4, 6, 8 triangles can make a tetrahedron and up.
6, 8, 9, 10 quadrilaterals can make a cube and up.
12, 16, 18, 20 pentagons can make a tetartoid or dodecahedron and up.
7, 8, 9, 10 hexagons can make make a Szilassi toroid and up.
12, 24 heptagons can make a heptagonal dodecahedron or Klein quartic 3-torus (shown below).
4, 6, 12, 7, 12, ... what is next in this sequence?
Could it possibly be 15, with Foster graph F040, which I call the Moving Day graph after Loyd's Moving Day puzzle? If so, how can those octagons be made planar to contain a 3d-printable space? Or is it some other graph? Perhaps 24 octagons can be linked together in a way similar to the 24-cell?
For nonagons, is it Foster F060A, or the Biggs-Smith graph?
Anything between 12 and 24 for heptagons?