Polyhedra on an Equilateral Hyperboloid

Dirk Huylebrouck
Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (2012)
Pages 231–238 Regular papers


The paper gives examples of polyhedra inscribed in a (equilateral) hyperboloid, because this surface can be seen as negative curvature counterpart of a sphere. There is a historical argument: Kepler used polyhedra inscribed in spheres as a model for the orbits of planets, and so, since many comets follow a hyperbolic path, their orbits might be compared to polyhedra in hyperboloids. Thus, square, penta-, hexa- and heptagrammic crossed prisms are proposed, as well as an (extended) 'hyperbolic cuboctahedron' with a recognizable hourglass aspect. The approach is not exhaustive, but it might nevertheless inspire poetic mathematicians to develop a 'Comet Mysterium', or artists to make creations using the hyperboloid.