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.
Here are some photographs from during and after my talk, taken by Carlo Séquin.