Tensegrities
The subject of tensegrities was first considered by J.C. Maxwell in Maxwell (1864), who started to investigate first questions regarding force-loads for frameworks. Nowadays tensegrities are one of the leading directions of study in modern theory of rigidity (see, e.g., Connelly (1993) for further information). Let us recall several standard definitions.
Definition 1.1
Fix a positive integer d. Let \(G=(V,E)\) be an arbitrary graph without loops and multiple edges. Let it have n vertices \(v_1,\ldots , v_n\).
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A configuration is a finite collection P of n labeled points \((p_1,p_2,\ldots ,p_n)\), where each point \(p_i\) is in a fixed Euclidean space \(\mathbb {R}^d\).
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The realization of G with straight edges, induced by mapping \(v_j\) to \(p_j\) is called a tensegrity framework and it is denoted as G(P). (Here we allow the realization to have self-intersections).
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A stress w on a framework is an assignment of real scalars \(w_{i,j}\) (called tensions) to its edges \(p_ip_j\).
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A stress w is called a self-stress if at every vertex \(p_i\) we have
$$\begin{aligned} \sum \limits _{\{j|j\ne i\}} w_{i,j}(p_j-p_i)=0. \end{aligned}$$
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A pair (G(P), w) is a tensegrity if w is a self-stress for the framework G(P).
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If \(w_{i,j} {<} 0\) then we call the edge \(p_ip_j\) a cable, if \(w_{i,j} {>} 0\) we call it a strut.
Configuration Space of Tensegrities and its Stratification
Denote by \(B_d(G)=(\mathbb {R}^d)^n\) the configuration space of all tensegrity frameworks. Let W(n) denote the linear space with coordinates \(w_{i,j}\) where \(1\le i,j\le n\). It is clear that \(\dim W(n)=n^2\).
Definition 1.2
Consider a framework \(G(P)\in B_d(G)\) and denote by W(G, P) the linear subspace of W(n) of all possible self-stresses for G(P). The space W(G, P) is the fiber at P.
Definition 1.3
Fibers \(W(G,P_1)\) and \(W(G,P_2)\) are said to be equivalent if there exists a homeomorphism \(\xi :W(G,P_1)\rightarrow W(G,P_2)\), such that for any \(w\in W(G,P_1)\) we have
$$\begin{aligned} {{\mathrm{sgn}}}\big (\xi (w_{i,j})\big )={{\mathrm{sgn}}}\big (w_{i,j}\big ) \end{aligned}$$
for every coordinate \(w_{i,j}\) of w. Here \({{\mathrm{sgn}}}\) denotes the standard sign function.
The described equivalence relation gives us a stratification of \(B_d(G)=(\mathbb {R}^d)^n.\) A stratum is by definition a maximal connected component of \(B_d(G)\) with equivalent fibers. In Doray et al. (2010) we prove that all strata are semialgebraic sets.
Remark
According to all known examples the majority of the strata of codimension k are intersections of the strata of codimension 1 see e.g. Doray et al. (2010), Karpenkov et al. (2013). So the most important case to study is the codimension 1 case.
Remark
A stratification of a subgraph is a substratification of the original graph (i.e., each stratum for a subgraph is the union of certain strata for the original graph). Hence the case of complete graphs \(K_n\) is universal. This is straightforward as each extra edge contributes at least as much to dimensions of the fibers of a stratum as to the dimensions of the fibers of adjacent strata (locally).
Example 1.4
Let us consider a simple example of \(B_1(K_{3})\), namely we study tensegrities for a complete graph on three vertices and its realizations in the line. We assume that the line has a coordinate, so each point of \(B_1(K_3)\) is associated with three coordinates \((x_1,x_2,x_3)\).
First we study a particular case \(x_1=0\), which we denote by \(B_1^0(K_3)\). The set \(B_1^0(K_3)\) has the following stratification (see Fig. 1, Left):
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1 stratum of codimension two: the origin. Here all three vertices coincide and the dimension of the fiber is 3.
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6 strata of codimension one: some pair of vertices coincide. The dimension of fiber is two.
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6 connected components of full dimension correspond to triple of distinct vertices. The dimension of a fiber is one.
It is clear that the dimension of the fiber for \((x_1,x_2,x_3)\) coincides with the dimension of the fiber \((0,x_2-x_1,x_3-x_1)\) for every \(x_1,x_2,x_3\). Therefore, we have
$$\begin{aligned} B_1(K_3)=B^0_1(K_3)\times \mathbb {R}^1. \end{aligned}$$
Example 1.5
Let now \(G_{1,2-3}\) be the graph on three vertices \(v_1,v_2,v_3\) with the only edge connecting \(v_2\) and \(v_3\). Then we have 1 stratum of codimension one defined by \(p_2=p_3\) (with fibers of dimension 1) and two connected components in the complement, i.e., where \(p_2\ne p_3\) (with fibers of dimension 0). As in previous example we have
$$\begin{aligned} B_1(G_{1,2-3})=B^0_1(G_{1,2-3})\times \mathbb {R}^1, \end{aligned}$$
where \(B^0_1(G_{1,2-3})\) is the stratification of the section \(x_1=0\) (see Fig. 1, Right).