\(f\)-vectors of 3-polytopes symmetric under rotations and rotary reflections (Q2659225)

The \(f\)-vector of a three-dimensional convex polytope \(P\) can be defined as \((f_0,f_2)\), where \(f_0\) is the number vertices and \(f_2\) the number of facets of \(P\); the number of edges follows from \(f_0,f_2\) by the Euler relation. The \(f\)-vectors of all \(3\)-polytopes are known since E. Steinitz. The present paper determines the \(f\)-vectors of all \(3\)-polytopes which have a given symmetry group (a finite subgroup of the orthogonal group) which does not contain a reflection. These are the axis-rotation group \(C_n\), the dihedral rotation group \(D_d\), the rotation groups of the regular tetrahedron, octahedron, or isosahedron, and the rotary reflection group \(G_d\), generated by a product of a rotation and a reflection. In each case, the authors completely determine the possible \(f\)-vectors, described by congruences and, in a few cases, also by inequalities. Necessary conditions can be obtained by some general procedures. The main difficulty consists in constructing polytopes with given symmetry group and \(f\)-vector. The constructive procedures extend ideas of Steinitz. As tools, several operations on symmetric polytopes are developed, and a longer list of special symmetric polytopes is used. The proofs are nicely illustrated by polytope pictures and tables.