## 8 Polarity for Rigidity
### Sheet Structures and Polarity for Rigidity in \(\mathbb{R}^{3}\)
: _A jointed-sheetwork is a bipartite incidence graph \(G=(A,B;I)\) with an assignment \(P^{i}\) of weighted plane segments (3-extensors) to the vertices in \(A\) and an assignment \(p_{j}\) of weighted points in projective space (1-extensors) to the vertices in \(B\) such that \(P^{i}\wedge p_{j}=0\) (the point lies on the plane) for each pair \((i,j)\in I\)._  
Since these jointed-sheetworks include bar-joint frameworks, all the same gaps in combinatorial characterisations of independence and infinitesimal rigidity in \(\mathbb{R}^{3}\) remain. Notice that the entire presentation here was thoroughly projective, so the theory must immediately include points, edges, and faces at infinity. It can also be presented with point centers of motion for sheets and plane momenta for joints. These have not yet been explored in appropriate detail. With the same projective lens, there is an immediate transfer of infinitesimal and static rigidity of shetworks to Minkowski space both via polarity in that space and also with a direct transfer of the rigidity analysis of each of the structures from Euclidean space. Similarly, the infinitesimal and static rigidity of sheetworks transfer sheets to bar-joint equivalence classes of sheets, in all projective metrics, including spherical frameworks. In \(\mathbb{R}^{3}\), polarity takes a pure condition on points for bar-joint frameworks to a polar pure condition on faces for sheet structures. Analogously, there will be a pure condition for point and sheet structures with variables for both points and faces in the polynomial. Note that polarity within spherical space (and hyperbolic space) takes distance constraints to angle constraints [69]. This is very different than polarity in the Euclidean space, followed by direct transfers. We return to this connection in Section 8.4 for a special class of frameworks: triangulated spheres. Figure 37: The opposite faces of a convex octahedron form an isostatic sheetwork (**a**), with the octahedral framework as a model (**b**). A polar is the sheetwork on opposite vertices of the cube (**c**), with the tetrahedral framework as a model (**d**). Figures are adapted from [114]. **Remark 6**. : _There are a number of avenues for further exploration of sheetworks. Some of these might be recognized in actual models, if we have sufficient vision. Figure 38 illustrates transformations among the various forms of sheetworks. We end with a few areas for further extensions._  
1. _There is a complete geometric theory of infinitesimal motions of sheetworks with projective centers of motion. Each sheet has a point centre in the sheet, and the two centres on sheets at a shared edge satisfy a compatibility condition which is the polar of the condition for bar-joint frameworks. Does this offer additional insights?_
2. _What happens with points, lines and sheets at infinity?_
3. _What about four compactness sheets--the polar of four coplanar points of a tetrahedron. The polar will be four sheets through a single point, but with six specified hinge lines through this point. What will the infinitesimal motions look like?_
4. _Are there any examples of sheetworks with finite motions where sheets remain as coplanar sheets? Even the polar of the double banana--with two sheets joining two bodies--only has finite motions which warp these joining sheets. For example, the polar of the double banana shown in Figure 39b,c only has infinitesimal motions which bend the two sheets (d), though as a triangulated model it does have a finite motion._
5. _Consider the polar of_ \(\mathrm{K}_{4,6}\) _with the four points coplanar. The four points become four sheets which are co-punctual, with no hinges (edges) among them. As the polar of_ \(6\)_-valent vertices, these four sheets are hexagonal sheets. These sheets must meet in_ \(6\) _four-valent sheets. This type of geometry has yet to be explored._
6. _The analogue of tensegrity frameworks are slotted sheetworks.