Let $S$ be a Riemann Surface of genus 2. Is there a picture in the literature for a regular tiling of $S$ by 12 heptagons (where three heptagons meet at each vertex)? Also, apart from the obvious restriction given by the Euler characteristic $2-2g=f-nf/2+nf/v$ (where $g$ is the genus, $f$ is the number of faces, $v$ is the number of faces meeting at each vertex) are there any obstructions for the existence of a tiling of a surface of genus $g$ by $n$-gons (where the same number $v$ of $n$-gons meet at each vertex).


I know that such a tiling exists (for a surface of genus 2 by heptagons), but I am unable to make a drawing. The construction goes like this (thanks to Mladen Bestvina): Start with a sphere. View this as a octahedron so that you have 8 triangles. 
Pick a hexagon (start from the top go down, left, down come back up from the opposite side)
We fatten up these 6 sides to look like circles with one ray coming from the middle. You should think of these as the quotient of 2 triangles attached along one edge <|> via rotation by 180 degrees (so you get two edges meeting in a point A and 2 edges plus a ray meeting in the other point B). We call the center of the circle C so that the ray goes from C to B.  

You have to view the point $B\_i$ (of the $i$-th side, $i=1,2,3,4,5,6$) as being attached to $A\_{i+1}$ for $i=1,2,3,4,5,6$ (of course mod 6) so each of these vertices has valence $5+2=7$.
Now you take the double cover with fixed points $C\_1,...,C\_6$ and you get a surface of genus 2 with 28 triangles (28=2x8+2x6 where 8 is the number of triangles and 6 are the circles with a ray that unwrap in to two triangles). The vertices still have valence 7 (because the cover is étale here). Take the dual tiling and you are done!


This is just a curiosity. I tried asking the first part of this on stackexchange with no luck.
I am also aware of the beautiful pictures of the famous tiling of a surface of genus 3 by 24 heptagons but you can not use that to obtain the tiling of a genus two surface in an obvious way.



regular tiling of a surface of genus 2 by heptagons
Addressing the second question positively, this is the Main Theorem of this paper:


*Edmonds, Allan L.; Ewing, John H.; Kulkarni, Ravi S.*, [**Regular tessellations of surfaces and (p,q,2)-triangle groups**](http://dx.doi.org/10.2307/2007049), Ann. Math. (2) 116, 113-132 (1982). [ZBL0497.57001](https://zbmath.org/?q=an:0497.57001). [MR0662119](https://mathscinet.ams.org/mathscinet-getitem?mr=662119) 


In fact, they show this also works for non-orientable surfaces, except for the non-existence of a degree 3 triangulation of the projective plane. 


