On Roli's Cube

Abstract

First described in 2014, Roli's cube $\mathcal{R}$ is a chiral $4$-polytope, faithfully realized in Euclidean $4$-space (a situation earlier thought to be impossible). Here we describe $\mathcal{R}$ in a new way, determine its minimal regular cover, and reveal connections to the M\"{o}bius-Kantor configuration.