Polyhedral Sculptures with Hyperbolic Paraboloids

This paper describes the results of our experiments with gluing together partial hyperbolic paraboloids, or hypars. We make a paper model of each hypar by folding a polygonal piece of paper along concentric polygons in an alternating fashion. Gluing several hypars together along edges, we obtain a beautiful collection of closed, curved surfaces which we call hyparhedra. Our main examples are given in Figure 6. We present an algorithm that constructs a hyparhedron given any polyhedron. The surface represents each face of the polyhedron by a “hat” of hypars. For a Platonic solid, the corners of the hypars include the vertices of the polyhedron and its dual, and one can easily reconstruct the input polyhedron from the hyparhedron. More generally, the hyparhedron captures the combinatorial topology of any polyhedron. We also present several possibilities for generalization.