  The 60 real vectors derived from the vertices of a 600-cell are shown to
yield a number of pentagon inequalities that are satisfied by realistic
noncontextual theories but violated by quantum mechanics. The replicas of these
inequalities cover Hilbert space so densely that every real four-dimensional
vector violates at least one of them. It is pointed out that this set of 60
vectors contains numerous "N-gons" (generalizations of pentagons) that may be
of interest in connection with demonstrations of contextuality.
