F -Vectors Of 3-Polytopes Symmetric Under Rotations And Rotary Reflections

The f-vector of a polytope consists of the numbers of its i-dimensional faces. An open field of study is the characterization of all possible f-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e., to characterize f-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize Steinitz's approach. (C) 2021 Elsevier B.V. All rights reserved.