Suppose you want a collection of convex polygons in $3$-space such that, when you glue them together edge-to-edge, you get 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!

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!

Suppose you want a collection of convex polygons in 3-space such that, when you glue them together edge-to-edge, you get 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!

Suppose you want a collection of convex polygons in $3$-space such that, when you glue them together edge-to-edge, you get 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!