This tutorial gives an insight into the start of a new application used to treat polyhedral surfaces in R3. To convert existing Wavefront obj files to polymake application surface format we included a simple obj2surf script.
A polyhedral surface consist of its combinatorial structure given by the FACETS and a GEOMETRIC_REALIZATION. So a file of a cube contains at least the following:
The application polytope deals with the simplest kind of polyhedral surfaces, that is, polyhedral 3-spheres. To convert a polytope to a surface use the client poly2surf:
To verify that this really was a sphere you may calculate its EULER_CHARACTERISTIC and GENUS
The following example is an equivelar surface as described by McMullen, Schulz, and Wills, and later by Joswig & Schröder contained in the Schlegel diagram of neighborly cubical polytopes. The graph of the surface is the graph of a five dimensional cube. We may calculate the GENUS of the surface and have a look at the MORSE_MATCHING_CRITICAL_FACE_VECTOR. We observe that the MORSE_MATCHING is optimal, since it has exactly one critical node, 2×GENUS critical edges, and one critical two dimensional cell:
To verify that this really is an equivelar surface of type (4,q), that is, all polygons are quadrilaterals and all vertices have the same degree, we ask polymake for the POLYGON_SIZES and VERTEX_SIZES:
We see that all 40 polygons are quadrilaterals and all the vertices have degree five. Similarly to the application topaz there are several visualization methods related to the MORSE_MATCHING. The corresponding polymake commands are: