Why are atom spherical in shape ?

I am learning about structure of atom, in which i saw right from J.J. Thomsom atomic model to modern nuclear atomic model all are spherical in shape.

I have seen how different discoveries help to develop the atomic model but i have't came through a single discovery which proves that atoms are spherical in shape.

So, Why atoms are spherical in shape?

Answer

Isolated atoms are spherically symmetric.

The s-orbitals are spherically symmetric.

Probabilty contour plots of the three p-orbitals are dumbbell shape individually; however, adding the probability functions of all three results in a spherically symmetrical distribution.

Likewise, the sum of the probabililty functions of each subshell of orbitals (such as all three 2p orbitals, all five 3d orbitals, etc.) is also spherically symmetric.

For the above reason it is apparent that atoms with filled or half filled valence shells will have spherical symmetry, but what about the remaining atoms?

In 1965, Journal of Chemical Eduction published an article "Shapes of Atoms" volume 42 page 145, stating that atoms with filled or half filled valence shells are spherical, while the remainder are other shapes. A few months later a correction was published at volume 42 page 397, explaining that the original article failed to consider the fact that a valence electron in a isolated atom does not occupy a particular orbital; instead the wave function of the electon is a "hybrid state" of all the orbitals of the valence subshell because they are degenerate.

See also Ruslan's comments below regarding the second J. Chem. Ed. article, that the proper description would be: the atom is in a "mixed state".