Universal violation of pentagon inequalities in four-state systems
  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. 
Mordecai Waegell and P. K. Aravind
quant-ph