# Minimal surfaces for planar octagons and nonagons

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?

I believe I have an answer to one of your many sub-questions, and here it goes:

No, the minimum for octagons is neither 15 nor 24. This guy has 12. Each face is one of the following shapes: The geometric symmetry is relatively low ($$\bar4$$ in Hermann–Mauguin notation).

The construction details are below:

Take this one (left), overlap is with a tetrahedron (right): I mean, like this: Then subtract the tetrahedron from the other one. Going against the customs of this site and my own habits, I decided to add another answer to the same question, because I can't think of a better place where to put this. Strictly speaking, it is an answer to a different question, which the OP did not ask, but probably had in mind.

Behold the figure made of twelve 11-gons. Pity it has many pairs of faces share more than one edge, and consequently many pairs of faces not sharing an edge, and hence is not the much sought-after hypothetical continuation to the series of polyhedra in which every face meets every other (tetrahedron $$\to$$ Szilassi $$\to\;???$$).

Each face is one of the following shapes: The geometric symmetry, believe it or not, is again $$\bar4$$.

The construction details are below:

First we take a prism made out of a concave quadrilateral. Then we add another such beam in a criss-cross fashion, so that their sharp edges cut into each other a bit. And finally we cut a hole of this shape in a tetrahedron. Topologically this is a sphere with six handles, just as it should be.