Icosahedral coincidence rotations

TL;DR

This paper investigates the coincidence problem for 3D structures with icosahedral symmetry, providing a quaternion-based parametric description and a comprehensive list of rotations with indices below 100.

Contribution

It introduces a novel quaternion-based framework for describing coincidence rotations and characterizes possible indices, offering a complete enumeration for Sigma < 100.

Findings

Characterization of coincidence indices for icosahedral symmetry

Complete list of rotations with Sigma < 100

Quaternion-based parametric description of rotations

Abstract

The coincidence problem for three-dimensional discrete structures with icosahedral symmetry is reinvestigated. We present a parametric description of the coincidence rotations based on special quaternions, called icosian numbers. In particular, we give a characterization of the possible coincidence indices (Sigma-factors) and present a complete list of the possible rotations with Sigma < 100.