 # E8 and the Platonic solids

The classifications of root systems and simple Lie algebras are the same: the classical types An, Bn, Cn, and Dn, (n=1,2,3....) and the exceptional ones G2, F4, E6, E7, and E8. Types An, Dn, E6, E7 and E8 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 A5 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 3-dimensional 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 n-gon in the plane. If we allow symmetries in 3-space we get the dihedral group of order 2n.

This gives all finite subgroups of SO(3): Cn, Dn, 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:

 Group Root System Cyclic group of order n A2n-1 Dihedral Group of order 2n Dn+2 Symmetry group of tetrahedron E6 Symmetry group of cube/octahedron E7 Symmetry group of icosahedron/dodecahedron E8
The amazing thing is that this is more than a coincidence: there is a deep connection between the entries in each row. This was observed by John McKay and is called the McKay Correspondence.

## 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.

• The first composition is the real numbers R. This is an ordered field.
• Add the complex number   i = √-1   to make the complex numbers C. This is a two-dimensional algebra. It is a field, but not ordered.
• Next make the Hamltonian quaternions H by adding j, which satisfies the famous questions ij = -ji, i2 = j2 = -1. This is four dimensional. It is a division algebra, but not commutative. Note: ij is usually denoted k, and the properties of k can be deduced from the properties of i and j.
• Finally the octonions O are eight dimensional. This is a non-associative algebra.
These algebras give rise to the exceptional Lie algebras. For example G2 is the automorphism group of the octonions. The other exceptional Lie algebras F4, E7, E8 and E8 are obtained from these composition algebras via "Freudenthal's magic square". See the article by John Baez in the Bulletin of the American Mathematical Society 2002 for a nice exposition of this subject.