  It is shown how the 300 rays associated with the antipodal pairs of vertices
of a 120-cell (a four-dimensional regular polytope) can be used to give
numerous "parity proofs" of the Kochen-Specker theorem ruling out the existence
of noncontextual hidden variables theories. The symmetries of the 120-cell are
exploited to give a simple construction of its Kochen-Specker diagram, which is
exhibited in the form of a "basis table" showing all the orthogonalities
between its rays. The basis table consists of 675 bases (a basis being a set of
four mutually orthogonal rays), but all the bases can be written down from the
few listed in this paper using some simple rules. The basis table is shown to
contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at
the low end to 41 bases at the high end. Some explicit examples of these proofs
are given, and their implications are discussed.
