This paper gives an elementary account of the "Penrose dodecahedron," a set of 40 states of a spin-3/2 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 spin-3/2 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. (C) 1999 American Association of Physics Teachers. [References: 15]
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