Generic infinitesimal rigidity for rotational groups in the plane

TL;DR

This paper provides combinatorial criteria for determining infinitesimal rigidity of symmetric frameworks in the plane under various rotational groups, extending previous results to new group orders and cases.

Contribution

It introduces new combinatorial characterisations for symmetry-generic infinitesimal rigidity for rotational groups of specific orders, including cases with joints at the rotation center.

Findings

Characterisations given in terms of sparsity counts on group-labelled quotient graphs.

Extension of previous results to groups of order 4, 6, and odd orders between 5 and 1000.

Sparsity counts are insufficient for even order groups of at least 8.

Abstract

In this paper we establish combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks in the Euclidean plane for rotational groups of order 4 and 6, and of odd order between 5 and 1000, where a joint may lie at the centre of rotation. This extends the corresponding results for these groups in the free action case obtained by R. Ikeshita and S. Tanigawa in 2015, and our recent results for the reflection group and the rotational groups of order 2 and 3 in the non-free action case. The characterisations are given in terms of sparsity counts on the corresponding group-labelled quotient graphs, and are obtained via symmetry-adapted versions of recursive Henneberg-type graph constructions. For rotational groups of even order at least 8, we show that the sparsity counts alone are not sufficient for symmetry-generic infinitesimal rigidity.