Regular compound tessellations of the hyperbolic plane

Abstract

Kepler, in his Harmonice mundi of 1619, extended the idea of a regular polyhedron in at least two directions. Observing that two equal regular tetrahedra can interpenetrate in such a way that their twelve edges are the diagonals of the six faces of a cube, he called this combination stella octangida . It is occasionally found in nature as twinned crystals of tetrahedrite, Cu 10 (Zn, Fe, Cu) 2 Sb 4 S 13 . In addition to this ‘compound’ of two tetrahedra inscribed in a cube, there are several other compound polyhedra, the prettiest being the compound of five tetrahedra inscribed in a dodecahedron. The icosahedral group of rotations may be described as the alternating group on these five tetrahedra. Kepler observed also that the tessellation of squares (or regular hexagons, or equilateral triangles), filling and covering the Euclidean plane, may be regarded as an infinite analogue of the spherical tessellations which are ‘blown-up’ versions of the Platonic solids. Putting these two ideas together, one naturally regards the compound polyhedra as compound tessellations of the sphere. The analogous compound tessellations of the Euclidean plane (18 two-parameter families of them) were enumerated in 1948. The present paper describes many compound tessellations of the hyperbolic plane: five one-parameter families and seventeen isolated cases. It is conjectured that this list is complete, but there remains the possibility that a few more isolated cases may still be discovered.