Symmetries of polyhedral structures

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Most of my research deals with various aspects of symmetry in mathematics. In particular, I explore mathematical objects that have many symmetries. The techniques I employ involve an interplay between geometry, combinatorics and algebra. Of great interest to me are polyhedra and higher dimensional polyhedra-like structures called polytopes. Regular polyhedra have been studied since antiquity. I also study graphs, especially those that are in some ways related to polytopes. Because of their high degree of symmetries, the graphs and polytopes I explore also have great aesthetic appeal. For many years I have been involved with investigating properties of chiral polytopes. Chiral polytopes have maximal symmetry by rotation (in contrast with regular polytopes which have maximal symmetry by reflection) and have great potential for applications in crystallography, biology, chemistry and physics.