## 4 Results
The term \"tensegrity\" used for the above examples in the references [6, 8, 12, 15, 16, 17, 18, 20, 37, 41, 42, 64, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,  
Two zero eigenvalues in \(\mathbf{L}\) are responsible for the existence of two mechanisms (\(M\)) (Figure 7):  
\[ \begin{array}{cccccccccccc}\mathbf{x}_{15}=&\{-0.24&0.24&-0.27&0.27&-0.27&-0.27&- 0.24&-0.24&0.24\\ &-0.24&0.24&0.24&-0.21&0.27&-0.21&-0.27\}\\ \end{array} \]  
\[ \begin{array}{cccccccccccc}\mathbf{x}_{16}=&\{-0.12&0.12&0.26&-0.26&0.26&0.26& -0.12&-0.12&0.12\\ &-0.12&0.12&0.12&-0.49&-0.26&-0.49&-0.26\}\end{array} .\]  
One zero eigenvalue presented in \(\mathbf{M}\) is responsible for the existence of a self-stress state (\(S\)):  
\[ \begin{array}{cccccccccccc}\mathbf{y}_{16}=&\{0.38&0.19&0.13&0.19&0.38&0.38&0. 27&0.38&0.19\\ &&0.13&0.19&-0.13&-0.13&-0.27&-0.27\}\end{array} .\]  
In this case, all eigenvalues of the matrix \((\mathbf{K}_{L}+\mathbf{K}_{G})\) are positive:  
\[\mathbf{O}=\text{diag}[\ \ 2.83&2.69&2.42&2.07&1.86&1.43&1.13&1.10&0.85&0.64&0.43&0.28&0.14&0.13&0.05],\]  
which means that mechanisms are balanced by the self-stress state--the mechanisms are infinitesimal. For this truss the system of compression members (thick line) is discontinuous (\(D\)) and included inside the set of elements in tension (\(I\)), which are cables (\(C\)). This is a pure tensegrity with two mechanisms stabilized by one self-stress. A selection of pure tensegrities found in the literature is shown in Figure 8:  
* the truss (\(T\)) with nine elements (\(n=9\)) and nine d.o.f. (\(m=9\)) [6,20] (Figure 8a),  
Figure 6: Second pure tensegrity truss. Figure 7: Infinitesimal mechanisms of the second pure tensegrity truss: (**a**) \(\mathbf{x}_{15}\), (**b**) \(\mathbf{x}_{16}\). * the truss (\(T\)) with nine elements (\(n=9\)) and nine d.o.f. (\(m=9\)) [4; 8; 9; 35; 38; 39; 41; 74; 85; 86; 88] (Figure 8b),
* the truss (\(T\)) with 12 elements (\(n=12\)) and 13 d.o.f. (\(m=13\)) [4; 38; 41; 70] (Figure 8c),
* the truss (\(T\)) with 12 elements (\(n=12\)) and 13 d.o.f. (\(m=13\)) [41] (Figure 8d). For all the trusses there is one zero eigenvalue in matrix \(\mathbf{M}\) which is responsible for the existence of the one self-stress state (\(S\)) shown in Figure 8. For these trusses the system of compression members (thick red line) is discontinuous (\(D\)) and included inside the set of components in tension (\(I\)), which are cables (\(C\)). For the first two cases (Figure 8a,b) there is one zero eigenvalue in matrix \(\mathbf{L}\) responsible for the existence of one mechanism (\(M\)). For the last two cases (Figure 8c,d) two mechanisms are identified. All mechanisms (Figure 9) are infinitesimal because eigenvalues of (\(\mathbf{K}_{L}+\mathbf{K}_{G}\)) are positive. The analyzed trusses are pure tensegrities with infinitesimal mechanisms stabilized by one self-stress. According to the features presented above the tensegrities presented in Figure 8 are promising as unit cells for the applications of 2D metamaterials with unusual mechanical properties [30]. Figure 8: Values of self-stress state for four pure tensegrity trusses (**a–d**). An interesting, as well as instructive, example is presented in Figure 9(a). It is the simplest truss (\(T\)) with two elements \((n=2)\) and two d.o.f. \((m=2)\)[6,8,20,64,77]\(\mathbf{q}=\left[ \begin{array}{cc}q_{3}&q_{4}\end{array} \right]^{T}\). Here, the compatibility matrix is square \(\mathbf{B}\big{(}\in\mathbb{R}^{2\times 2}\big{)}\), thus the matrices \(\mathbf{L}\) and \(\mathbf{M}\big{(}\in\mathbb{R}^{2\times 2}\big{)}\) are the same:  
\[\mathbf{L}=\mathbf{M}=\mathrm{diag}\Big{[} \begin{array}{cc}2&0\end{array}  \Big{]}.\]  
The zero eigenvalue in \(\mathbf{L}\) and \(\mathbf{M}\) is responsible for the existence of the mechanism (\(M\)) (Figure 9(b)):  
\[\mathbf{x}_{2}=\Big{\{} \begin{array}{cc}0&-1\end{array} \Big{\}}.\]  
and the self-stress state (\(S\)):  
\[\mathbf{y}_{2}=\Big{\{} \begin{array}{cc}0.707&0.707\end{array}