Chiral polytopes from hyperbolic honeycombs

Abstract

Abstract polytopes are partially ordered structures which generalize the notion of polyhedra in a combinatorial sense. This concept includes all of the classical regular polytopes as well as many other well-known configurations. Chiral polytopes are abstract polytopes with maximal rotational symmetry which lack reflexive symmetry. We employ hyperbolic geometry to derive toroidal abstract polytopes of type {6, 3, 4} and {6, 3, 5} which are either regular or chiral. Their rotation groups are projective linear groups over finite rings.