One can say a symmetric and uniform distribution of n points on a 2D circle can be simply obtained by using n identical circular sectors.
We can also say that for a 3-dimensional sphere, there exists a symmteric and uniform distribution of 4 points, which forms a tetrahedron.
While there are 5 platonic solids in 3D, I was guessing that a 'good symmetric' distribution can only be achieved with these 5 orientations.
Here are my questions.
- Is it possible to place 5 points on a 3D sphere so that they are symmetrical? How about n points?
(By 'symmetrical', it has to be possible to rotate the sphere to match the previous 'shape' but the name of the points should be different)
- What happens if the problem is extended to a k-dimensional sphere?
You may change/modify the definition of symmetrical distribution if that makes this question clearer.
P.S. I am sorry but my english is not that good, so it might be hard to understand my words here. Also, I would prefer elementary explanations as I am a high school student though advanced solutions are also welcomed. (I will try my best to understand the solutions!)