## 10 Body-Hinge Frameworks
### Block and Hole Polyhedra
The swapped structure \(\overline{\mathcal{P}}=(\{B\},\mathcal{H}_{\overline{\mathcal{P}}})\) has one block, which we often think of as a rigid ground, and the rest is a bar-joint framework on the edges of the polyhedron. These maps give a variant of the isomorphism between the motion assignments of \(G^{M}(\mathcal{P})\) and the equilibrium stresses of \(G_{\mathcal{S}}(\overline{\mathcal{P}})\). Such 'panel discs' are encountered implicitly in a number of studies such as [96], as well as some recent work on structures built on quad-graphs [142] in discrete differential geometry. A more recent exploration of these types of frameworks is given by Cruicksbank et al. [143]. A graph is \((3,6)\)_-tight_ if it satisfies the natural conditions from the Maxwell count: \(|E|=3|V|-6\) and for every subgraph with at least 3 vertices, \(|E^{\prime}|\leq 3|V^{\prime}|-6\). **Theorem 23** (Cruicksbank, Kitson, and Power [143]). : _Let \(G_{\mathrm{S}}(\mathcal{P})\) be the static framework graph with a single block and finitely many holes, or, a single hole and finitely many blocks. Then the following statements are equivalent:_  
1. \(G_{\mathrm{S}}(\mathcal{P})\) _is generically isostatic in_ \(\mathbb{P}^{3}\)_;_
2. \(G_{\mathrm{S}}(\mathcal{P})\) _is_ \((3,6)\)_-tight;_
3. \(G_{\mathrm{S}}(\mathcal{P})\) _is constructible from_ \(K_{3}\) _by vertex splitting operations and isostatic block substitution._  
In [108], Finbow-Singh and Whiteley conjectured that (2) is equivalent to (1) for all separated block and hole frameworks. If we develop the pure condition for an isostatic block and hole polyhedron \(\mathcal{P}\), as a bar-joint framework, there will be a factor for each block, which may depend on which generically isostatic graph was inserted. If we factor out these _block factors_, we are left with  
Figure 62: Examples of block and hole polyhedra, and their associated graphs for tracking stresses \(G_{\mathrm{S}}(\mathcal{P})\) and tracking hinge motions \(G^{M}(\mathcal{P})\). Figure by Elissa Ross, reused with permission [107]. a form of pure condition for the triangulated surface--the _surface polynomial_\(T(\mathcal{P})\). This was observed earlier in Example 21. In this example, we observed the surface polynomial was the same after we swapped. **Conjecture 3** ([107] Conjecture 5.1). : _Given a generically isostatic block and hole polyhedron \(\mathcal{P}\), the surface polynomial of \(\mathcal{P}\) is the same as the surface polynomial of the swapped polyhedron \(T(\mathcal{P})=\bar{T}(\overline{\mathcal{P}})\)._  
The second paper [108] describes ways of demonstrating that a block and hole polyhedron is, at least generically, isostatic by a range of inductive constructions, as well as corresponding reduction processes for locating a simple isostatic base structure from which to induct up to the desired example. These inductions are combinatorial, with an emphasis on vertex splitting (Figure 64). Therefore they will apply over a range of projective realisations, frameworks, and metrics. The inductive constructions in that paper can be applied to a much broader array of spatial frameworks than block and hole polyhedra. This is worth exploring in the future, but is not sufficiently projective to take up more space in this paper. We mention that a class of examples in that paper, called 'towers', were sufficiently transparent to provide initial examples of how infinitesimal motions of one part of a framework would transmit through to infinitesimal motions elsewhere: a form of _mathematical allostery_[126]. They also provided illustrative examples for motions of finite and infinite tubes, which also occur in biology of proteins [144]. Figure 64: Given a pair of edges from a shared vertex (**a**), a vertex split opens this up to a pair of triangles, with one new vertex (**b**). If we are working in part of a triangulated surface, this expands the triangulated surface, preserving infinitesimal rigidity in 3-space.