Antonio Montero

An (n+1)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar (n+1)-toroids with at most n flag-orbits; in particular, we discuss a classification of 2-orbit toroids of arbitrary dimension.

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 \(n ≥2\). As a consequence, we prove that for every \(d ≥3\)there exist infinitely many chiral d-polytopes.

Voltage operations extend traditional geometric and combinatorial operations (such as medial, truncation, prism, and pyramid over a polytope) to operations on maniplexes, maps, polytopes, and hypertopes. In classical operations, the symmetries of the original object remain in the resulting one, but sometimes additional symmetries are created; the same situation arises with voltage operations. We characterise the automorphisms of the new object that are derived from the original one and use this to bound the number of flag orbits (under the action of its automorphism group) of the new object in terms of the original one. The conditions under which the automorphism group of the original object is the same as the automorphism group of the resulting object are given. We also look at the cases where there is additional symmetry which can be accurately described due to the symmetries of the operation itself.