All hyperbolic Coxeter $n$-cubes

TL;DR

This paper proves the non-existence of hyperbolic Coxeter n-cubes for dimensions six and higher, and classifies all such cubes in dimensions five and below using combinatorial and algebraic methods.

Contribution

It establishes a complete classification of hyperbolic Coxeter n-cubes for n ≤ 5 and proves their non-existence for n ≥ 6, filling a significant gap in geometric polyhedron theory.

Findings

No hyperbolic Coxeter n-cubes for n ≥ 6.

Complete classification of hyperbolic Coxeter n-cubes for n ≤ 5.

Methods implemented in Mathematica for combinatorial and algebraic analysis.

Abstract

Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a full classification for $n\leq 5$. Our methods, which are essentially of combinatorial and algebraic nature, can be (and have been successfully) implemented in a symbolic computation software such as Mathematica$^\circledR$.