# regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably very well-known (even if I couldn't find any reference).

Actually, if we define a regular graph as a planar graph which (drawn on a sphere) has "faces" all surrounded by $n\ge 3$ edges and vertices all with $k\ge 3$ edges, there exist only the five regular graphs corresponding to the usual platonic solids, and now this result is a pure combinatorial result, using only $S+F=A+2$.

All this implies that the usual metric proofs of the impossibility of more than 5 platonic solids are quite misleading, but is there a general theory?

For example, with a similar definition of a regular graph on a surface of genus $g$, we get $S+F=A+2-2g$, so $nF/k+F=nF/2 +2-2g$, implying for $g=2$, $n=3$ and $k=7$ the uniqueness of a graph with 12 vertices, 42 edges and 28 triangles. It is not completely obvious that such a graph is impossible, but I am almost sure of it...

See "regular tessellations of surfaces and $(p, q, 2)$ triangle groups (Edmonds, Ewing, Kulkarni, annals of math, 1982) and references therein.

Two supplements: A reference to an article coauthored by Igor (only tangentially related to the question); and images of two hyperbolic dodecahedra from a Wolfram demo, differing in positive/negative $\alpha$, where $r \to r (r . r)^\alpha$ for points $r$ on the dodecahedron surface.

Igor Rivin, Craig D. Hodgson. "A characterization of compact convex polyhedra in hyperbolic 3-space." Inventiones mathematicae. 1993, Volume 111, Issue 1, pp 77-111. (Springer link)

(Images created in Wolfram Demo Project: Hyperbolization of Dodecahedron by Michael Trott.)

• Actually, this was my thesis, and the reference should be I. Rivin and C. D. Hodgson (monumental set of editorial screwups), but thanks for the mention! – Igor Rivin Mar 4 '15 at 1:48
• @Igor: Swapped coauthors. :-) – Joseph O'Rourke Mar 4 '15 at 14:10
• Joseph, you are my hero :) – Igor Rivin Mar 4 '15 at 14:46