Each point represents a possible electron location, forming a probability cloud.
Your ideas connect to the frontiers of physics! The Hilbert-Pólya conjecture is a fascinating, unproven idea that links these two fields.
It suggests that the energy levels (eigenvalues) of a hypothetical quantum system might perfectly correspond to the zeros of the Riemann zeta function—one of the biggest unsolved problems in mathematics.
If true, it would mean there's a deep, hidden link between the laws of quantum physics (like the atom you're seeing) and the fundamental nature of prime numbers. A true "many-body" simulation like you suggested is too complex for a browser, but I've added a chart for a simpler chaotic system (the Lorenz Attractor) to show the "fractal trace" concept you mentioned.