XII.—Isohedral and Isogonal Generalizations of the Regular Polyhedra

0.1. In Max Brückner's discussion of isohedral ( gleichflächig ) and isogonal ( gleicheckig ) polyhedra ( Vielecke und Vielflache , pp. 140 ff.) two faces of a polyhedron are defined to be equal when they are either directly or inversely congruent, and the dihedral angles at corresponding edges are equal; two vertices are equal when the spherical polygons, which they form on unit spheres with centres at the vertices, are either directly or inversely congruent, and the lengths of corresponding edges are equal. With these definitions he shows that all possible isogonal polyhedra are obtained by truncating the corners and edges of the regular n -sided prism, the octahedron and hexahedron, and the icosahedron and dodecahedron, in a manner similar to that in which the semi-regular polyhedra are obtained from the regular polyhedra. The semi-regular or Archimedean bodies are indeed just special varieties of the more general isogonal polyhedra. The isohedral polyhedra are the polar reciprocals of the isogonal polyhedra, and are obtained from the regular double-pyramid and the regular polyhedra by the reciprocal constructions, “pointing” the faces and edges.