The orbitals come from the quantum numbers needed to solve the Schrödinger equation.

The **principal quantum number** #n#, determines the energy of the orbital. Allowed values are #n = 1, 2, 3, 4, …#

The **azimuthal quantum number** #l#, determines the shape of the orbital. Allowed values are integers from #0 " to " n#.

#l = 3# corresponds to an #f# orbital.

The **magnetic quantum number** #m_l#, determines how many orbitals there are in the subshell. Allowed values are integers from #–l " to " +l#.

If #l = 3#, the allowed values of #m_l# are #"-3, -2, -1, 0, 1, 2, 3"#.

So there are seven #f# orbitals in a subshell.

Four of the #4f# orbitals look like eight-leafed clovers, and three look like #p# orbitals with two doughnuts around their middle.

(from chemistry.caddell.org)