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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!

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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.)

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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.)

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  • $\begingroup$ Is it obvious that a triangulation will minimize the number of vertices of a genus g surface built of convex faces? In any case, thanks for the answer! $\endgroup$
    – Qurious
    Commented Aug 22, 2017 at 3:36
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    $\begingroup$ @Qurious: Allowing faces that are quads, pentagons, etc., doesn't change anything, because each quad could be partitioned into two triangles, each pentagon into three triangles, etc., and the number of vertices remains the same. So if you are minimizing vertices, it is appropriate to look at triangulations. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 22, 2017 at 10:56

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