Classifying edges and faces as manifold or non-manifold elements in 4D orthogonal pseudo-polytopes

Abstract

This article presents our experimental results for classifying edges and faces as manifold or non-manifold elements in 4D Orthogonal Pseudo-Polytopes (4D-OPP's). For faces in 4D-OPP's we propose a condition to classify them as manifold or non-manifold. For the edges' analysis in 4D-OPP's we have developed two approaches: 1) The analogy between incident (manifold and non-manifold) edges to a vertex in 3D Orthogonal Pseudo-Polyhedra (3D-OPP's) with incident (manifold and non-manifold) faces to a edge in 4D-OPP's; and 2) The extension of the concept of "cones of faces" (which is applied for classifying a vertex in 3D-OPP's as manifold or non-manifold) to "hypercones of volumes" for classifying an edge as manifold or non-manifold in 4D-OPP's. Both approaches have provided the same results, which present that there are eight types of edges in 4D-OPP's. Finally, the generalizations for classifying the n-3 and the n-2 dimensional boundary elements for n-dimensional Orthogonal Pseudo-Polytopes as manifold or nonmanifold elements is also presented.