The combinatorially regular polyhedra of index 2

Summary We investigate polyhedral realizations of regular maps with self-intersections in E³, whose symmetry group is a subgroup of index 2 in their automorphism group. We show that there are exactly 5 such polyhedra. The polyhedral sets have been more or less known for about 100 years; but the fact that they are realizations of regular maps is new in at least one case, a self-dual icosahedron of genus 11. Our polyhedra are closely related to the 5 regular compounds, which can be interpreted as discontinuous polyhedral realizations of regular maps.The author was born on March 5, 1937; so exactly half a century after Otto Haupt.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.