Can solutions to Thomson's problem have pentagons?

Thomson's problem asks for the minimum-energy configuration for $$N$$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/thethomsonproblem2 ). Given a configuration, you can view it as a polyhedron, given by the convex hull. It seems that all known optimal solutions have only square and triangular faces, and it's "obvious" that for large $$N$$, it approaches a geodesic sphere with only triangular faces.

Question: Has it been proved that the optimum never has pentagonal faces? It doesn't seem like it should be hard to prove (some argument that for sufficiently large $$N$$, there is a maximum area for a face, so that it is approximately flat; and then arguing that given a pentagonal face, you can turn it into squares/triangles with less energy), but couldn't find anything in literature.

• Wait, the dodecahedron is not a minimum energy configuration? Commented Feb 6, 2021 at 2:50
• @M.Winter Surprisingly, it seems that the minimal energy for 20 points is not related to the dodecahedron. Nor are the vertices of the cube the optimal arrangement for 8 points. A 2003 survey article by Atiyah & Sutcliffe includes citations and nice illustrations of the polyhedra (arxiv.org/abs/math-ph/0303071). Commented Feb 6, 2021 at 5:50
• @M.Winter I find it disappointingly unasthetic too! I can't help feeling that, if we can't even get a pentagon on a dodecahedron, then surely they can't show up anywhere else. Dodecahedra are the only places that pentagons belong! :) Commented Feb 6, 2021 at 7:29
• @BrianHopkins. Did you notice the following sentence of the introduction ? A particularly interesting application of polyhedra in biology is provided by the structure of spherical shells, such as HIV which is built around a trivalent polyhedron with icosahedral symmetry. A few years later, the authors would have changed HIV into Coronavirus. Commented Feb 6, 2021 at 10:09
• I guess a related conjecture: among all the optimal solutions to Thomson's problem, only a finite number of them have squares. The same idea that, eventually, everything becomes triangle packings. Commented Feb 8, 2021 at 0:32

Technically, there are best known configurations with pentagons and even with hexagons -- if we allow these polygons to loop around multiple faces. Such configurations may be read from the Wikipedia list via the presence of a proper ($$C_n$$) five- or sixfold rotational symmetry element in the listed point group. For example, the rigorously proved solution of a regular icosahedron for twelve points has a pentagon looping around the five faces touching each vertex. All configurations listed by Wikipedia having a proper $$n$$-fold symmetry axis with $$n\in\{5,6\}$$ have a number of points $$\equiv2\bmod n$$ (such as $$7$$ points with fivefold symmetry or $$14$$ with sixfold symmetry), indicating that the polygonal loops surround multiple faces with a central vertex at both poles of the rotational axis. Most commonly, the pentagons or hexagons are the bases of pyramids whose triangular lateral faces (instead of the base) are among the faces of the polyhedron.
Of some relevance for the absence of single faces having five or more sides is the instability that emerges from placing too many points on the edge of a hemisphere. On a hemisphere of unit radius, four identically signed unit charges equally spaced around the edge give a net repulsive energy of $$1+2\sqrt2\approx3.83$$ units; if we move one charge to the top of the hemisphere and redistribute the rest equally around the edge the repulsive energy becomes $$(3/2)\sqrt2+\sqrt3\approx3.85$$ units. Thus the square configuration is (barely) stable in this case (we must use a section larger than a hemisphere to realize the familiar tetrahedral pattern). With five points, however, the pentagonal planar configuration gives $$\approx6.88$$ units of repulsive energy while the square pyramid, with a point at the apex and thus only square and triangular faces, gives $$\approx6.66$$ units -- the square pyramid has become more stable. Wikipedia configurations that include five points on a great circle (or any small circle, for that matter) are possible only because the poles are plugged up by additional points.