ADE or the omnipresence of the Platonic solids. (Q2918938)

The author introduces the groups of symmetries of regular polygons and those of regular polyhedra. She gives a proof of Euler's theorem for polyhedra, which she then uses to classify the regular convex polyhedra. Subsequently she describes the rotational symmetries of the five regular convex polyhedra, the Platonic solids.NEWLINENEWLINE The irreducible finite reflection groups are classified as groups of type \(A_n\), \(B_n\), \(C_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\), \(F_4\), \(G_2\), \(I_3\), \(I_4\). The groups \(A_n\), \(D_n\), \(E_6\), \(E_7\), and \(E_8\) satisfy the crystallographic condition. The author displays the Coxeter diagrams of these groups and later points out, that there are certain similarities between those diagrams.NEWLINENEWLINE These topics are essential in recent research on the geometry of singularities. The author describes simple singularities that are related to the Platonic solids, and she hints at a connection of these singularities with others that are \(ADE\)-classified. She is particularly interested in the relation between geometric structures and quantum field theory. In the end she gives a list of books to study for anybody who got fascinated by the subject.NEWLINENEWLINEFor the entire collection see [Zbl 1222.00035].