Symmetric matroid polytopes and their generation

The study of polyhedra within the framework of oriented matroids has become a natural approach. Methods for enumerating combinatorial types of convex polytopes inductively within the Euclidean setting alone have not been established. In contrast, the oriented matroid concept allows one to generate matroid polytopes inductively. Matroid polytopes, when not interesting in their own right as topological balls with certain sphere properties, form an intermediate structure to search for realizable convex spheres. We provide in this article an interesting class of self-polar 3-spheres that stimulated research in this area. What are effective methods of generating matroid polytopes with prescribed properties? Having in mind open problems for which a corresponding solution is still open, we present the class of 3spheres of Gábor Gévay that were found independently by other authors as well. We discuss two new algorithmical methods of David Bremner and of Jürgen Bokowski for generating matroid polytopes that were tested in this context.