1. Configuration:

Symmetry: Prismatic tensegrity structures interested here have dihedral symmetry with 2n symmetry operations: there is a single major n-fold rotation axis, which we assume is the vertical, z-axis, and n 2-fold rotation axes perpendicular to this. Any node can be transform to any other by one and only one of these symmetry operations. Hence, there are totally 2n nodes, and numbered 1, 2, ..., 2n according to the order of the symmetry operations in dihedral group. The nodes 1 to n and n+1 to 2n lie in two different horizontal planes perpendicular to z-axis.

Connectivity: Every node is connected by three different types of members: two horizontal cables, one vertical cable and one strut. The horizontal cables are connected by the nodes i and i+h in the same horizontal plane, while the vertical cables and struts are connected by the nodes in different planes. For convenience, fix the connectivity of struts to be connected by node i and n+i in different planes, and assume that vertical cables are connected by i+v and n+i +v . Thus, connectivity of the structure is denoted by the variables h for horizontal cables and v for vertical cables.

Connectivity of the structure is h=1 and v=1.
Nodes {1,2,3} and {4,5,6} are in different horizontal planes.

2. Self-equilibrium:

The nodes (members of the same type) can be taken to each other by a symmetry operation of dihedral group, and the members of the same type have the same axial forces and lengths, and therefore, the same force densities. Hence, it is sufficient to consider only one node instead of the whole structure to determine the force densities in self-equilibrium and then the configuration.

The self-equilibrium equations of one node in every direction can be written in a matrix form as Ax=0, wher A is a 3-by-3 matrix, which is determined by the force densities and the reducible representation matrices [2], and x denotes the coordinates of this node in three-dimensional space. Making A singular to let x have non-trivial solutions, we can determine the force densities and then configuration of the structure (by x and symmetry operations).

Stability (with minimality of energy): A structure can be stable if and only if it can return to its original configuration subject to small disturbances. In other words, the structure is stable if its current configuration has strictly minimum energy such that any disturbances will lead to strictly increase of energy, because structure intends to stay in the state with lower energy. Stability is related to level of self-stress and material properties.

Prestress Stability: Assume that the stiffness of members is infinite compared to level of self-stress. If the structure is stable in this case, then it is said to be prestress stable, which can be considered as a special case of stability. Hence, stability implies prestress stability, but the inverse is not always correct.

Super Stability: This is an unconditional stability, regardless of level of self-stress and material properties. Thus, super stability implies stability. It is the strongest criterion among these three and we obviously prefer this one in the design of tensegrity structures. For the prismatic tensegrity structures with dihedral symmetry, they are super stable if the horizontal cables are connected by adjacent nodes; i.e., nodes i and i+1.

Divisibility: Dependent on the connectivity, some prismatic tensegrity structures can be physically and mechanically divided into several identical substructures. Because the substructures can have relative motions, the original structure is obviously not stable.

[1] Connelly, R. and Terrell, M., 1995. Globally rigid symmetric tensegrities. Structural Topology, 21, 59--78.
[2] J.Y. Zhang, S.D. Guest and M. Ohsaki, Symmetric prismatic tensegrity structures. Proc. of Symposium of IASS, Beijing, China, Oct., 2006. <pdf>(corrected version) and <ppt>.
[3] J.Y. Zhang, S.D. Guest and M. Ohsaki, Symmetric prismatic tensegrity structures: Part I. configuration and stability. to be submitted to Int. J. Solids & Structures <pdf>.
[4] J.Y. Zhang, S.D. Guest and M. Ohsaki, Symmetric prismatic tensegrity structures: Part II. symmetry-adapted formulation. in preparation.