Geometric and Algebraic Approach to Polytopes

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Symmetry, a frequently recurring theme in mathematics, is at core of this proposal. The last three decades have seen a remarkable revival of interest in geometric and combinatorial structures and their symmetry. The general area of the proposed research is in discrete and combinatorial geometry and interaction between geometry and algebra. The questions posed and the methods proposed will capitalize on the recent rapid developments of three areas that have largely developed independently: the study of maps, the theory of geometric and abstract polytopes (combinatorial objects that locally have structure of classical polytopes or tessellations) and thin geometries. My contribution to the proposed research is in the use of classical euclidean and hyperbolic geometry and algebra. Many of the proposed projects will involve junior researchers. The geometric problems in the proposed research are dealing with realizations of highly regular polyhedral structures in euclidean and hyperbolic 3-spaces which satisfy certain specific conditions that should make them interesting to mathematicians as well as to, for example, chemists or structural engineers. I propose to investigate, with the aim to classify, certain discrete polyhedra with high degree of symmetry. This in fact comprises several different projects depending on the type and degree of symmetry and the ambient space. Some examples are regular polyhedra (polyhedra with regular faces, which are not necessarily planar or finite, and isomorphic vertex-figures) in hyperbolic 3-space; uniform polyhedra (those that have regular facets and isomorphic vertex-figures), and hereditary polyhedra (polyhedra that inherit all symmetries from its facets) in euclidean 3-space. I also propose to continue the research contained in a series of recently published papers on the classification of tessellations on compact euclidean space-forms (flat closed 3-manifolds). More precisely, we analyze the orbit-space of quotients of euclidean tessellations by a fixed-point free subgroup of euclidean isometries. The regular tessellations are only possible on torus for which the classification has been completed for all ranks. The work on other space-forms has been only partially completed. However, much of the work remains to be done in order to complete the classification for non-orientable manifolds. Over the past 25 years there has been considerable interest in chiral polytopes, the abstract structures with basic combinatorial properties of classical polytopes that are maximally symmetric by rotations but are not symmetric by reflections. Much of the published work has centred on regular maps. Recent work by myself and collaborators in extending this concept to thin geometries has opened numerous new questions that I propose to at least partially answer. Of particular interest is the classification of toroidal residually connected thin geometries.