TITLE: Minimal approximations of surfaces by convex polygons
QUESTION [9 upvotes]: Suppose you want a collection of convex polygons in $3$-space such that, when you glue them together edge-to-edge, you obtain an orientable surface of genus $g$. What is the fewest number of polygons you need? Is this a known result? I've done some searching, and there's a bunch of literature on polygonal surfaces/polygonal meshes, but I haven't found an answer to my question yet.
I'm pretty sure that for $g > 2$, you can do it with $6g$ rectangles, essentially by gluing together a bunch of triangular prisms. Similarly, the best I've found for the torus is $9$ rectangles. Is this the best possible, and is there an easy way to see that? This seems like a natural enough question that I'd be a little surprised if it hasn't been addressed.
Does the answer change if we don't require that the polygons be glued together edge-to-edge?
Thanks in advance!

REPLY [5 votes]: I interpret your question as seeking polyhedra that realize genus-$g$ surfaces and have the fewest number of convex faces.
A related goal has seen
considerable work: the same problem but minimizing the number
of vertices, so-called vertex-minimal triangulations.
For example, this paper finds all the $g=3$ and $g=4$ vertex-minimal
triangulations, and finds a $12$-vertex $g=5$ example:

Hougardy, Stefan, Frank H. Lutz, and Mariano Zelke. "Surface realization with the intersection segment functional." Experimental Mathematics 19.1 (2010): 79-92. (arXiv abs.)
  
            
  


This work has subsequently been extended:
Brehm, Ulrich, and Undine Leopold. "Polyhedral Embeddings and Immersions of Many Triangulated 2-Manifolds with Few Vertices." arXiv:1603.04877 (2016).
(arXiv abs.)