Skip to main content
SpringerLink
Log in

Simple and Combined Symmetries

  • Chapter
Symmetry through the Eyes of a Chemist

  • 1995 Accesses

Abstract

The simplest and most common of all symmetries is bilateral symmetry, yet at first sight, it does not appear as overwhelmingly important in chemistry as in our every-day life. The human body has bilateral symmetry, except for the asymmetric location of some internal organs.

Beauty is the first test… Godfrey Harold Hardy (1877–1947) [1]

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This passage is in French in both the German original and English translation of Mann’s The Magic Mountain (see, References).

  2. 2.

    This plant has been used to extract physiologically important alkaloids. One of the derivatives has become an important medicine that dilates blood vessels in the brain. Cavinton has been a popular drug for improving memory.

  3. 3.

    We are grateful to Professor Alan L. Mackay (London) for the English translation.

References

  1. G. H. Hardy, A Mathematician’s Apology, Cambridge University Press, Cambridge, 1941.

    Google Scholar 

  2. T. Mann, The Magic Mountain. The cited passage is in French both in the original German (see, e.g., T. Mann, Der Zauberberg. S. Fischer Verlag, Frankfurt am Main, 1960; 1974, p. 477; the book was originally published by S. Fischer Verlag, Berlin, 1924) and its English translation (see, e.g., T. Mann, The Magic Mountain. Translated from the German by H. T. Lowe-Porter. Alfred A. Knopf, New York, 1946, pp. 342–343). The English translation cited in our text was kindly provided by Dr. Jack M. Davis, Professor of English, University of Connecticut, Storrs, 1984.

    Google Scholar 

  3. Mann, The Magic Mountain, pp. 276–277 (German edition, pp. 386–387).

    Google Scholar 

  4. J. Kepler, Strena, seu De Nive Sexangula, 1611. English translation by L. L. Whyte, The Six-cornered Snowflake, Clarendon Press, Oxford, 1966.

    Google Scholar 

  5. S. P. Springer, G. Deutsch, Left Brain, Right Brain, Freeman & Co., San Francisco, 1981; J. B. Hellige, Hemispheric Asymmetry. What’s Right and What’s Left, Harvard University Press, Cambridge, MA, 1993.

    Google Scholar 

  6. H. Weyl, Symmetry, Princeton University Press, Princeton, New Jersey, 1952, p. 9.

    Google Scholar 

  7. E. Häckel (Haeckel), Kunstformen der Natur. Vols. 1-10, Verlag des Bibliographischen Instituts, Leipzig, 1899–1904.

    Google Scholar 

  8. M. Hargittai, “Hawaiian flowers with fivefold summetry” in I. Hargittai, ed., Fivefold Symmetry. World Scientific, Singapore, 1992 pp. 529–541.

    Google Scholar 

  9. L. L. Whyte, “Foreword.” In J. Kepler (ed.), The Six-Cornered Snowflake, pp. v–vii, p. vi.

    Google Scholar 

  10. G. Taubes, “The Snowflake Enigma.” Discover 1984, 5(1), 74–78, p. 75.

    Google Scholar 

  11. Ibid.

    Google Scholar 

  12. D. McLachlan, “The Symmetry of Dendritic Snow Crystals.” Proc. Natl. Acad. Sci. 1957, 43, 143–151.

    Article  CAS  Google Scholar 

  13. Ibid.

    Google Scholar 

  14. W. A. Bentley, W. J. Humphreys, Snow Crystals. McGraw-Hill, New York and London, 1931.

    Google Scholar 

  15. J. Nittmann, H. E. Stanley, “Tip Splitting without Interfacial-Tension and Dendritic Growth-Patterns Arising from Molecular Anisotropy” Nature 1986, 321, 663–668; S. Kai, ed., Pattern Formation in Complex Dissipative Systems, World Scientific, Singapore, 1992; Y. Furukawa, W. Shimada, “3-Dimensional Pattern-Formation During Growth of Ice Dendrites – Its Relation to Universal Law of Dendritic Growth.” J. Crystal. Growth 1993, 128, 234–239; R. Kobayashi, “Modeling and Numerical Simulations of Dendritic Crystal-Growth.” Physica D 1993, 63, 410–423.

    Article  Google Scholar 

  16. D. A. Tomalia, “Birth of a New Macromolecular Architecture: Dendrimers as Quantized Building Blocks for Nanoscale Synthetic Organic Chemistry.” Aldrichimica Acta 2004, 37(2), 39–57.

    CAS  Google Scholar 

  17. D. A. Tomalia, H. D. Durst, “Genealogically Directed Synthesis – Starburst Cascade Dendrimers and Hyperbranched Structures.” Top. Curr. Chem. 1993, 165, 193–313.

    Article  CAS  Google Scholar 

  18. Mann, The Magic Mountain, p. 480.

    Google Scholar 

  19. Attributed to M. Polányi. Private communication from Professor W. Jim Neidhardt, New Jersey Institute of Technology, Newark, New Jersey, 1984.

    Google Scholar 

  20. A. L. Mackay, “Generalised Crystallography.” Izvj. Jugosl. Cent. Kristallogr. 1975, 10, 15–36; A. L. Mackay, personal communication, 1982.

    Google Scholar 

  21. J. Needham, Lu Gwei-Djen, “The Earliest Snow Crystal Observations.” Weather 1961, 16, 319–327.

    Article  Google Scholar 

  22. Ibid.

    Google Scholar 

  23. Ibid.

    Google Scholar 

  24. J. Kepler, The Six-cornered Snowflake.

    Google Scholar 

  25. U. Nakaya, Snow (in Japanese), Iwanami-Shoten Publ. Co., Tokyo, 1938 (latest printing, 1987).

    Google Scholar 

  26. G. Hellmann, Schneekrystalle: Beobachtungen und Studien, Mückenberger, Berlin, 1893.

    Google Scholar 

  27. T. Stamp, C. Stamp, William Scoresby; Arctic Scientist, Caedmon of Whitby Press, Whitby, North Yorkshire, 1976.

    Google Scholar 

  28. Nakaya, Snow

    Google Scholar 

  29. Stamp, Stamp, William Scoresby

    Google Scholar 

  30. Bentley, Humphreys, Snow Crystals.

    Google Scholar 

  31. Ibid.

    Google Scholar 

  32. Nakaya, Snow.

    Google Scholar 

  33. Ibid.

    Google Scholar 

  34. K. G. Libbrecht, The Snowflake. Voyageur Press, Stillwater, Minnesota, 2003.

    Google Scholar 

  35. W. J. Broad, “Snowflakes as Big as Frisbees? Could Be.” The New York Times 2007, March 20, pp. F1; F4.

    Google Scholar 

  36. J. Reston, International Herald Tribune, Thursday, May 7, p. 4, 1981.

    Google Scholar 

  37. D. Y. Curtin, I. C. Paul, “Chemical Consequences of the Polar Axis in Organic Solid-State Chemistry.” Chem. Rev. 1981, 81, 525–541.

    Article  CAS  Google Scholar 

  38. P. Groth, Chemische Kristallographie. 5 Volumes. Verlag von Wilhelm Engelmann, Leipzig, 1906–1919.

    Google Scholar 

  39. Curtin, Paul, Chem. Rev. 525–541.

    Google Scholar 

  40. G. N. Desiraju, Crystal Engineering: The Design of Organic Solids, Elsevier, Amsterdam, 1989.

    Google Scholar 

  41. L. Kelvin, Baltimore Lectures, C. J. Clay and Sons, London, 1904.

    Google Scholar 

  42. J. Applequist, “Optical Activity: Biot’s Bequest.” Am. Sci. 1987, 75, 59–68.

    Google Scholar 

  43. L. Pasteur, Researches on the Molecular Asymmetry of Natural Organic Products. Alembic Club Reprints No. 14. W. F. Clay, Edinburgh, 1897, p. 21.

    Google Scholar 

  44. G. Wald, “The Origin of Optical Activity.” Ann. NY Acad. Sci. 1957, 69, 352–368.

    Article  CAS  Google Scholar 

  45. E. Fischer, “Einfluss der Configuration auf die Wirkung der Enzyme.” Ber. Deutschen Chem. Ges. 1894, 27, 2985–2993.

    Article  CAS  Google Scholar 

  46. J. M. Bijvoet, A. F. Peerdeman, A. J. van Bommel, “Determination of the Absolute Configuration of optically Active Compounds by Means of X-Rays.” Nature 1951, 168, 271–272.

    Article  CAS  Google Scholar 

  47. L. L. Whyte, “Chirality.” Leonardo 1975, 8, 245–248; “Chirality.” Nature 1958, 182, 198.

    Article  Google Scholar 

  48. V. Prelog, “Chirality in Chemistry” (Nobel lecture). Science 1976, 193, 17–24.

    Article  CAS  Google Scholar 

  49. I. Hargittai, B. Hargittai, “Prelog Centennial: Vladimir Prelog (1906–1998).” Structural Chemistry 2006, 17, 1–2.

    Article  CAS  Google Scholar 

  50. M. Gardner, The New Ambidextrous Universe. Symmetry and Asymmetry from Mirror Reflections to Superstrings, Third Revised Edition, W. H. Freeman and Co., New York, 1990.

    Google Scholar 

  51. P. Curie, “Sur la symétrie dans les phénomenes physiques, symétrie d’un champ eléctrique et d’un champ magnétique.” J. Phys. (Paris) 1894, 3, 393–415

    Google Scholar 

  52. A. V. Shubnikov, Simmetriya i antisimmetriya konechnikh figure (in Russian, Symmetry and Antisymmetry of Finite Figures). Izd. Akad. Nauk S.S.S.R., Moscow, 1951.

    Google Scholar 

  53. M. Curie, Pierre Curie, With the Autobiographical Notes of Marie Curie. Dover Publications, New York, 1963.

    Google Scholar 

  54. A. V. Shubnikov, “On the Works of Pierre Curie on Symmetry.” In I. Hargittai, B. K. Vainshtein, eds., Crystal Symmetries. Shubnikov Centennial Papers, Pergamon Press, Oxford, 1988, p. 357–364. This is the English translation of the Russian original, A. V. Shubnikov, Usp. Fiz. Nauk 1956, 59, 591–602.

    Google Scholar 

  55. I. Stewart, M. Golubitsky, Fearful Symmetry. Is God a Geometer?, Blackwell, Oxford, 1992.

    Google Scholar 

  56. M. Curie, Pierre Curie.

    Google Scholar 

  57. Prelog, Science, 17–24.

    Google Scholar 

  58. See, e.g., L. E. Orgel, The Origins of Life: Molecules and Natural Selection, John Wiley & Sons, New York, London, Sydney, Toronto, 1973; J. D. Bernal, The Origin of Life, The World Publ. Co., Cleveland and New York, 1967.

    Google Scholar 

  59. Prelog, Science, 17–24.

    Google Scholar 

  60. S. F. Mason, Chemical Evolution: Origins of the Elements, Molecules and Living Systems, Oxford University Press, England, 1991.

    Google Scholar 

  61. A. Szabó-Nagy, L. Keszthelyi, “Demonstration of the Parity-Violating Energy Difference Between Enantiomers.” Proc. Natl. Acad. Sci. USA 1999, 96, 4252–4255.

    Article  Google Scholar 

  62. J. B. S. Haldane, “Pasteur and Cosmic Asymmetry.” Nature 1960, 185, 87.

    Article  Google Scholar 

  63. L. Pasteur, C. R. Acad. Sci. Paris, June 1, 1874.

    Google Scholar 

  64. Orgel, The Origins of Life.

    Google Scholar 

  65. Ibid.

    Google Scholar 

  66. L. Carrol, Through the Looking Glass and what Alice found there; See, e.g., in The Complete Illustrated Works of Lewis Carrol. Chancellor Press, London, 1982, p. 127.

    Google Scholar 

  67. See, e. g., G. W. Muller, “Thalidomide: From Tragedy to New Drug Discovery.” Chemtech 1997, 27(1), 21–25.

    CAS  Google Scholar 

  68. See, e. g., E. L. Eliel, “Louis Pasteur and Modern Industrial Stereochemistry.” Croatica Chemica Acta 1996, 69, 519–533.

    CAS  Google Scholar 

  69. P. Ahlberg, “The Nobel Prize in Chemistry.” In Les Prix Nobel—The Nobel Prizes 2001, Almquist & Wiksell International, Stockholm, 2002, p. 20.

    Google Scholar 

  70. See, e.g., S. C. Stinson, “Chiral Drugs.” Chem. Eng. News 2000, October 23, 55–78; S. C. Stinson, “Chiral Pharmaceuticals.” Ibid. 2001, October 1, 79–97; A. M. Rouhi, “Chiral Business.” Ibid. 2003, May 5, 45–55; A. M. Rouhi, “Chirality at Work.” Ibid. 2003, May 5, 56–61; R. Winder, “Pure enzymes.” Chemistry & Industry. 2006, June 5, 18–19; C. O’Driscoll, “Reflective Work.” Ibid. 2007, May 7, 22–25.

    Google Scholar 

  71. A. V. Shubnikov, V. A. Koptsik, Symmetry in Science and Art, Plenum Press, New York and London, 1974. Russian original: Simmetriya v nauke i isskustve, Nauka, Moscow, 1972.

    Google Scholar 

  72. Ibid.

    Google Scholar 

  73. F. A. L. Anet, S. S. Miura, J. Siegel, K. Mislow, “La-Coupe-Du-Roi and Its Relevance to Stereochemistry – Combination of 2 Homochiral Molecules to Give an Achiral Product.” J. Am. Chem. Soc. 1983, 105, 1419–1426.

    Article  CAS  Google Scholar 

  74. M. Cinquini, F. Cozzi, F. Sannicoló, A. Sironi, “Bisection of an Achiral Molecule into Homochiral Halves – The 1st Chemical Analog of La Coupe du Roi.” J. Am. Chem. Soc. 1988, 110, 4363–4364.

    Article  CAS  Google Scholar 

  75. Ibid.

    Google Scholar 

  76. Anet et al., J. Am. Chem. Soc. 1419–1426.

    Google Scholar 

  77. Cinquini et al. J. Am. Chem. Soc. 4363–4364.

    Google Scholar 

  78. Anet et al. J. Am. Chem. Soc. 1419–1426.

    Article  CAS  Google Scholar 

  79. H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, New York, 1973.

    Google Scholar 

  80. Weyl, Symmetry, p. 74.

    Google Scholar 

  81. Coxeter, Regular Polytopes.

    Google Scholar 

  82. N. V. Belov, Ocherki po strukturnoi mineralogii (in Russian, Notes on structural mineralogy), Nedra, Moscow, 1976.

    Google Scholar 

  83. Häckel, Kunstformen der Natur.

    Google Scholar 

  84. J. Kepler, Mysterium cosmographicum, 1595.

    Google Scholar 

  85. Häckel, kunstformen der Natur.

    Google Scholar 

  86. A. Koestler, The Sleepwalkers, The Universal Library, Grosset and Dunlap, New York, 1963, p. 252.

    Google Scholar 

  87. Kepler, Mysterium Cosmographicum.

    Google Scholar 

  88. T. Saito, A. Yoshikawa, T. Yamagata, H. Imoto, K. Unoura, “Synthesis, Structure, and Electronic Properties of Octakis(□3-sulfido)hexakis (triethylphophine) hexatungsten as a Tungsten Analogue of the Molecular Model for Superconducting Chevrel Phases” Inorg. Chem. 1989, 28, 3588–3592.

    Article  CAS  Google Scholar 

  89. Ibid.

    Google Scholar 

  90. A. Müller, P. Kögerler, A. W. M. Dress, Giant Metal-Oxide-Based Spheres and Their Topology: from Pentagonal Building Blocks to Keplerates and Unusual Spin Systems.” Coord. Chem. Rev. 2001, 222, 193–218.

    Article  Google Scholar 

  91. Coxeter, Regular Polytopes; L. Fejes Tóth, Regular Figures, Pergamon Press, New York, 1964.

    Google Scholar 

  92. Ibid. and H. M. Cundy, A. P. Rollett, Mathematical Models, Clarendon Press, Oxford, 1961; M. J. Wenninger, Polyhedron Models, Cambridge University Press, New York, 1971; P. Pearce, S. Pearce, Polyhedra Primer, Van Nostrand Reinhold Co., New York, 1978.

    Google Scholar 

  93. Cundy, Rollett, Mathematical Models.

    Google Scholar 

  94. 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 

  95. W. M. Meier, D. H. Olson, Atlas of Zeolite Structure Types, Third Revised Edition, Butterworth-Heinemann, London, 1992.

    Google Scholar 

  96. B. Beagley, J. O. Titiloye, “Modeling the Similarities and Differences between the Sodalite Cages (Beta-Cages) in the Generic Materials - Sodalite, Zeolites of Type-A, and Zeolites with Faujasite Frameworks” Struct.Chem. 1992, 3, 429–448.

    Article  CAS  Google Scholar 

  97. Ibid.

    Google Scholar 

  98. W. P. Schaefer, “The Snub Cube in the Glanville Courtyard of the Beckman Institute at the California Institute of Technology.” Chemical Intelligencer 1996, 2(4), 48–50.

    Google Scholar 

  99. I. Hargittai, “Imperial Cuboctahedron.” Math. Intell. 1993, 15(1), 58–59.

    Google Scholar 

  100. S. Alvarez, “Polyhedra in (Inorganic) Chemistry.” Dalton Trans. 2005, 2209–2233.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Budapest University of Technology and Economics, 91, H-1521, Budapest, Hungary

    Magdolna Hargittai & István Hargittai

Authors
  1. Magdolna Hargittai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. István Hargittai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Magdolna Hargittai .

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Hargittai, M., Hargittai, I. (2009). Simple and Combined Symmetries. In: Symmetry through the Eyes of a Chemist. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5628-4_2

Download citation

Publish with us

Policies and ethics

Search

Navigation