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The Penrose Dodecahedron Revisited, Padmanabhan Aravind, Jordan Massad
The Penrose Dodecahedron Revisited, Padmanabhan Aravind, Jordan Massad
Padmanabhan K. Aravind
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 …
Friction Anisotropy And Asymmetry Of A Compliant Monolayer Induced By A Small Molecular Tilt, Nancy Burnham, M Liley, D. Gourdon, Dimitrios Stamou, U Meseth, T Fischer, C Lautz, H Stahlberg, H Vogel, C Duschl
Friction Anisotropy And Asymmetry Of A Compliant Monolayer Induced By A Small Molecular Tilt, Nancy Burnham, M Liley, D. Gourdon, Dimitrios Stamou, U Meseth, T Fischer, C Lautz, H Stahlberg, H Vogel, C Duschl
Nancy A. Burnham
Lateral force microscopy in the wearless regime was used to study the friction behavior of a lipid monolayer on mica. In the monolayer, condensed domains with long-range orientational order of the lipid molecules were present. The domains revealed unexpectedly strong friction anisotropies and non-negligible friction asymmetries. The angular dependency of these effects correlated well with the tilt direction of the alkyl chains of the monolayer, as determined by electron diffraction and Brewster angle microscopy. The molecular tilt causing these frictional effects was less than 15 degrees, demonstrating that even small molecular tilts can make a major contribution to friction.