# Tiling of genus 2 surface by 8 pentagons

In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons".

Question 1: What does this tiling look like?

Question 2: Is it always possible to tile a genus $$n$$ surface by $$f$$ regular $$n$$-gons with interior angle $$\pi/v$$ (so that $$v$$ faces meet at every vertex) as long as the restriction given by the Euler characteristic $$\chi=2-2n= f-nf/2+nf/v$$ is satisfied? Answering this question, Igor Rivin says yes, but it seems his argument only shows that regular hyperbolic $$n$$-gons of interior angle smaller than the interior angle of a euclidean regular $$n$$-gon exist.

Ian answered the second question as asked, but in case you meant to ask a different question: there is not always a symmetric tiling by regular polygons of the given type, even if those restrictions hold. For instance, there is no tiling of the genus 2 surface by heptagons meeting 3 at a vertex so that the symmetries permute and rotate the heptagons in all possible ways. Said differently, the Hurwitz bound of $$168(g-1)$$ on the number of symmetries of a genus $$g$$ surface is not achieved for $$g=2$$.