Chiral Extensions of Regular Toroids

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the n-dimensional torus ( \(n \geqslant 2\) $n⩾2$ ). As a consequence, we prove that for every \(d \geqslant 3\) $d⩾3$ there exist infinitely many chiral d-polytopes.