Symmetric Frameworks on Surfaces

Wall, Joe and Nixon, Anthony and Schulze, Bernd (2024) Symmetric Frameworks on Surfaces. PhD thesis, Lancaster University.

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Abstract

We present a study of combinatorial constructions that are related to understanding the structure of bar-joint frameworks that are restricted to a subspace of Rd. There are two such restrictions of Rd we approach. We combine two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in R3 restricted to move on a surface. In Chapter 3 necessary combinatorial conditions are given for a symmetric framework on the sphere, cylinder, cone, elliptical cylinder and ellipsoid to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In Chapter 4 we focus exclusively on the cylinder. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts. Motivated by applications where boundary conditions play a significant role, one may generalise and consider linearly constrained frameworks where some vertices are constrained to move on fixed affine subspaces. Additional to Chapter 3, the necessary combinatorial conditions are given for a symmetric linearly constrained framework in Rd to be isostatic under a choice finite point group symmetries. In Chapter 5, we consider linearly constrained frameworks in the plane, and the case of rotation symmetry groups whose order is either 2 or odd. These conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts. To conclude there is a short chapter in which we suggest ways for this research to be furthered.