## 7 Pure Conditions
### Pure Conditions: Basic Theorems  
In the next two subsections we present a number of results from White and Whiteley [85]. The goal is to compute a single polynomial in the projective coordinates of the vertices of a generically isostatic graph, which is zero if and only if the corresponding framework has an equilibrium stress, and the rank of the rigidity matrix drops in rank. The idea is to square up the rigidity matrix so we can use the determinant to generate the desired polynomial. There are two ways to square this matrix up: add rows or delete columns. In [85], White and Whiteley add rows, because this gives a tool to prove that, in the end, the polynomial does not depend on which columns are deleted, or equivalently, that any good choice of added rows generates a simple factor depending only on those added rows, leaving a single polynomial \(C(G)\) which depends on the graph but not on the added rows or on the columns deleted. For an isostatic graph \(G=(V,E)\) realised generically in \(\mathbb{P}^{d}\), a _tie-down_\(T\) of a framework \((G,p)\) in \(\mathbb{P}^{d}\) is a set of \(n=\binom{d+1}{2}\) bars of the form \(ax\) with \(a\in V\) and \(x\
otin V\) where \(m(x)=0\) for every infinitesimal motion \(m\) and each such bar adds a row to the rigidity matrix (which is nonzero only in the columns corresponding to \(a\)). The tie-down bars are chosen to remove all infinitesimal motions and hence pin the framework. The following matrix shows the rows of a _basic tie-down_ of \(G\) in \(\mathbb{P}^{d}\): \(M_{G}(T)\) with \(d+(d-1)+\cdots+1=\binom{d+1}{2}\) rows. Figure 25: The design of the Hartford roof as a half-octahedral tetrahedral truss (**a**) and an image after the collapse from a snow load (**b**), shortly after the sports fans left the arena [101; 102]. We now have a sequence of steps drawn from [85] to complete this analysis and prove there is a unique pure condition for an isostatic graph. 1. The first step is a lemma from [85]. **Lemma 1**. : _A framework \((G,p)\) in general position in \(\mathbb{P}^{d}\) is isostatic if and only if there exists a tie-down \(T\) which produces an invertible extended rigidity matrix \(R(G,p,T)\)._
2. If we represent the tie-down bars of a framework by 2-extensors, we can construct a square \(\binom{d+1}{2}\times\binom{d+1}{2}\) matrix with determinant \(C(T)\) in the bracket algebra which is non-zero if and only if the tie-down will not support an equilibrium stress (the tie-down rows are independent). These are the _non-degenerate tie-downs_ with \(C(T)\
eq 0\). 3. For \(v_{i}\in V\), let \(\alpha_{i}\) be the number of tie-down bars incident to \(v_{i}\), and assume that we have reindexed so that \(\alpha_{1}\geq\alpha_{2}\geq\ldots\geq\alpha_{m}\). Then \(C(T)\
eq 0\) if and only if \[\sum_{i=1}^{k}\alpha_{i}\leq dk-\binom{k}{2}\text{ for all }k,\;1\leq k\leq n-1.\]  
Figure 26: Possible patterns of non-degenerate tie-downs of an isostatic framework in \(d=3\). As the figure indicates, we can index the tie-downs by their sequence of attachments. Suppose \(G\) is isostatic in \(\mathbb{P}^{d}\) and \(T\) is a non-degenerate tie-down. Then the determinant of the extended rigidity matrix \(R(G,p,T)\) equals an element \(C(G,T)\) of the bracket ring \(B\) on the set of vertices of \(G\cup T\)[85]. 5. For a non-degenerate tie-down \(T\), the polynomial \(C(T)\) is a factor of the larger determinant \(C(G,T)\) so that \(C(G,T)=C(T)C_{T}(G)\), for some bracket polynomial \(C_{T}(G)\). 6. For two non-degenerate tie-downs \(T,T^{\prime}\) the residual factors \(C_{T}(G)=C_{T^{\prime}}(G)\), so there is a unique pure condition \(C(G)\). This uses a lemma that moves one tie-down edge at a time along an edge of \(G\), provided the moves preserve the non-degeneracy of the tie-down. **Theorem 8** (White and Whiteley [85]). : _Suppose \(G\) is isostatic in \(\mathbb{P}^{d}\).