The Eisenlohr-Farris algorithm for fully transitive polyhedra

The purpose of this note is to present a method for classifying three-dimensional polyhedra in terms of their symmetry groups. This method is constructive, and it is described in terms of the conjugation classes of crystallographic groups in \(\mathbb{E}^3\). For each class of groups \(\Gamma\) the method can generate without duplication all polyhedra in three-dimensional space on which \(\Gamma\) acts fully-transitively. It was proposed by J. M. Eisenlohr and S. L. Farris for generating every fully transitive polyhedron in \(\mathbb{E}^d\). We also illustrate how the method can be applied in the Euclidean space \(\mathbb{E}^3\).