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?

• A 144 octagon solution by Andrew Weimholt exists. It likely isn't minimal. Aug 2, 2018 at 23:08

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.

• Wow. That's quite amazing. Oct 18, 2018 at 1:28
• Awesome! Could you share the code for generating the faces of this surface? Oct 24, 2018 at 3:11
• @shamisen What code? It's a file in GeoGebra. Then again, Ed did actually publish some code: community.wolfram.com/groups/-/m/t/… Oct 24, 2018 at 7:00
• @IvanNeretin Thanks :) Oct 24, 2018 at 13:00
• A friend of mine produced a better rendering: drive.google.com/file/d/1e0YySTDN-xwyfE6kGHamkx3Nbu4OaSrO/view Nov 3, 2018 at 9:43

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:

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.