The Penrose dodecahedron revisited

This paper gives an elementary account of the ‘‘Penrose dodecahedron,’’ a set of 40 states of a spin3 2 particle used by Zimba and Penrose @Stud. Hist. Phil. Sci. 24, 697–720 ~1993!# to give a proof of Bell’s nonlocality theorem. The Penrose rays are constructed here from the rotation operator of a spin2 particle and the geometry of a dodecahedron, and their orthogonality properties are derived and illustrated from a couple of different viewpoints. After recalling how the proof of Bell’s theorem can be reduced to a coloring problem on the Penrose rays, a ‘‘proof-tree’’ argument is used to establish the noncolorability of the Penrose rays and hence prove Bell’s theorem. © 1999