M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 63
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Side-Effects of DecorationsShubnikov and Koptsik analyzed the influence of the various spacegroups of bands and networks on people’s perception of movement[53]. A one-sided band decoration without a polar axis induces nofeeling of movement (see, e.g., Figure 8-4b). On the other hand,bands with polar axes (an example is shown in Figure 8-4a) conveythe impression as if inducing left-bound or right-bound motion. Afurther pattern of band decoration with no symmetry plane is shown inFigure 8-35.
A symmetry plane conveys the impression of preventingmotion perpendicular to it. They are supposed to induce calmnessand thus may be recommended for decorating the walls of halls forserious meetings. On the other hand, the walls of dancing halls shouldprobably be decorated with patterns of rotational symmetry only. Twowall decorations, both from the Alhambra so rich in such patterns, areshown in Figure 8-36. The pattern of the Escher-like airline advertisement in Figure 8-37 conveys a strong feeling of motion, both leftbound and right-bound. There are no symmetry planes “preventing”8.5. Two-Dimensional Space Groups407Figure 8-34.
The 17 two-dimensional plane groups by George Pólya [52].Figure 8-35. Ancient band decoration in the Tarxien Temple in Malta, from 3600–2500 BCE, with no symmetry planes induces the feeling of motion (photograph bythe authors).4088 Space-Group SymmetriesFigure 8-36. Wall decorations with symmetry planes (left) and without (right) inthe Alhambra, Granada, Spain (photographs by the authors).Figure 8-37.
Airline advertisement near O’Hare Airport, Chicago (photograph bythe authors).left-bond or right-bound motion. The effects are enhanced by thedifferent coloring of the birds flying in opposite directions. Up anddown motion is perceived as if being hindered; there seem to be horizontal symmetry planes (although they are approximate only, dueto the birds’ tails of the same color tilting in one direction, that is,nonsymmetrically).8.5.3. MoirésThe so-called Moiré patterns are created by superimposing infinite planar patterns. The resulting pattern is a new two-dimensionalnetwork.
The simplest cases are illustrated in Figure 8-38. Considerfirst two identical systems of lines on transparent paper being superimposed. The starting and resulting systems have a period of λ and d,8.5. Two-Dimensional Space Groups409Figure 8-38. Moiré patterns from the superposition of two patterns at increasingangles. Top: two line systems; Bottom: two circle systems [54].respectively and they are superimposed at an angle ⌰.
These parameters have the following relationship:λ = 2d sin(⌰/2).This expression is well known as the Bragg law in X-ray diffractionof crystals, where λ is the wave length of the X-rays, d is the distancebetween atomic layers in the crystal, and ⌰/2 is the angle at which theX-rays hit the atomic layer. The patterns produced in Figure 8-38 havetwofold rotation axes perpendicular to their plane. Thus the superposition at angle ⌰ will produce the same result as at angles 180 + ⌰.Figure 8-38 (bottom) shows the interference of two identical infinitesystems of small circles at two different angles.4108 Space-Group SymmetriesMoirés patterns occur in the most diverse natural phenomena andoften occure in artistic creations as well [55]. They can be treated byfairly simple mathematics [56].References1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.J.
D. Bernal, The Origin of Life. Weidenfeld and Nicolson, London, 1967.L. B. Young, The Mystery of Matter. Oxford University Press, New York, 1965.F. J. Budden, The Fascination of Groups. University Press, Cambridge, 1972.Ibid.A. V. Shubnikov, V. A. Koptsik, Symmetry in Science and Art, Plenum Press,New York and London, 1974. Russian original: Simmetriya v nauke i iskusstve,Nauka, Moscow, 1972.I.
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Sherk, eds., Springer-Verlag, New York, 1982.Hargittai, Lengyel, J. Chem. Educ. 1033–1034.Hargittai, Symmetrie in Geistes- und Naturwissenschaf ; Crowe, The GeometricVein. The Coxeter Fertschrift.Shubnikov, Koptsik, Symmetry in Science and Art.Ibid.See, e.g., Q. Liu, W. Liu, Z.-M. Cui, W.-G. Song, L.-J. Wan, “Synthesis andCharacterization of 3D Double Branched K Junction Carbon Nanotubes andNanorods.” Carbon 2007, 45, 268–273; X.
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4.P. Corradini, “The Role of the Discovery and Investigation of StereoregularPolymers in Macromolecular Chemistry.” In S. Carra, F. Parisi, I. Pasquon,P. Pino, eds., Giulio Natta: Present significance of his scientific contribution.Editrice di Chimica, Milan, 182, pp. 126–146, p.136.L. Pauling, “The Discovery of the Alpha Helix.” Chem. Intell. 1996, 2(1), 32–38 (posthumous publication).Corradini, Giulio Natta, p. 134.References41118. J.
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Kh. S. Mamedov, I. R. Amiraslanov, G. N. Nadzhafov, A. A. Muzhaliev, Decorations Remember (in Azerbaijani), Azerneshr, Baku, 1981.47. I. Amiraslan, H. Necefoglu, Azerbajani Tessellations, Published by HacaliNecefoglu, Kars, Turkey, 2006.48. See, also, I. Hargittai, “Crystal Structures and Culture: In Memoriam KhuduMamedov (1927–1988).” Struct. Chem. 2007, 18, 535–536.49. Mamedov et al., Decorations Remember, Hargittai, Struct.
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