Skip to main content
SpringerLink
Log in

Rhombellane space filling

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

A space-filling polyhedron is a polyhedron that can tessellate the 3D space. The complete bipartite graph K2.3 is the graph representation of [1,1,1]-propellane, a synthesized molecule, or rather of its reduced form, appearing in the polymer called staffane, with all rings being rhombs/squares. Further, the complete bipartite graphs K2.n represent generalized [1,..,1n]-propellanes, in the following named rhombellanes; they are involved in the space-filling within rhombic arrays. This paper presents construction of some crystals and quasicrystals consisting of rhombellanes, and their characterization in crystallographic terms (by connectivity and ring signature) and in topological terms (by Omega polynomial).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. V.A. Blatov, O. Delgado-Friedrichs, M. O’Keeffe, D.M. Proserpio, Acta Crystallogr. A 63, 418 (2007)

    Article  CAS  PubMed  Google Scholar 

  2. B. Grünbaum, G.C. Shephard, Bull. Am. Math. Soc. 3, 951 (1980)

    Article  Google Scholar 

  3. M. Gardner, The Sixth Book of Mathematical Games from Scientific American (University of Chicago Press, Chicago, 1984)

    Google Scholar 

  4. H. Steinhaus, Mathematical Snapshots, 3rd edn. (Dover, New York, 1999), pp. 185–190

    Google Scholar 

  5. N.W. Johnson, Uniform Polytopes (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  6. M. Goldberg, Geom. Dedicata. 8, 491 (1979)

    Article  Google Scholar 

  7. D. Weaire, R. Phelan, Philos. Mag. Let. 69, 107 (1994)

    Article  CAS  Google Scholar 

  8. T. Aste, D. Weaire, The Pursuit of Perfect Packing, 2nd edn. (Taylor & Francis, CRC Press, New York, London, 2008)

    Google Scholar 

  9. T.C. Hales, Ann. Math. 162, 1065 (2005)

    Article  Google Scholar 

  10. F.C. Frank, J.S. Kasper, Acta Crystallogr. 11, 184 (1958)

    Article  CAS  Google Scholar 

  11. F.C. Frank, J.S. Kasper, Acta Crystallogr. 12, 483 (1959)

    Article  CAS  Google Scholar 

  12. J. Kepler, The Six-Cornered Snowflake (Clarendon, Oxford, 1966)

    Google Scholar 

  13. R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974)

    Google Scholar 

  14. J.H. Conway, S. Torquato, Proc. Natl. Acad. Sci. 103, 10612 (2006)

    Article  CAS  PubMed  Google Scholar 

  15. M.V. Diudea, Multi-Shell Polyhedral Clusters (Springer, New York, 2018)

    Book  Google Scholar 

  16. E. Steinitz, Polyeder und Raumeinteilungen, Encyclopädie der mathematischen Wissenschaften, vol. 3 (B.G. Teubner Verlag, Leipzig, 1922), pp. 1–139

    Google Scholar 

  17. K.B. Wiberg, F.H. Walker, J. Am. Chem. Soc. 104, 5239 (1982)

    Article  CAS  Google Scholar 

  18. P. Kazynsky, J. Michl, J. Am. Chem. Soc. 110, 5225 (1988)

    Article  Google Scholar 

  19. M.V. Diudea, Iran. J. Math. Chem. 9, 1 (2018)

    Google Scholar 

  20. M.V. Diudea, Iran. J. Math. Chem. 9, 167 (2018)

    Google Scholar 

  21. B. Szefler, P. Czeleń, M.V. Diudea, Studia Univ. “Babes-Bolyai”. Chemia 63, 7 (2018)

    Google Scholar 

  22. F. Harary, Graph Theory (Addison-Wesley, Reading, 1969)

    Book  Google Scholar 

  23. P.J. Steinhardt, Endeavour 14, 112 (1990)

    Article  Google Scholar 

  24. T.T. Luo, H.L. Tsai, S.L. Yang, Y.H. Liu, R.D. Yadav, C.C. Su, C.H. Ueng, L.G. Lin, K.L. Lu, Angew. Chem. Int. Ed. 44, 6063 (2005)

    Article  CAS  Google Scholar 

  25. S. Bhattacharya, M. Gnanavel, A.J. Bhattacharya, S. Natarajan, Cryst. Growth Des. 14, 310 (2014)

    Article  CAS  Google Scholar 

  26. M. O’Keeffe, M.A. Peskov, S.J. Ramsden, O.M. Yaghi, Accts. Chem. Res. 41, 1782 (2008)

    Article  CAS  Google Scholar 

  27. D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53, 1951 (1984)

    Article  CAS  Google Scholar 

  28. M.V. Diudea, A. Pîrvan-Moldovan, R. Pop, M. Medeleanu, MATCH Commun. Math. Comput. Chem. 80, 835 (2018)

    Google Scholar 

  29. L. Euler, Novi Comm. Acad. Sci. Petrop. 4, 109 (1752–53)

  30. E. Schulte, Acta Cryst. A 70, 203 (2014)

    Article  CAS  Google Scholar 

  31. M.V. Diudea, M. Topan, A. Graovac, J. Chem. Inf. Comput. Sci. 34, 1072 (1994)

    Article  CAS  Google Scholar 

  32. C.L. Nagy, M.V. Diudea, MATCH Commun. Math. Comput. Chem. 77, 479 (2017)

    Google Scholar 

  33. M.V. Diudea, Nanomolecules and Nanostructures—Polynomials and Indices, MCM, 10 (University of Kragujevac, Serbia, 2010)

  34. Wolfram Mathematica, Version 10.4, (Champaign, IL, 2017)

  35. http://rcsr.anu.edu.au/nets. Accessed June 2017

  36. P.E. John, A.E. Vizitiu, S. Cigher, M.V. Diudea, MATCH Commun. Math. Comput. Chem. 57, 479 (2007)

    Google Scholar 

  37. D.Ž. Djoković, J. Combin. Theory. Ser. B 14, 263 (1973)

    Article  Google Scholar 

  38. P.M. Winkler, Discrete Appl. Math. 8, 209 (1984)

    Article  Google Scholar 

  39. M.V. Diudea, S. Klavžar, Acta Chim. Slov. 57, 565 (2010)

    CAS  PubMed  Google Scholar 

  40. S. Klavžar, MATCH Commun. Math. Comput. Chem. 5, 217 (2008)

    Google Scholar 

  41. M.V. Diudea, Carpath. J. Math. 22, 43 (2006)

    Google Scholar 

  42. C.L. Nagy, M.V. Diudea, Nano Studio Software (”Babes-Bolyai” University, Cluj, 2009)

    Google Scholar 

Download references

Acknowledgements

This work was supported by a Grant of the Romanian National Authority for Scientific Research and Innovation, CCCDI—UEFISCDI, Project Number 8/2015, acronym GEMNS).

Author information

Authors and Affiliations

  1. Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babeș-Bolyai University, 400028, Cluj-Napoca, Romania

    Mircea V. Diudea & Csaba L. Nagy

Authors
  1. Mircea V. Diudea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Csaba L. Nagy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mircea V. Diudea.

Appendix

Appendix

Tables 1, 2, 3, 4, 5 and 6.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Diudea, M.V., Nagy, C.L. Rhombellane space filling. J Math Chem 57, 473–483 (2019). https://doi.org/10.1007/s10910-018-0959-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-018-0959-5

Keywords

Search

Navigation