Low energy points on the sphere and the real projective plane

We present a generalization of a family of points on ${\mathbb{S}}^2$, the Diamond ensemble, containing collections of $N$ points on ${\mathbb{S}}^2$ with very small logarithmic energy for all $N\in \mathbb{N}$. We extend this construction to the real projective plane ${\mathbb{R}\mathbb{P}}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.