# Symmetric distribution of points on a sphere

Background:

The VSEPR theory provides a useful guide to predicting molecular geometries. One implication of the theory is that the orientation of ligands in three-dimensional space is such that the seperation between each pair of them is maximum.

This is analogous to the mathematical problem of symmetrically distributing points on the surface of a sphere. With 2, 3, 4, or 6 points, the solutions are trivial. The required distributions would be linear, triangular, tetrahedral, and octahedral respectively. An issue arises when we attempt to distribute 5 points on the surface of a sphere.

We could try to arrange the points such that they lie on the vertices of a regular polyhedron inscribed in the sphere. However, from my knowledge of Euclidean geometry, I believe that such a shape with 5 vertices simply cannot exist.

Chemists suggest an approximately symmetric distribution such as placing the points on the vertices of a square pyramid. But from a mathematical perspective, there ought to be a better solution.

Question: What is the nearest approximation to a symmetric distribution which we can achieve? Can the approximation be generalized to any other number of points which cannot be distributed at the vertices of a platonic solid?

The definition of a symmetrical distribution here is that any non-labelled point on the sphere is indistinguishable from another such point. A nearly symmetrical distribution implies the least variation in the relative distances between the points (distances measured along the surface).

The ambiguity in the definition of symmetry makes the question more difficult to answer. Any additional insights or numerical methods to solve the problem are also welcome.

## 2 Answers

The answer depends on your definition of best approximation of a symmetric distribution. Personally, I will vote for a "triangular bi-pyramid" because

1. The optimal packing of 5 equal spheres inside another sphere is degenerate. In one of the degenerate configurations, the centers are arranged in a triangular bi-pyramid. (see wiki entry of sphere packing in a sphere and refs there, please note that the maximum radius of inner spheres is the same for $$n = 5$$ and $$6$$).

2. It is the unique configuration which minimize the 5 electron case of Thomson's problem. (R.E. Schawrtz, 2012, preprint on arxiv).

3. It is also the configuration which maximizes the volume of corresponding inscribed polyhedron (see refs. in this answer).

If you are only after symmetry (which you are probably not), then you have the following options:

• for every $$n$$ you can arrange the points in the shape of a regular $$n$$-gon. This arrangement is "flat", but the points are indistinguishable as desired.
• for $$2n$$ points, you can arrange the points in the shape of two regular $$n$$-gons and position them "on top of each other", either as the vertices of an $$n$$-gonal prism, or an $$n$$-gonal antiprism.
• Then there are the (vertices of the) 18 uniform polyhedra. With these you can find symmetric arrangements for $$n\in\{4,6,8,12,20,24,30,48,60,120\}$$.

And that is basically it. You can more generally look at the class of vertex-transitive polyhedra, which is the technical term for all the possible ways to arrange $$n$$ points in a symmetric fashion, but each of these has a version as a uniform polyhedron that I already listed above.

In particular, the only way to arrange five points symmetrically is to arrange them in a (flat) regular pentagon. If you want the points to be spread apart as much as possible, then the solution is a triangular bipyramid as explained by achille, which is not symmetric in your sense. Maybe, symmetry is not the only thing you are after?

I do not know of a reasonable way to talk about "almost symmetric".