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.

Partitioned Binary Relations

We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented) Temperley-Lieb diagrams. We construct a one-parameter deformation of the category of partitioned binary relations and show that it gives rise to classical one-parameter deformations of partition, Brauer and Temperley-Lieb categories. Finally, we describe a factorization of partitioned binary relations into a product of certain idempotents and pairs of usual binary relations.