Symmetric versions of Laman's Theorem

TL;DR

This paper extends Laman's Theorem to symmetric frameworks with 3-fold rotational symmetry, providing new criteria and characterizations for isostatic structures under symmetry constraints.

Contribution

It proves that symmetry-based restrictions, combined with Laman conditions, are sufficient for isostaticity in $C_3$ symmetric frameworks and establishes symmetric versions of classical theorems.

Findings

Symmetric Laman conditions are sufficient for $C_3$-symmetric frameworks.

Symmetric versions of Henneberg's and Crapo's Theorems are established.

Results extend to other symmetry groups like $C_2$ and $C_s$.

Abstract

Recent work has shown that if an isostatic bar and joint framework possesses non-trivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are `fixed' by various symmetry operations of the framework. For the group $C_3$ which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed in [4] that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with $C_3$ symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's Theorem and Crapo's Theorem for $C_3$ which provide alternate characterizations of `generically' isostatic graphs with $C_3$ symmetry. As shown in [19], our techniques can be extended to establish analogous results for the symmetry groups $C_2$ and $C_s$ which are generated by a half-turn and a reflection in the plane, respectively.