Department of Mathematics

Regular Complex Polytopes

The regular complex polytopes are generalisations of the platonic solids to complex Euclidean space. They consist of points, edges, faces, etc, which satisfy certain combinatorial properties (which have further been generalised to abstract regular polytopes).

The regular complex polytopes have been completely classified via their symmetry groups, which are (Shephard-Todd) finite reflection groups (which have been further generalised to Coxeter groups). The points of the regular complex polytope can be constructed directly from the abstract symmetry group (by considering its irreducible representations).

In the case of those which are real, the edges, faces, etc (and their incidence relationships) can be determined by considering the convex hull of the points.

For the remaining regular complex polytopes it is an open question whether they are determined by their points (i.e., can the edges, faces, etc, be inferred) - if not this would be equivalent to new symmetries of the points (which are not symmetries of what are called the flags). Another question is in what sense are some highly symmetric configurations of points (which also come from finite reflection groups) semi-regular?

Researcher at The University of Auckland