Projective polytope

A projective polytope is a tesselation of the real projective plane.

Hemi-polytopes


Since the real projective plane is equivalent to the sphere with antipodal points identified, certain spherical polytopes naturally give rise to projective polytopes by identifying antipodal elements. This is a quotient of the original polytope.

For example using the cube we can build a hemicube, by identifying opposite elements. Since the cube has 6 faces the hemicube 3 faces, each corresponding to a pair of opposite faces in the original cube. Vertices and edges are identified as well leading to half as many of each in the hemicube.

This idea works well with a particular set of polytopes, namely those in which every element has an antipodal element. However the concept can be generalized by selecting a Petrie polygon over which the polytope is symmetric and performing the quotient by that symmetry. For example the cube has a skew hexagon as its Petrie polygon and the fundamental domain of the hemicube is also a hexagon.

When this is done on a regular polytope the result is regular as well. We give the hemipolytope the Schläfli symbol $X_{n}$ where $X$ is the Schläfli symbol of the original polytope and $n$ is half the number of edges in the Petrie polygon. In the example of the hemicube the Schläfli symbol is $$\{4,3\}_3$$.