2022/10/12 by Anthony Nixon, Nixon, Anthony, Bernd Schulze +3
Biochemistry, Genetics and Molecular Biology · Engineering · #05C70 #20C35 #52C25 #Advanced Materials and Mechanics #Cellular Mechanics and Interactions #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2210.06060
openalex publication_date 2022/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A bar-joint framework (G,p) is the combination of a finite simple graph G=(V,E) and a placement p:V→ ℝd. The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines 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 particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. 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.