This projects depicts atomic orbitals, or more precisely the electron probability density functions of the energy levels of the hydrogen atom. The python code used to generate these pictures was adapted from these notebooks.

The hydrogen atom is composed of a proton, relatively heavy and positively charged, and of an electron, relatively light and negatively charged. Naively, the electron "orbits" the proton. But quantum mechanics teaches us that the energy eigenstates of the electron (the states in which the electron has a deterministic energy) are delocalised around the proton. There is only a probability density of finding the electron at any given position near the proton. The pictures below depict this probability density for various energy eigenstates, by sampling points from the corresponding probability density.

Constructing a basis for the electron states require choosing an arbitrary direction, traditionally called the z-axis. The states are then labeled by three quantum numbers n, l and m. 

n is a strictly positive integer that determines the average distance of the electron to the proton. The pictures below are rescaled by n for practical reasons, so larger n orbit do not appear bigger, even if they should be. 

l is a positive integer < n, corresponding to the angular momentum of the electron around the proton.

m is an integer comprised between -l and l (included), corresponding to the projection of the angular momentum along the z-axis.

The corresponding probability density functions are all rotationally symmetric around the z axis, so in the series of pictures below, we show only its projection on the x-z plane (projecting on the y-z plane would produce identical pictures). Similarly, the density functions of the states with opposite values of m are identical. We show sequentially the probability density functions corresponding to the states with (n, l, m) in lexicographic order, up to n = 7, omitting negative values of m, i.e.: (1, 0, 0), (2, 0, 0), (2, 1, 0), (2, 1, 1), (3, 0, 0), (3, 1, 0), (3, 1, 1), (3, 2, 0), ... It helps to keep in mind that any state of the form (n, 0, 0) is spherically symmetrical, so looks like concentric circles in the x-z plane projections below, while the other states do not.

Here are a few pictures of n = 7 orbits from random directions.

Series with fixed n and l, and m running from l to 0, viewed from a random direction. 

n = 5, l = 4

n = 7, l = 4

n = 8, l = 4

Dall-e's interpretation of atomic orbitals.