  We show the existence of families of periodic polyhedra in spaces of constant
curvature whose fundamental domains can be obtained by attaching prisms and
antiprisms to Archimedean solids. These polyhedra have constant discrete
curvature and are weakly regular in the sense that all faces are congruent
regular polygons and all vertex figures are congruent as well. Some of our
examples have stronger conformal or metric regularity. The polyhedra are
invariant under either a group generated by reflections at the faces of a
Platonic solid, or a group generated by transformations that are reflections at
the faces of a Platonic solid, followed by a rotation about an axis
perpendicular to the respective face. In particular, suitable quotients will be
compact polyhedral surfaces in (possibly non-compact) spaceforms.
