Minimal surfaces for planar octagons and nonagons

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.