  It is demonstrated that the set of 40 states of a spin-3/2 particle used by
Zimba and Penrose to give proofs of the Kochen-Specker and Bell theorems is
identical (i.e., unitarily equivalent) in CP(3) to the set of 40 rays derived
from the vertices of the Witting polytope, which is a regular complex polytope
in C(4). The Witting polytope actually has two different apparitions in
projective spaces of different dimensions: it appears in CP(3) as the Penrose
dodecahedron and in RP(7) (after an initial inflation into R(8)) as a set of
rays associated with the root vectors of the Lie algebra E8. The interest of
these apparitions is that they provide proofs of the Kochen-Specker theorem,
but of very different types: while the proofs provided by the Penrose
dodecahedron are complex (in both senses of the word), those provided by the E8
system are real and easy to grasp (being parity proofs that take no more than
simple counting to verify). The different proofs it provides in different
settings would seem to justify calling the Witting polytope a "quantum
chameleon", and we raise (but leave unanswered) the question of whether it is
the only object of this type.
