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Symmetry of Structures That Can Be Approximated by Chains of Regular Tetrahedra

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Abstract

The noncrystallographic symmetries of chains of regular tetrahedra are determined by mapping the system of algebraic geometry and topology designs to the structural level. It has been shown that the basic structural unit of such a chain is a tetrablock: a seven-vertex linear aggregation over faces of four regular tetrahedra, which is implemented in linear (right- and left-handed) and planar versions. The symmetry groups of linear and planar tetrablocks are isomorphic, respectively, to the projective special linear group PSL(2, 7) of order 168 and the projective general linear group PGL(2, 7) of order 336. A class of structures formed by an assembly of tetrablocks having no common tetrahedra is introduced. Examples of tetrablock assembly over common face, leading to a Boerdijk–Coxeter helix, an α helix, and a helix used as one of collagen models are presented.

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REFERENCES

  1. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: Mechanics (Fizmatlit, Moscow, 2004; Pergamon, Oxford, 1976).

  2. E. Cartan, Geometry of Riemannian Spaces (Math Sci., Brookline, 1983).

    MATH  Google Scholar 

  3. J. H. Mason, Math. Gaz. 56, 194 (1972).

    Article  Google Scholar 

  4. G. Ringel, Map Color Theorem (Mir, Moscow, 1977) [in Russian].

  5. H. S. M. Coxeter, Regular Polytopes (Dover, New York, 1973).

    MATH  Google Scholar 

  6. M. Görner, arXiv:1406.2827v3 [math.GT].

  7. W. P. Thurston, Classical Quantum Gravity 15, 2545 (1998).

    Article  MathSciNet  ADS  Google Scholar 

  8. I. Agol; http://citeseerx.ist.psu.edu/viewdoc/download?doi10.1.1.121.3407&reprep1&typepdf

  9. M. Samoylovich and A. Talis, Acta Crystallogr. A 70, 186 (2014).

    Article  Google Scholar 

  10. A. L. Mackay, THEOCHEM 336, 293 (1995).

    Article  Google Scholar 

  11. C. Zheng, R. Hoffmann, and D. R. Nelson, J. Am. Chem. Soc. 112, 3784 (1990).

    Article  Google Scholar 

  12. H. Babiker and S. Janeczko, Commun. Inf. Syst. 15, 331 (2015).

    MathSciNet  Google Scholar 

  13. M. Samoylovich and A. Talis, Arxiv1606.01237.

  14. A. L. Rabinovich and A. L. Talis, Obozr. Prikl. Prom. Mat. 25 (1), 53 (2018).

    Google Scholar 

  15. A. L. Talis and A. L. Rabinovich, Obozr. Prikl. Prom. Mat. 25 (1), 56 (2018).

    Google Scholar 

  16. A. L. Talis and A. L. Rabinovich, Obozr. Prikl. Prom. Mat. 25 (3), 280 (2018).

    Google Scholar 

  17. A. L. Rabinovich and A. L. Talis, Obozr. Prikl. Prom. Mat. 25 (3), 275 (2018).

    Google Scholar 

  18. A. L. Talis and A. L. Rabinovich, Obozr. Prikl. Prom. Mat. 25 (2), 191 (2018).

    Google Scholar 

  19. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups (Springer, New York, 1980; Mir, Moscow, 1990).

  20. P. Martin and D. Singerman, Eur. J. Combinatorics 33, 1619 (2012).

    Article  Google Scholar 

  21. J. H. Conway, R. T. Curtis, S. P. Norton, et al., Atlas of Finite Groups (Clarendon, Oxford, 1985).

    MATH  Google Scholar 

  22. A. T. White, Proc. London Math. Soc. s3-70, 33 (1995).

  23. U. Brehm and W. Kühnel, Eur. J. Combinatorics 29, 1843 (2008).

    Article  Google Scholar 

  24. A. Altshuler, Discrete Math. 1, 211 (1971).

    Article  MathSciNet  Google Scholar 

  25. A. Császár, Acta Sci. Math. Univ. Szegediensis 13, 140 (1949).

    Google Scholar 

  26. J. Bokowski and A. Eggert, Struct. Topol., No. 17, 59 (1991).

  27. L. Szilassi, 3rd Int. Conf. APLIMAT-2004: A Plenary Lecture, Department of Mathematics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 2004, p. 173.

  28. M. Stojanović, Kragujevac J. Math. 41, 203 (2017).

    Google Scholar 

  29. R. Mosseri, D. P. DiVincenzo, J. F. Sadoc, and M. H. Brodsky, Phys. Rev. B 32, 3974 (1985).

    Article  MathSciNet  ADS  Google Scholar 

  30. C. H. Li and B. Xia, arXiv:1408.0350v3 [math.GR].

  31. A. Ceulemans, R. B. King, S. A. Bovin, et al., J. Math. Chem. 26, 101 (1999).

    Article  MathSciNet  Google Scholar 

  32. V. A. Koptsik, Comput. Math. Appl. 16, 407 (1988).

    Article  MathSciNet  Google Scholar 

  33. A. Talis and V. Kraposhin, Acta Crystallogr. A 70, 616 (2014).

    Article  Google Scholar 

  34. T. Shioda, Comment. Math. Univ. St. Paul 42, 61 (1993).

    MathSciNet  Google Scholar 

  35. E. A. Lord, A. L. Mackay, and S. Ranganathan, New Geometries for New Materials (Cambridge Univ. Press, Cambridge, 2006).

    MATH  Google Scholar 

  36. H. Nyman, C. E. Carroll, and B. G. Hyde, Z. Kristallogr. 196, 39 (1991).

    Article  Google Scholar 

  37. G. Sadler, F. Fang, J. Kovacs, and K. Irwin, arXiv:1302.1174v1 [math.MG].

  38. S. Kumar and M. Bansal, Biophys. J. 75, 1935 (1998).

    Article  ADS  Google Scholar 

  39. L. Pauling, R. V. Corey, and H. R. Branson, Proc. Natl. Acad. Sci. USA 37, 205 (1951).

    Article  ADS  Google Scholar 

  40. J. F. Sadoc and N. Rivier, Eur. Phys. J. B 12, 309 (1999).

    Article  ADS  Google Scholar 

  41. K. Beck and B. Brodsky, J. Struct. Biol. 122, 17 (1998).

    Article  Google Scholar 

  42. W. Steurer and S. Deloudi, Crystallography of Quasicrystals. Concepts, Methods and Structures (Springer, Berlin, 2009).

    MATH  Google Scholar 

  43. R. Zandi, D. Reguera, R. F. Bruinsma, et al., Proc. Natl. Acad. Sci. USA 101, 15556 (2004).

    Article  ADS  Google Scholar 

  44. E. Lijnen, A. Ceulemans, P. W. Fowler, and M. Deza, J. Math. Chem. 42, 617 (2007).

    Article  MathSciNet  Google Scholar 

  45. R. B. King, Croat. Chem. Acta 73, 993 (2000).

    Google Scholar 

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ACKNOWLEDGMENTS

The study was supported from the federal budget: state contract nos. 0085-2019-0004 (registration number AAAA-A18-118012590359-8, Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences) and 0221-2017-0050 (registration number АААА-А17-117031710039-3, Institute of Biology, Karelian Research Centre, Russian Academy of Sciences).

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Authors and Affiliations

  1. Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, 119991, Moscow, Russia

    A. L. Talis

  2. Institute of Biology, Karelian Research Centre, Russian Academy of Sciences, 185910, Petrozavodsk, Russia

    A. L. Rabinovich

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  1. A. L. Talis
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  2. A. L. Rabinovich
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Correspondence to A. L. Talis.

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Translated by Yu. Sin’kov

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Talis, A.L., Rabinovich, A.L. Symmetry of Structures That Can Be Approximated by Chains of Regular Tetrahedra. Crystallogr. Rep. 64, 367–375 (2019). https://doi.org/10.1134/S106377451903026X

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