The familiar polyhedron used as an architectural element created by the intersection of two cylinders of the same radius and their axes lying in the same plane and being perpendicular to each other was studied as early as ca 250 BCE by Archimedes [1]. Many different polyhedral forms can be created by symmetrical intersection of cylinders in the same way. A description of three unique groups of cylindrical polyhedra is described based on the symmetry axes of the five regular planar forms of polyhedra.