  The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a
real 8-dimensional Hilbert space that contains a large number of parity proofs
of the Kochen-Specker theorem. After introducing the rays in a triacontagonal
representation due to Coxeter, we present their Kochen-Specker diagram in the
form of a "basis table" showing all 2025 bases (i.e., sets of eight mutually
orthogonal rays) formed by the rays. Only a few of the bases are actually
listed, but simple rules are given, based on the symmetries of E8, for
obtaining all the other bases from the ones shown. The basis table is an object
of great interest because all the parity proofs of E8 can be exhibited as
subsets of it. We show how the triacontagonal representation of E8 facilitates
the identification of substructures that are more easily searched for their
parity proofs. We have found hundreds of different types of parity proofs,
ranging from 9 bases (or contexts) at the low end to 35 bases at the high end,
and involving projectors of various ranks and multiplicities. After giving an
overview of the proofs we found, we present a few concrete examples of the
proofs that illustrate both their generic features as well as some of their
more unusual properties. In particular, we present a proof involving 34 rays
and 9 bases that appears to be the most compact parity proof found to date in 8
dimensions.
