construction supplies a common concurrency point at no extra cost: two lines of a flat pencil already share the point of the pencil.

Katoh–Tanigawa realize the base of their induction directly, at one and two bodies (lemma 23.36). Asking the hinges at a body to pass through one point has content only at degree three or more, so on two bodies it has content only when three or more edges join them. lemma 29.21 handles such a class by placing both bodies on a single panel at a fixed normal, with two independent hinges through a common point of the panel (lemma 29.6) making the two bodies rigid (lemma 23.30); each further edge reuses one of the two hinges. It assembles this with the edgeless and single-edge cases into the base case of theorem 29.15.

29.5 Pencil cycles: degree-two concurrency is automatic

Katoh–Tanigawa’s cycle realization (lemma 23.80, KT Lemma 5.4) places a cycle of bodies with each hinge the meet of two adjacent panels. Every body has degree two, and at a body of degree at most two the concurrency requirement costs nothing: two lines lying in a common plane always meet in a point of that plane. The following records this projective fact in the containment model, so that the cycle realization — indeed any hinge-coplanar realization (definition 23.63) restricted to its degree-two bodies — already admits the per-body concurrency point a pencil realization asks for. The same fact accounts for two features of the nondegeneracy conditions in section 29.8. A pencil hub there is a body of degree at least three, since only there does passing through one point constrain the hinges. At a body of degree two, whose two hinges pass through a common point for free, a further condition keeps them from coinciding. theorem 29.15 does not call on the two pencil realizations of a cycle built below; they give a direct example of the same fact, on the cycle realization Katoh–Tanigawa already use.

The range \(3 \le m \le 4\) is exactly where the shared-normal family exists at \(d = 3\) (lemma 23.62), so the triangle and the quadrilateral are both realized. Since every body of a cycle has degree two, the concurrency fact above (lemma 29.8) supplies each body’s point directly, alongside the shared-normal family’s panel data, giving a pencil realization outright.

The coplanar realization below is its corollary, discarding the per-body concurrency point: a hinge-coplanar panel realization needs only the panel data, which the pencil realization above already supplies, built directly from the cyclic family of shared panel normals rather than through the meet model of the original cycle realization.

29.6 The two-pencil extension step

Katoh–Tanigawa’s induction extends a hinge-coplanar realization across a re-added edge by supplying a supporting extensor in the meet of the two endpoint panels (p. 670): for any two normals the common perp \(n_u^{\perp } \cap n_v^{\perp } \subseteq K^4\) has dimension at least \(4 - 2 = 2\), so it contains a nonzero hinge lying in both panels, with no transversality of the normals needed. On the pencil condition the re-added hinge must additionally pass through the concurrency point of each endpoint. Where in the coplanar model a common hinge always exists, the pencil model asks for one through two prescribed points as well, and this constrains the two bodies: a hinge lying in both panels and passing through both points confines each point to the common perp, so each concurrency point must lie not only in its own panel but in the other body’s panel too. The following existence step assumes exactly these two cross-incidences.

The two cross-incidences are not only sufficient but necessary, so the extension hinge exists precisely when the two bodies’ concurrency points are mutually panel-incident.

These cross-incidences are the chapter’s first condition that the coplanar model’s extension step does not need, and three later arguments turn on them. Katoh–Tanigawa reach a graph that is not minimal by adding edges to a realization of a minimal spanning subgraph. For pencil realizations the cross-incidences of the added edges cannot always be met, so we run the induction on every multigraph (section 29.7). The cut-edge case meets the cross-incidences of its crossing edge by first repositioning one side along a projective automorphism (lemma 29.18). At \(K_4\) the necessity direction of lemma 29.12 forces every vertex’s panel normal into a single hyperplane, so no simple realization of \(K_4\) can give all four vertices independent panel normals (section 29.8).

29.7 Reduction on all spanning multigraphs

This subsection states the induction behind theorem 29.34 and proves three of its five cases. Katoh–Tanigawa’s induction runs over minimal graphs, those from which no edge can be deleted without changing the deficiency (theorem 21.56). They reach any other graph by realizing a minimal spanning subgraph with the same deficiency and adding the remaining edges, each with a hinge in the meet of its two panels (theorem 23.66). The added edges need not raise the rank, since the subgraph already attains the deficiency rank.

For pencil realizations this step fails, and the failure lies in the smaller graph. An added edge needs its two cross-incidences (lemma 29.12), and imposing them can lower the rank of the smaller graph. Take \(K_4\) and a \(4\)-cycle in it, both of deficiency \(0\). A pencil realization of the \(4\)-cycle has rank \(14\) plus the dimension of the span of its four hinges, so it attains its deficiency rank \(18\) only when the hinges are linearly independent. The cross-incidences of the two diagonals, with those the cycle edges satisfy already, put all four points in every panel, so the points span at most a plane. If they span a plane, every panel is that plane, and the hinges lie in the three-dimensional space of its lines. If they span a line, at least two edges of the cycle join bodies with different points, and both of their hinges are that line. If they coincide, all four hinges pass through one point. In each case the hinges span at most three dimensions, so no pencil realization of the \(4\)-cycle at its deficiency rank admits the diagonals.

We therefore run the induction on every multigraph. Its case analysis still needs a vertex of degree two when there is no proper rigid subgraph. Katoh–Tanigawa find one by an edge count that uses minimality (KT Lemmas 4.5 and 4.6, pp. 663–664). Here a counting fact takes its place: a loopless multigraph whose vertices all have degree at least three has a proper rigid subgraph (lemma 29.14). So a graph without one has a vertex of degree at most two, and two-edge-connectivity forces that degree to be exactly two. The statement the induction proves is the following.

We take the five cases in the order of the lemmas below. The loop case (i) is lemma 29.16. The cut-edge case (iii) is lemma 29.19, after two lemmas on projective automorphisms. The base case (ii) is lemma 29.21. The contraction and split cases, (iv) and (v), are left for later subsections. Section 29.8 strengthens the statement that the induction proves. Section 29.9 then proves both cases by one argument, which rests on section 30. That argument and section 30 contract proper rigid subgraphs, using lemma 29.22: contraction leaves the deficiency unchanged.

The cut-edge case moves one side of a realization by a projective automorphism. The next lemma shows that such a move keeps a pencil realization at the deficiency rank, and the one after it chooses the automorphism.

29.8 Nondegenerate realizations and the conditioned pair

Katoh and Tanigawa’s induction statement asks for more than the rank. When the graph is simple, their realization must also be nonparallel, that is, no two of its panels are parallel (KT p. 668; KT Theorem 5.5, p. 670). This subsection defines a pencil analogue, the nondegenerate pencil realization, and strengthens the statement of the induction to include it. The strengthened statement, the conditioned pair, asks for a nondegenerate realization at the deficiency rank under two conditions on the graph. We show that neither condition can be dropped. The subsection ends with the reduction of theorem 29.15 for the conditioned pair. Throughout, \(k = 2\), so panel normals and concurrency points lie in \(K^4\).

The pencil condition constrains the hinges at a body only when the body has three or more of them (section 29.2). The next definition calls such a body a hub. It asks for independence among the normals and among the points around each body.

The first condition makes the hinge of every link the line through the points of its two ends (lemma 22.18). The other two are limited by the incidences of a pencil realization. The hinge of an edge \(vw\) is nonzero, lies in the panel of \(w\) and passes through \(q_v\), so \(q_v \cdot n_w = 0\) (lemma 29.12). Thus the normals of \(v\) and its neighbours lie in the three-dimensional space \(q_v^{\perp }\), and their points lie in the panel \(n_v^{\perp }\), also of dimension three. Hence the normals of a closed hub-neighbourhood are independent only if it has at most three members. At \(K_4\) every body is a hub and every closed hub-neighbourhood has four members, so \(K_4\) is not nondegeneracy-feasible. Likewise, the points of a closed neighbourhood are independent only if it has at most three members, which rules out every hub of a simple graph. The third condition is asked only at bodies that are not hubs. At such a body it follows from the first condition and \(q_v \neq 0\), unless the body has two distinct neighbours. There it asks that the two hinges at the body be distinct lines. The first two conditions do not imply this. The path on three bodies has a pencil realization whose three points are distinct but collinear, and both of its hinges are then the same line.

In the next definition, generic names only the property stated, not an algebraic independence of coordinates.

Keeping only the first condition of definition 29.23 gives a weaker notion, which every simple multigraph with at least one body satisfies at the deficiency rank (theorem 29.34). We do not add this condition to definition 29.2, for two reasons. It would break the symmetry noted after lemma 29.5. Also, some multigraphs would have no pencil realization at the deficiency rank. Two bodies joined by two edges have deficiency rank \(D = 6\), which needs two independent hinges. At distinct points both hinges are the line through the two points, and the rank is \(5\).

A generic pencil realization is adjacent-distinct, and an adjacent-distinct one is in particular a pencil realization at the deficiency rank.

The induction statement asks for a generic realization under two conditions. Simplicity alone, the condition of KT Theorem 5.5, is not enough, since \(K_4\) is simple and not nondegeneracy-feasible. Nondegeneracy-feasibility alone is not enough either. Two bodies joined by two edges are nondegeneracy-feasible. Neither is a hub, so it suffices to take two distinct panels, two distinct points on their common line, and that line as both hinges. By the count above, none of their nondegenerate realizations attains the deficiency rank. So the statement asks for both conditions.

We now prove the reduction for the conditioned pair, beginning with the three cases that section 29.7 proves for the bare statement.

29.9 Reduction to the main component

This subsection proves the two hypotheses of theorem 29.30, the contraction and split cases for the conditioned pair, and with them the pencil conjecture. At a simple two-edge-connected multigraph, both cases follow from two statements: the multigraph has an adjacent-distinct pencil realization at the deficiency rank, and a generic one when it is nondegeneracy-feasible. These are the main-component statements, proved in section 30. A loopless two-edge-connected multigraph that is not simple has two parallel edges, and we contract them, as Katoh and Tanigawa do (KT Lemma 6.2).

The idea is to build pencil realizations by lifting planar pictures. Place each body \(v\) at a point \(q_v = (x_v, y_v)\) of the plane, with adjacent bodies at distinct points and no closed neighbourhood on a line. Such a picture is admissible (definition 30.1). Lift \(v\) to the point \(p_v = (x_v, y_v, z_v, 1)\) of \(K^4\) at a height \(z_v\) (definition 30.6). Take \(p_v\) as the concurrency point of \(v\), and the line through \(p_u\) and \(p_v\) as the hinge of an edge \(uv\). Every hinge then passes through the points of its two ends. The hinges at \(v\) lie in a common panel exactly when the points of the closed neighbourhood of \(v\) lie in a plane. So the lifted configuration is a pencil realization when every closed neighbourhood is coplanar. Over a fixed picture \(q\) this is a linear condition on the heights (lemma 30.12), and the heights that satisfy it form a linear space \(L(q)\), the lifting space (definition 30.2).

Over the admissible pictures at which \(\dim L(q)\) is least, these configurations form a vector bundle. Its closure is the main component of the configuration space (remark 30.19). In section 30 we show that, at every simple connected multigraph whose bodies all have degree at least two, a general configuration of the main component has the deficiency rank (theorem 30.139). The proof is a strong induction on the number of bodies. A lifted configuration at the deficiency rank is an adjacent-distinct pencil realization (lemma 30.11), which gives the first statement. The second does not follow in this way. When two pencil hubs have three common neighbours, as in \(K_{2,3}\), no configuration over a general picture is nondegenerate (lemma 30.127). We prove the second statement instead inside the induction of theorem 29.15, from the conditioned pair at smaller multigraphs (theorem 30.137).