Exploring the List of Smallest Right-Angled Hyperbolic Polyhedra

Abstract

An algorithm for determining the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented using the program Orb to compute volumes. The smallest 825 polyhedra in the list, as computed by Orb have been determined. However, because the numerical reliability of Orb has not been definitively proved, this list should be considered conjectural.

Keywords:

• Hyperbolic geometry
• polyhedral
• right-angled

Acknowledgments

Thanks to the referees and editors at Experimental Mathematics for the helpful suggestions and for the images in Figure 1. This work was completed at the California State University, Maritime.

Notes

1 See the video Not Knot for a guided tour of ℍ³ tiled by right-angled hyperbolic dodecahedra [Gunn and Maxwell 91].

2 In 1967, A. Pogorelov formulated the combinatorial conditions which are necessary and sufficient for a right-angled hyperbolic polyhedron with a given 1-skeleton to exist [Pogorelov 67]. However, the result assumed the existence of an acute-angled polyhedra with the given combinatorics. This assumption was proved to be true in 1970 by E.M. Andreev [Andreev 70]. His result, which has come to be known as Andreev’s Theorem, precisely characterizes when an combinatorial polyhedron admits a geometric realization in hyperbolic space with non-obtuse dihedral angle measures. And so, Pogorelov’s characterization of right-angled hyperbolic polyhedra arises as a corollary of Andreev’s Theorem. A correct proof of Andreev’s Theorem can be found in [Roeder et al. 07] and a useful generalization can be found in [Rivin and Hodgson 93].