On Roli's Cube

TL;DR

This paper introduces a new perspective on Roli's cube, a chiral 4-polytope, detailing its structure, minimal regular cover, and links to the M"{o}bius-Kantor configuration, challenging previous assumptions about its realizability.

Contribution

It provides a novel description of Roli's cube, determines its minimal regular cover, and uncovers its connection to the M"{o}bius-Kantor configuration.

Findings

Roli's cube is a faithfully realized chiral 4-polytope in Euclidean 4-space.

The minimal regular cover of Roli's cube is identified.

Connections between Roli's cube and the M"{o}bius-Kantor configuration are established.

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.