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The Kochen-Specker theorem states that a 3-dimensional complex Euclideanspace admits a finite configuration of projective lines such that thecorresponding quantum observables (the orthogonal projectors) cannot beassigned with 0 and 1 values in a classically consistent way. This paper showsthat the irreducible root systems of exceptional and of non-crystallographictypes are useful in constructing such configurations in other dimensions. Thecases $E_6$ and $E_7$ lead to new examples, while $F_4$, $E_8$, and $H_4$,yield a new interpretation of the known ones. The described configurations havean additional property: they are saturated, i.e. the tuples of mutuallyorthogonal lines, being partially ordered by inclusion, yield a poset with allmaximal elements having the same cardinality (the dimension of space).

Exceptional and Non-crystallographic Root Systems and the Kochen-Specker Theorem