Regular projective polyhedra with planar faces. I
A regular projective polyhedron is defined as a faithful realization in \(\mathbb{P}^3\) of a regular combinatorial polytope of rank 3. The study of regular projective polyhedra in \(\mathbb{P}^3\) is equivalent to the study of finite regular polyhedra in \(\mathbb{R}^4\), but the authors choose to study these polyhedra directly in \(\mathbb{P}^3\). More precisely, this paper deals only with Platonic projective polyhedra, i.e. the projective polyhedra which are realized as an embedded surface. Moreover it is assumed that faces are planar and the vertex figures are skew. After recalling some of the terminology of polyhedra in the projective context, the authors classify these polyhedra in two steps: \(\bullet\) A list of the Platonic polyhedra is given: an infinite family of tori \(4,4;r\), two polyhedra \(4,6;\pi/4\) and \(6,4;\pi/4\) related to the skew \(\{4,6\}\) in \(\mathbb{R}^4\), two polyhedra \(4,8;\pi/8\) and \(8,4;\pi/8\) related to the 12-cell. \(\bullet\) A trigonometric equation, the waist equation, similar to the one used by Coxeter in his work on regular skew polyhedra in 3 and 4 dimensions, is used to establish that the list is complete.
scientific article; zbMATH DE number 1436165