## 8 Polarity for Rigidity
### Cauchy, Alexandrov and Polarity  
There is a cluster of rigidity theorems which were initially proven for convex triangulated spheres in \(\mathbb{R}^{3}\)[117, 118], but have generalisations to a broader class of structures in \(\mathbb{P}^{3}\). There are higher dimensional extensions [97] with the 2-dimensional faces of a convex polytope in \(\mathbb{P}^{d}\) triangulated in their planes giving infinitesimal rigidity for the entire convex polytope \(\mathbb{P}^{d}\)[119]. The combinatorial analogue of Cauchy's Theorem, which says that all triangulated spheres are generically rigid (proven by vertex splitting from a triangle [120]), transfers directly to bivariate splines on triangulated spheres (see Section 11 and [121]). **Definition 5**. : _A strictly convex polyhedral framework is a framework formed by_  
1. _placing a joint at each vertex of a strictly convex polyhedron,_
2. _placing a bar along each edge of the polyhedron._  
We note that the graph \(G=(V,E)\) of any triangulation of the sphere, by Euler's formula, satisfies \(|E|=3|V|-6\). Thus, such frameworks are isostatic in \(\mathbb{P}^{3}\) if and only if there is no non-zero equilibrium stress. The proofs in [97, 118] show there is only the all zero equilibrium stress. Cauchy's original proof was for a related theorem about the global uniqueness of convex triangulated polyhedra, within the class of all convex triangulated polyhedra. We give an infinitesimal rigidity version for an extended class of polyhedra which is found in the book of Alexandrov [122] and was reworked with statics in [97]. **Theorem 14** (Alexandrov [97, 122]). : _A strictly convex polyhedral framework in \(\mathbb{P}^{3}\) with joints at the vertices and bars on the natural edges, and additional bars to triangulate each face polygon which is not already a triangle, is isostatic (Figure 41)._  
Alexandrov further extended this geometric result by adding additional vertices along the original convex edges of the polyhedron, and ensuring that these vertices are included in isostatic frameworks in both of the faces at this edge [97, 122]. This preserves infinitesimal rigidity in \(\mathbb{P}^{3}\). Note that the geometric polar of a triangulated spherical polyhedron (or simplicial polyhedron in \(\mathbb{P}^{3}\)) is a hinged sheetwork on a simple polyhedron, which fits within Alexandrov's Theorem. As theorems about the infinitesimal rigidity of bar-joint frameworks, the results of Cauchy and Alexandrov transfer directly to spherical, hyperbolic, Minkowski and de Sitter spaces. They also extend to include vertices and edges at infinity. Some variants of Cauchy's Theorem include sending a vertex to infinity, creating an open polyhedron with edges fanning out to infinity. We can extend these theorems to convex simplicial polytopes in higher dimensions. The proof connects the 3-dimensional result to the vertex figure viewed as a cone of a convex  
Figure 41: A strictly convex triangulated polyhedron is isostatic (**a**). A more general strictly convex polyhedron (**b**) is isostatic if all faces are triangulated (**c**). polytope of the next lower dimension [97]. It is also related to Alexandrov's theorem in the sense that within their 3-space, the cells of a 4-polytope are statically rigid frameworks, once all 2-faces are triangulated. **Definition 6**. : _A strictly convex 4-polytopial framework is a bar-joint framework in \(\mathbb{P}^{4}\) built on a strictly convex 4-polytope by_  
1. _placing a joint at each vertex of the polytope,_
2. _placing a bar on each edge of the 4-polytope._  
We can give the 4-space analogue of Theorem 14. **Theorem 15** (Whiteley [97]). : _A strictly convex 4-polytopial framework, with all 2-faces triangulated, is infinitesimally rigid in \(\mathbb{P}^{4}\)._  
Clearly it follows that the graph \(G=(V,E)\) of the convex 4-polytope satisfies \(|E|\geq 4|V|-10\). As Kalai observed [119], this proves a case of the lower bound theorem for 4-polytopes.