Advertisement

Group-theoretical Foundations for the Concept of Mandalas as Diagrammatical Expressions for Characterizing Symmetries of Stereoisomers

Article

Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order to reinforce the spread of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (H\)G and left coset representations (LCR) G(/H) for the group G and its subgroup H, a regular body of G-symmetry is defined as a diagrammatical expression for a right regular representation (C 1\)G, which is an extreme case of RCRs. The |G| substitution positions of the regular body as a reference are numbered in accord with the numbering of the elements of G and segmented into |G|/|H| of H-segments, which are governed by the RCR (H\)G. By regarding each H-segment as a substitution position, the H-segmented regular body is reduced into a reduced regular body, which can be regarded as a secondary skeleton for generating a molecule. The reference regular body (or H-segmented one) is operated by every symmetry operations of G to generate regular bodies (or H-segmented ones), which are placed on the vertices of a hypothetical regular body of G-symmetry. The resulting diagram (a nested regular body) is called a mandala (or a reduced mandala), which is a diagrammatical expression for specifying the G-symmetry of a molecule. The effect of a K-subduction on the regular bodies of a mandala (or a reduced mandala) results in the K-assemblage of the mandala (or the reduced mandala), where the resulting K-assemblies governed by the LCR G(/K) construct a |G|/|K|-membered orbit, which corresponds to a molecule of K-symmetry. The sphericity of the RCR (or the LCR) is used to characterize symmetrical properties of substitution positions and those of stereoisomers. The fixed-point vector for each mandala (or reduced mandala) in terms of row view and the number of fixed points of K-assembled mandalas (or K-assembled reduced mandalas) in terms of column view are compared to accomplish combinatorial enumeration of stereoisomers. The relationship between a mandala and a reordered multiplication table is discussed.

Keywords

mandala regular body segmentation assemblage sphericity coset representation 

References

  1. 1.
    Mislow K., Raban M. (1967) Top. Stereochem. 1: 1–38CrossRefGoogle Scholar
  2. 2.
    Eliel E.L. (1980) J. Chem. Educ. 57: 52–55CrossRefGoogle Scholar
  3. 3.
    Mislow K., Siegel J. (1984) J. Am. Chem. Soc. 106: 3319–3328CrossRefGoogle Scholar
  4. 4.
    Eliel E., Wilen S.H. (1994). Stereochemistry of Organic Compounds. Wiley, New YorkGoogle Scholar
  5. 5.
    North N. (1998). Principles and Applications of Stereochemistry. Stanley Thornes, CheltenhamGoogle Scholar
  6. 6.
    Morris D.G. (2001). Stereochemistry. Royal Society of Chemistry, CambridgeGoogle Scholar
  7. 7.
    Eames J., Peach J.M. (2003). Stereochemistry at a Glance. Blackwell, OxfordGoogle Scholar
  8. 8.
    Sheehan J. (1968) Can. J. Math. 20: 1068–1076Google Scholar
  9. 9.
    Brocas J. (1986) J. Am. Chem. Soc. 108: 1135–1145CrossRefGoogle Scholar
  10. 10.
    Hässelbarth W. (1985) Theor. Chim. Acta 67: 339–367CrossRefGoogle Scholar
  11. 11.
    Mead C.A. (1987) J. Am. Chem. Soc. 109: 2130–2137CrossRefGoogle Scholar
  12. 12.
    Fujita S. (1991). Symmetry and Combinatorial Enumeration in Chemistry. Springer-Verlag, Berlin-HeidelbergGoogle Scholar
  13. 13.
    Fujita S. (1990) Bull. Chem. Soc. Jpn. 63: 315–327CrossRefGoogle Scholar
  14. 14.
    Fujita S. (1990) J. Am. Chem. Soc. 112: 3390–3397CrossRefGoogle Scholar
  15. 15.
    Fujita S. (1989) Theor. Chim. Acta 76: 247–268CrossRefGoogle Scholar
  16. 16.
    Fujita S. (1990) J. Math. Chem. 5: 121–156CrossRefGoogle Scholar
  17. 17.
    Fujita S. (1990) Bull. Chem. Soc. Jpn. 63: 203–215CrossRefGoogle Scholar
  18. 18.
    Fujita S. (2001) Bull. Chem. Soc. Jpn. 74: 1585–1603CrossRefGoogle Scholar
  19. 19.
    Fujita S. (2002) Chem. Rec. 2: 164–176CrossRefGoogle Scholar
  20. 20.
    Fujita S. (2002) Bull. Chem. Soc. Jpn. 75: 1863–1883CrossRefGoogle Scholar
  21. 21.
    Mead C.A. (1992) J. Am. Chem. Soc. 114: 4018–4019Google Scholar
  22. 22.
    El-Basil S. (1999). Combinatorial Organic Chemistry, An Educational Approach. Nova Scientific, New YorkGoogle Scholar
  23. 23.
    El-Basil S. (2002) MATCH Commun. Math. Comput. Chem. 46: 7–23Google Scholar
  24. 24.
    Fujita S. (2002) J. Org. Chem. 67: 6055–6063CrossRefGoogle Scholar
  25. 25.
    Fujita S. (2004) J. Comput. Chem. Jpn. 3: 113–120CrossRefGoogle Scholar
  26. 26.
    Fujita S., Chem. Educ. J. 8 (2005) Registration No. 8–8.Google Scholar
  27. 27.
    S. Fujita, Chem. Educ. J. 8 (2005) Registration No. 8–9.Google Scholar
  28. 28.
    Fujita S., El-Basil S. (2002) MATCH Commun. Math. Comput. Chem. 46: 121–135Google Scholar
  29. 29.
    Fujita S., El-Basil S. (2003) J. Math. Chem. 33: 255–277CrossRefGoogle Scholar
  30. 30.
    Fujita S., El-Basil S. (2004) J. Math. Chem. 36: 211–229CrossRefGoogle Scholar
  31. 31.
    Fujita S. (2005) MATCH Commun. Math. Comput. Chem. 54: 251–300Google Scholar
  32. 32.
    Fujita S. (2006) MATCH Commun. Math. Comput. Chem. 55: 5–38Google Scholar
  33. 33.
    Fujita S. (2006) MATCH Commun. Math. Comput. Chem. 55: 237–270Google Scholar
  34. 34.
    Burnside W. (1911). Theory of Groups of Finite Order. 2nd ed. Cambridge University Press, CambridgeGoogle Scholar
  35. 35.
    Fujita S. (2005) Theor. Chem. Acc. 113: 73–79CrossRefGoogle Scholar
  36. 36.
    Fujita S. (2005) Theor. Chem. Acc. 113: 80–86CrossRefGoogle Scholar
  37. 37.
    Fujita S. (2006) Theor. Chem. Acc. 115: 37–53CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Chemistry and Materials TechnologyKyoto Institute of TechnologyKyotoJapan

Personalised recommendations