Polyhedrons the Faces of which are Special Quadric Patches

We seize an idea of Oswald Giering (see [1] and [2]), whoreplaced pairs of faces of a polyhedron by patches of hy-perbolic paraboloids and link up edge-quadrilaterals of apolyhedron with the pencil of quadrics determined by thatquadrilateral. Obviously only ruled quadrics can occur.There is a simple criterion for the existence of a ruled hy-perboloid of revolution through an arbitrarily given quadri-lateral. Especially, if a (not planar) quadrilateral allowsone symmetry, there exist two such hyperboloids of revo-lution through it, and if the quadrilateral happens to beequilateral, the pencil of quadrics through it contains eventhree hyperboloids of revolution with pairwise orthogonalaxes. To mention an example, for right double pyramids,as for example the octahedron, the axes of the hyper-boloids of revolution are, on one hand, located in the planeof the regular guiding polygon, and on the other, they areparallel to the symmetry axis of the double pyramid.Not only for platonic solids, but for all polyhedrons, whereone can define edge-quadrilaterals, their pairs of face-triangles can be replaced by quadric patches, and by thisone could generate roofing of architectural relevance. Es-pecially patches of hyperbolic paraboloids or, as we presenthere, patches of hyperboloids of revolution deliver versionsof such roofing, which are also practically simple to realize.