10 McKay Correspondence
The McKay correspondence links \(E_8\) to the binary icosahedral group and golden ratio.
10.1 Icosahedral Structure
\(h(E_8) = 30 = \) icosahedron edges
\(|2I| = 120\)
\(240 = 2 \times |2I| = \mathrm{rank}(E_8) \times h(E_8)\)
\(30 = p_2 \times N_{\mathrm{gen}} \times W = 2 \times 3 \times 5\)
Icosahedron Euler characteristic: \(V + F - E = 12 + 20 - 30 = 2 = p_2\)