What condition is necessary for an sp3 hybridized atom to be asymmetric?

An sp³ hybridized carbon atom is an atom that combined all of it's atomic orbitals, (1 s-orbital and 3 p-orbitals) into four sp³ hybridized molecular orbitals. An sp³ hybridized carbon atom will form four bonds, one with each sp³ orbital. The geometry of a carbon atom with four different substituents will be tetrahedral in nature. There is only one substitution pattern that leads to an asymmetric sp³ hybridized carbon atom.

A carbon center with the structure CH₄, has four of the same substituents and is symmetrical with two plans of symmetry. If we change one substituent, to CH₃Cl, we lose one plane of symmetry, but the other plane of symmetry still remains, therefore, it is not asymmetric. Changing the molecule to contain three different substituents, CH₂ClBr, creates a molecule that still has a single plane of symmetry. Therefore, the only way to have an asymmetric carbon center is to have four different substituents which will disrupt all planes of symmetry.

A molecule of the form CHClBrF, or simply with four different substituents, will be completely asymmetric as there are no planes of symmetry in this molecule.