A regular skew polyhedron is a polyhedron whose faces and vertex figures are regular skew polygons. There are
only three regular skew polyhedra in Euclidean three-space (Coxeter 1937, Garner
1967), the simplest of which is .
Garner (1967) considered regular skew polyhedra in hyperbolic space , and shows that there are exactly 32 which are derived from
honeycombs whose cells and vertex figures are derived from honeycombs whose cells
and vertex figures are not inscribed in equidistant surfaces.
Coxeter, H. S. M. "Regular Skew Polyhedra in Three and Four Dimensions." Proc. London Math. Soc.43, 33-62,
1937.Garner, C. W. L. "Regular Skew Polyhedra in Hyperbolic
Three-Space." Canad. J. Math.19, 1179-1186, 1967.
.
, and shows that there are exactly 32 which are derived from
honeycombs whose cells and vertex figures are derived from honeycombs whose cells
and vertex figures are not inscribed in equidistant surfaces.
