Although this may fall foul of the criticism that it perhaps it has not shaped a subject yet, I'll give it the benefit of the doubt that it may still shape a future subject. In a way the answer touches two answers already given: the Platonic solids and also the quaternions.
I am talking about the ADE classification, which appears in the theory of Lie algebras, finite subgroups of $SU(2)$ (McKay correspondence), representation theory of quivers (Gabriel's theorem), singularity theory (Du Val), classification of conformal field theories,...
