Growing super stable tensegrity frameworks

This paper discusses methods for growing tensegrity frameworks akin to what are now known as Henneberg constructions (HCs), which apply to bar-joint frameworks. In particular, this paper presents tensegrity framework versions of the three key HCs of vertex addition, edge splitting, and framework merging (where separate frameworks are combined into a larger framework). This is done for super stable tensegrity frameworks in an ambient 2-D or 3-D space. We start with the operation of adding a new vertex to an original super stable tensegrity framework, named vertex addition. We prove that the new tensegrity framework can be super stable as well if the new vertex is attached to the original framework by an appropriate number of members, which include struts or cables, with suitably assigned stresses. Edge splitting can be secured in R2 (R3) by adding a vertex joined to three (four) existing vertices, two of which are connected by a member, and then removing that member. This procedure, with appropriate selection of struts or cables, preserves super-stability. In d-dimensional ambient space, merging two super stable frameworks sharing at least d+1 vertices that are in general positions, we show that the resulting tensegrity framework is still super stable. Based on these results, we further investigate the strategies of merging two super stable tensegrity frameworks in IRd, (d ϵ {2, 3}) that share fewer than d+1 vertices, and show how they may be merged through the insertion of struts or cables as appropriate between the two structures, with a super stable structure resulting from the merge.