I thought of this while studying Chemical Bonding and the VSEPR theory, where we arrange surrounding atoms around a central atom to minimise repulsion between them. So I wondered how to mathematically go about it (in Chemistry, there are a bunch of other factors involved, like each atom might not be the same). I'm trying to arrange points around the origin so that the "repulsion" is minimised.
We know that for a given point, say the origin $O$, if we are to place $3$ points $A, B, C$ in a $3D$ space at a unit distance from $O$, such that the distance of the nearest neighbours from each point is equal, then the points form an equilateral triangle.
Likewise, if we are to place $4$ points $A, B, C, D$ at a unit distance from $O$, such that the distance of the nearest neighbours from each point is equal, then the points form a regular tetrahedron.
Similarly, if we are to place $6$ points $A, B, C, D, E, F$, such that the distance of the nearest neighbours from each point is equal, and the distance of the next-nearest neighbours from each point is equal, then we get a regular octahedron.
Again, for $8$ points, we get a cube.
But I just can't figure out what would be the arrangement for $5$ or $7$ points. Can anyone tell me the arrangement, along with how to construct it? Thanks.