E_{8} and the Platonic solidsThe classifications of root systems and simple Lie algebras are the same: the classical types A_{n}, B_{n}, C_{n}, and D_{n}, (n=1,2,3....) and the exceptional ones G_{2}, F_{4}, E_{6}, E_{7}, and E_{8}. Types A_{n}, D_{n}, E_{6}, E_{7} and E_{8} are called simply laced. The simply laced root systems have a connection to familiar geometric figures. There are exactly 5 Platonic solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. These solids have fascinating symmetry groups. The cube and octahedron are "dual" to each other, as are the icosahedron and the dodecahedron. Dual solids have the same symmetry group, so there are three symmetry groups here: the tetrahedron, cube and icoshedron. The symmetry group of the icosahedron is the most interesting one: it is the finite simple group A_{5} of order 60. The groups T (the symmetries of the tetrahedron), C (symmetries of the cube) and I (symmetries of the icosahedron) are subgroups of the group of rotations of 3dimensional space, which is known as SO(3). This Lie group is the symmetry group of the sphere. There are two other infinite families of finite subgroups of SO(3). The cyclic group of order n is the symmetry group of a regular ngon in the plane. If we allow symmetries in 3space we get the dihedral group of order 2n. This gives all finite subgroups of SO(3): C_{n}, D_{n}, T, C and I. See Hermann Weyl's beautiful classic book Symmetry. There is a remarkable coincidence between the subgroups of SO(3) and the simply laced root systems:
Why only a handful of exceptions?It may seem curious that there are only a few exceptional Lie groups. But there are other examples of mathematical objects with special properties, where only a small number possibilities actually occur. One example is the platonic solids. Another example is the algebraic structures that generalize the real numbers.The exceptional Lie algebras arise in connection with the composition algebras. There are four composition algebras containing the real numbers.
