Christmas Streak 15/88: This Is A Bit Repulsive

Geometry Level 3

Tom has a set of particles that repel each other with a force of \(F(r)\) whose strength is proportional to \(\frac{1}{r^2},\) where \(r\) is the distance of separation between any pair of particles.

If Tom puts 8 of these particles into a rigid, frictionless sphere and allows them to come to rest, they'll align to the vertices of a certain \(n\)-faced polyhedron.

What is \(n?\)

For example, if Tom puts 4 of the particles into the sphere, they'll come to rest at the vertices of a regular tetrahedron, as shown.

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