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.
@article{arxiv.1111.4713,
title = {Universal violation of pentagon inequalities in four-state systems},
author = {Mordecai Waegell and P. K. Aravind},
journal= {arXiv preprint arXiv:1111.4713},
year = {2011}
}