Simple and Combined Symmetries


The simplest and most common of all symmetries is bilateral symmetry. Yet at first sight it does not appear so overwhelmingly important in chemistry as in everyday life. The human body has bilateral symmetry, except for the asymmetric location of some internal organs. A unique description of the symmetry of the human body is given by Thomas Mann in The Magic Mountain[2],[1]as Hans Castorp is telling about his love to Clawdia Chauchat:

How bewitching the beauty of a human body, composed not of paint or stone, but of living, corruptible matter charged with the secret fevers of life and decay! Consider the wonderful symmetry of this structure: shoulders and hips and nipples swelling on either side of the breast, and ribs arranged in pairs, and the navel centered in the belly’s softness, and the dark sex between the thighs. Consider the shoulder blades moving beneath the silky skin of the back, and the backbone in its descent to the paired richness of the cool buttocks, and the great branching of vessels and nerves that passes from the torso to the arms by way of the arm pits, and how the structure of the arms corresponds to that of the legs!


Symmetry Plane Rotational Symmetry Polar Axis Bilateral Symmetry Symmetry Element 
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  1. [2-1]
    T. Mann, The Magic Mountain. The cited passage on p. 21 is in French both in the original German and its English translation. The English translation cited in our text was kindly provided by Dr. Jack M. Davis, Professor of English, The University of Connecticut, 1984.Google Scholar
  2. [2-2]
    J. Kepler, Strena, seu De Nive Sexangula (1611); English translation, The Six-Cornered Snowflake, Clarendon Press, Oxford (1966).Google Scholar
  3. [2-3]
    I. I. Shafranovskii, Kepler’s crystallographic ideas and his tract “The Six-Cornered Snowflake,” in Kepler, Four Hundred Years (A. Beer and P. Beer, eds.), Proceedings of conferences held in honor of Johannes Kepler, Vistas Astron. 18 (1975).Google Scholar
  4. [2-4]
    H. A. Sackeim, R. C. Gur, and M. C. Saucy, Science 202, 434 (1978).PubMedCrossRefGoogle Scholar
  5. [2-5]
    S. P. Springer and G. Deutsch, Left Brain, Right Brain, W. Freeman & Co., San Francisco (1981).Google Scholar
  6. [2-6]
    J. B. Hellige, Hemispheric Asymmetry. What’s Right and What’s Left, Harvard University Press, Cambridge, Massachusetts (1993).Google Scholar
  7. [2-7]
    A. V. Shubnikov and V. A. Koptsik, Symmetry in Science and Art, Plenum Press, New York (1974.); Russian original, Simmetriya v nauke i isskustve, Nauka, Moscow (1972).Google Scholar
  8. [2-8]
    R. Jastrow, Red Giants and White Dwarfs, W. W. Norton, New York (1979).Google Scholar
  9. [2-9]
    H. Weyl, Symmetry, Princeton University Press, Princeton, New Jersey (1952).Google Scholar
  10. [2-10]
    L. R. H. Appleton, American Indian Design and Decoration, Dover Publications, New York (1971).Google Scholar
  11. [2-11]
    W. Herzfielde, John Heartfield. Leben und Werk dargestellt von seinem Bruder, 3rd revised and expanded ed., VEB Verlag der Kunst, Dresden (1976).Google Scholar
  12. [2-12]
    M. Ickis, The Standard Book of Quilt Making and Collecting, Dover Publications, New York (1959).Google Scholar
  13. [2-13]
    E. Häckel, Kunstformen der Natur, Vols. 1–10, Verlag des Bibliographischen Instituts, Leipzig (1899-1904).Google Scholar
  14. [2-14]
    U. Nakaya, Snow, Iwanami-Shoten Publishing Co., Tokyo (1938) [in Japanese].Google Scholar
  15. [2-15]
    G. Taubes, Discover 1984 (January), 75.Google Scholar
  16. [2-161.
    D. McLachlan, Proc. Natl. Acad. Sci. U.S.A. 43, 143 (1957).PubMedCrossRefGoogle Scholar
  17. [2-17]
    W. A. Bentley and W. J. Humphreys, Snow Crystals, McGraw-Hill, New York (1931).Google Scholar
  18. [2-18]
    J. Nittmann and H. E. Stanley, Nature 321, 663 (1986).CrossRefGoogle Scholar
  19. [2-19]
    S. Kai (ed.), Pattern Formation in Complex Dissipative Systems, World Scientific, Singapore (1992).Google Scholar
  20. [2-20]
    S. Ohta and H. Honjo, in Santa Fe Institute’s Studies in the Sciences of Complexity, Addison-Wesley Publishing Co. in press.Google Scholar
  21. [2-21]
    Y. Furukawa and W. Shimada, J. Cryst. Growth 128, 234 (1993).CrossRefGoogle Scholar
  22. [2-22]
    R. Kobayashi, Physica D 63, 410 (1993).CrossRefGoogle Scholar
  23. [2-23]
    D. A. Tomalia and H. D. Durst, Top. Curr. Chem. 165, 193 (1993).CrossRefGoogle Scholar
  24. [2-24]
    Attributed to M. Polányi. Private communication from Professor W. Jim Neidhardt, New Jersey Institute of Technology, 1984.Google Scholar
  25. [2-25]
    A. L. Mackay, Jugosl. Cent. Kristallogr. 10, 5 (1975); A. L. Mackay, personal communication, 1982.Google Scholar
  26. [2-26]
    J. Needham and Lu Gwei-Djen, Whether 16, 319 (1961).Google Scholar
  27. [2-27]
    G. Hellmann, Schneekrystalle, Berlin (1893).Google Scholar
  28. [2-28]
    T. Stamp and C. Stamp, William Scoresby; Arctic Scientist, Caedmon of Whitby (1976).Google Scholar
  29. [2-29]
    J. Reston, International Herald Tribune, May 7, 1981, p. 4.Google Scholar
  30. [2-30]
    W. Engelhardt, Mathematischer Unterricht 9(2), 49 (1963).Google Scholar
  31. [2-31]
    E. A. Abbott, Flatland. A Romance of Many Dimensions, Barnes and Noble Books, 5th ed., New York (1983).Google Scholar
  32. [2-32]
    D. Y. Curtin and I. C. Paul, Chem. Rev. 81, 525 (1981).CrossRefGoogle Scholar
  33. [2-33]
    P. Groth, Chemische Kristallographie, 5 vols., Verlag von Wilhelm Engelmann, Leipzig (1906-1919).Google Scholar
  34. [2-34]
    G. N. Desiraju, Crystal Engineering: The Design of Organic Solids, Elsevier, Amsterdam (1989).Google Scholar
  35. [2-35]
    Lord Kelvin, Baltimore Lectures, C. J. Clay and Sons, London (1904).Google Scholar
  36. [2-36]
    R. N. Bracewell, The Galactic Club. Intelligent Life in theOuter Space, W. H. Freeman & Co., San Francisco (1975).Google Scholar
  37. [2-37]
    L. L. Whyte, Leonardo 8, 245 (1975); Nature 182, 198 (1958).CrossRefGoogle Scholar
  38. [2-38]
    I. Hargittai and M. Hargittai, Symmetry: A Unifying Concept, Shelter Publications, Bolinas, California (1994).Google Scholar
  39. [2-39]
    M. Gardner, The New Ambidextrous Universe. Symmetry and Asymmetry from Mirror Reflections to Superstrings, 3rd revised ed., W. H. Freeman and Co., New York (1990).Google Scholar
  40. [2-40]
    A. V. Shubnikov, Simmetriya i antisimmetriya konechnykh figur, Izd. Akad, Nauk SSSR, Moscow (1951).Google Scholar
  41. [2-41]
    “C’est la dissymétrie qui crée le phénomène”; P Curie, J. Phys. Paris 3, 393 (1894).Google Scholar
  42. [2-42]
    M. Curie, Pierre Curie, The Macmillan Company, New York (1929).Google Scholar
  43. [2-43]
    A. V. Shubnikov, in Shubnikov Centennial Papers (I. Hargittai and B. K. Vainshtein, eds.), Crystal Symmetries p. 357, Pergamon Press, Oxford (1988). [Originally appeared in Russian in Usp. Fiz. Nauk 59, 591 (1956).]Google Scholar
  44. [2-44]
    I. Stewart and M. Golubitsky, Fearful Symmetry. Is God a Geometer? Blackwell, Oxford (1992).Google Scholar
  45. [2-45]
    V. Prelog, Science 193, 17 (1976).PubMedCrossRefGoogle Scholar
  46. [2-46]
    L. E. Orgel, The Origins of Life: Molecules and Natural Selection, John Wiley & Sons, New York (1973).Google Scholar
  47. [2-47]
    J. D. Bernai, The Origin of Life, The World Publishing Co., Cleveland (1967).Google Scholar
  48. [2-48]
    J. B. S. Haldane, Nature 185, 87 (1960), citing L. Pasteur, C. R. Acad. Sci. Paris, June 1, 1874.CrossRefGoogle Scholar
  49. [2-49]
    R. C. Milton, S. C. F. Milton, and S. B. H. Kent, Science 256, 1445 (1992).PubMedCrossRefGoogle Scholar
  50. [2-50]
    G. Jung, Angew. Chem. Int. Ed. Engl. 31, 1457 (1992).CrossRefGoogle Scholar
  51. [2-51]
    K. F. Schmidt, Sci. News 143, 348 (1993).Google Scholar
  52. [2-52]
    R. Janoschek (ed.), Chirality—From Weak Bosons to the Alpha-Helix, Springer-Verlag, Berlin (1991).Google Scholar
  53. [2-53]
    M. Nógrádi, Stereoselective Synthesis. A Practical Approach, VCH, Weinheim (1995).Google Scholar
  54. [2-54]
    R. A. Aitken and S. N. Kilényi (eds.), Asymmetric Synthesis, Blackie Academic & Professional, London (1992).Google Scholar
  55. [2-55]
    H. Brunner and W. Zettlmeier, Handbook of Enantioselective Catalysis, Vols. I and II, VCH, Weinheim (1993).Google Scholar
  56. [2-56]
    S. C. Stinson, Chem. Eng. News, September 19, 1994, p. 38.Google Scholar
  57. [2-57]
    F. A. L. Anet, S. S. Miura, J. Siegel, and K. Mislow, J. Am. Chem. Soc. 105, 1419 (1983).CrossRefGoogle Scholar
  58. [2-58]
    M. Cinquini, F. Cozzi, F. Sannicoló and A. Sironi, J. Am. Chem. Soc. 110, 4363 (1988).Google Scholar
  59. [2-59]
    H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover Publications, New York (1973).Google Scholar
  60. [2-60]
    N. V. Belov, Ocherki po strukturnoi mineralogii, Nedra, Moscow (1976).Google Scholar
  61. [2-61]
    J. Kepler, Mysterium cosmographicum (1595).Google Scholar
  62. [2-62]
    A. Koestler, The Sleepwalkers, The Universal Library, Grosset and Dunlap, New York (1963), p. 252.Google Scholar
  63. [2-63]
    L. Fejes Tóth, Regular Figures, Pergamon Press, New York (1964).Google Scholar
  64. [2-64]
    H. M. Cundy and A. P. Rollett, Mathematical Models, Clarendon Press, Oxford (1961).Google Scholar
  65. [2-65]
    M. J. Wenninger, Polyhedron Models, Cambridge University Press, New York (1971).Google Scholar
  66. [2-66]
    P. Pearce and S. Pearce, Polyhedra Primer, Van Nostrand Reinhold, New York (1978).Google Scholar
  67. [2-67]
    N. Copernicus, De Revolutionibus Orbium Caelestium (1543), as cited in G. Kepes, The New Landscape in Art and Science, Theobald & Co., Chicago (1956).Google Scholar
  68. [2-68]
    W. M. Meier and D. H. Olson, Atlas of Zeolite Structure Types, 3rd revised ed., Butterworth-Heinemann, London (1992).Google Scholar
  69. [2-69]
    B. Beagley and J. O. Titiloye, Struct. Chem. 3, 429 (1992).CrossRefGoogle Scholar

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© Plenum Press 1995

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