My primary focus is on polytopes. These are geometric figures such as the polygons (in two dimensions), polyhedra in three dimensions, and higher-dimensional figures such as polychora (4D) and polytera (5D).
A polyhedron is a shape, such as a cube, with flat faces (polygons), edges (line segments), and vertices (points--these are the shape's "corners").
My current discoveries:
Heptagonal polyhedra (below)--the Small Supersemicupola and Great Supersemicupola, two unusual polyhedra having heptagonal and triangular faces and 7-fold symmetry--a rarity in the world of polyhedra which have only regular polygons for faces. These are the discoveries I'm most proud of.
Scaliform Polychora: These are polychora (4D versions of polyhedra) which have all edges of equal lengths and all vertices identical. So far, I've found 9 on my own and 2 additional ones which were co-discoveries.
Locally Convex Tetrahedral Polyhedra: Polyhedra having locally convex vertex figures and tetrahedral symmetry (same as a triangular pyramid).
Topologically Regular Polyhedra with Regular Faces: Polyhedra which are not regular, but which have only regular polygons for faces and also are topologically regular (if you disregard the actual positions of the faces, edges, and vertices and consider only how they attach together, they become regular like the Platonic Solids).
Symmetry groups in higher dimensions
If you don't know what a polyhedron is, I suggest you check out the wonderful beginner's introduction to the topic on George Hart's web site, at www.georgehart.com.
Near Misses to the Johnson Solids
{7}, {7/2}, and {7/3} are the three different types of regular heptagons. {7} is convex, the other two are different types of seven-pointed stars.
Both models are self-intersecting, which means that you can't see all of each polygon because the different faces pass through each other. For example, the blue triangles near the top of the Small SSC (on right) are almost completely visible except for the little bumps in the center of each one. In these images, it is fairly easy to see the triangles because stone cylinders are used to show the real edges. A line where two faces intersect but which is not a real edge of the polyhedron has no cylinder on it.
Neither polyhedron is a member of an infinite series. This means that supersemicupolas exist only with seven-fold symmetry. In contrast, the prisms and several related families of polyhedra can be generated based on any polygon, not just the heptagon.
I found both of these polyhedra in October 2005. They are also featured on Jim McNeill's polyhedron website here.
