Universal Alternating Semiregular Polytopes

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion. Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ , an alternating semiregular $(n+1)$ -polytope defined for any pair of regular $n$ -polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of ${\mathcal{P}}$ , ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ , and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$ -polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or ${\mathcal{Q}}$ .

Symmetry Groups Associated with Tilings of a Flat Torus

A flat torus E^2/Λ is the quotient of the Euclidean plane E^2 with a full rank lattice Λ generated by two linearly independent vectors v_1 and v_2. A motif-transitive tiling T of the plane whose symmetry group G contains translations with vectors v_1 and v_2 induces a tiling T^* of the flat torus. Using a sequence of injective maps, we realize T^* as a tiling T-of a round torus (the surface of a doughnut) in the Euclidean space E^3. This realization is obtained by embedding T^* into the Clifford torus S^1 × S^1 ⊆ E^4 and then stereographically projecting its image to E^3. We then associate two groups of isometries with the tiling T^* – the symmetry group G^* of T^* itself and the symmetry group G-of its Euclidean realization T-. This work provides a method to compute for G^* and G-using results from the theory of space forms, abstract polytopes, and transformation geometry. Furthermore, we present results on the color symmetry properties of the toroidal tiling T^* in relation with the color symmetry properties of the planar tiling T. As an application, we construct toroidal polyhedra from T-and use these geometric structures to model carbon nanotori and their structural analogs.

Minimal Covers of the Archimedean Tilings, Part II

In Part I of this paper the minimal regular covers of three Archimedean tilings were determined. However, the computations described in that work grow more complicated as the number of flag orbits of the tilings increases. In Part II, we develop a new technique in order to present the minimal regular covers of certain periodic abstract polytopes. We then use that technique to finish determining the minimal regular covers of the Archimedean tilings.