A theory of nets for polyhedra and polytopes related to incidence geometries (Q676702)

The authors investigate nets of polyhedra, and their generalizations (also for polytopes and combinatorial polytopes), from the viewpoint of incidence geometry, i.e., in terms of morphisms of geometries and the adjacency graph of faces. One of their main observations is that a great deal of this study only require the combinatorial structure (although in elementary geometry such investigations are based on metrical requirements). On the other hand, the metrical point of view is not overlooked, yielding interesting open problems and examples, also regarding toroidal polytopes and polyhedra which are, e.g., topologically equivalent to the Klein bottle. The arguments and also the references really combine polytope theory and incidence geometry in a fruitful manner, and the concept is even meaningful for geometries that are more general than polytopes.