Rhombellanic Crystals and Quasicrystals

Document Type : Research Paper

Author

Babes-Bolyai University, Cluj, Romania

Abstract

Design of some crystal and quasicrystal networks, based on rhombellane tiling,is presented. [1,1,1]Propellane,is a synthesized organic molecule; its hydrogenated form, the bicyclo[1.1.1]pentane,may be represented by the complete bipartite graph K2,3 which is the smallest rhombellane. Topology of translational and radial structures involving rhombellanes is described in terms of vertex symbol, connectivity sequence, ring sequence and map operations relating structures to their seeds. It is shown, by alternating sum of ranked substructures, that radial structures represent complex constructions of higher rank. Basic properties of rhombellanes, coloring included, are outlined.

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Main Subjects


1. K.B. Wiberg, F.H. Walker, [1.1.1]Propellane. J. Amer. Chem. Soc. 104
(19) (1982) 5239−5240.
2. P. Kazynsky, J.Michl, [n]Staffanes: a molecular-size tinkertoy construction
set for nanotechnology. Preparation of end-functionalized telomers and a
polymer of [1.1.1]propellane. J. Amer. Chem. Soc. 110 (15) (1988) 5225–
5226.
3. I. Hafner, T. Zitko, Relations among rhombic, Platonic and Archimedean
solids, Visual Math. 4 (2) (2002) 2(4).
4. M. V. Diudea, Multi-shell polyhedral clusters, Springer, Berlin, 2018.
5. E. Steinitz, Polyeder und Raumeinteilungen. Encyclopädie der
mathematischen Wissenschaften, B. G. Teubner Verlag, Vol. 3, 1922.
6. V. A. Blatov, M. O’Keeffe, D. M. Proserpio, Vertex-, face-, point-,
Schläfli-, and Delaney-symbols in nets, polyhedra and tilings:
recommended terminology, Cryst. Eng. Comm. 12 (2010) 44−48.
7. M. V. Diudea, Hypercube related polytopes, Iranian J. Math. Chem. 9 (1)
(2018) 1−8.
8. P. J. Steinhardt, Quasi–Crystals – A new form of matter, Endeavour 14
(1990) 112−116.
9. Reticular Chemistry Structure Resource, http://rcsr.anu.edu.au.
10. M. V. Diudea, M. Topan, A. Graovac, Molecular topology. 17. Layer
matrixes of walk degrees, J. Chem. Inf. Comput. Sci. 34 (5) (1994)
1072−1078.
11. M. V. Diudea, O. Ursu, Layer matrices and distance property descriptors.
Indian J. Chem. A 42 (6) (2003) 1283−1294.
12. C. L. Nagy, M. V. Diudea, Ring signature index, MATCH Commun. Math.
Comput. Chem. 77 (2) (2017) 479−492.
13. D. Shechtman, I. Blech, D. Gratias, J. W. Cahn, Metallic phase with long–
range orientational order and no translational symmetry, Phys. Rev. Lett. 53
(1984) 1951−1953.
14. E. Schulte, Polyhedra, complexes, nets and symmetry, Acta Cryst. A 70
(2014) 203−216.
15. L. Euler, Elementa doctrinae solidorum, Novi Comm. Acad. Scient. Imp.
Petrop. 4 (1752−1753) 109−160.
16. M. V. Diudea, Omega polynomial, Carpath. J. Math. 22 (2006) 43−47.
17. M. V. Diudea, S. Klavžar, Omega polynomial revisited, Acta Chem.
Sloven. 57 (2010) 565−570.
18. G. Chartrand, P. Zhang, Chromatic Graph Theory, CRC Press, Boca Raton,
FL, 2009.
19. D. W. Matula, G. Marble, J. D. Isaacson, in: R. Read (Ed.) Graph Theory
and Computing, Academic Press, New York, 1972, pp. 109−122.
20. N. Christofides, An algorithm for the chromatic number of a graph,
Computer J. 14 (1971) 38−39.
21. S. Pemmaraju, S. Skiena, Computational Discrete Mathematics:
Combinatorics and Graph Theory with Mathematica, Cambridge
University Press, Cambridge, 2003.
22. A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and
the Colorful Life of its Creators, Springer, New York, 2009.
23. J. Mycielski, Sur le coloriage des graphs, Colloq. Math. 3 (1955) 161−162.
24. A. A. Zykov, On some properties of linear complexes, Mat. Sbornik N. S.
(Russian) 24 (66) (1949) 163−188.
25. D. C. Fisher, P. A. McKenna, E. D. Boyer, Biclique parameters of
Mycielskians, Discrete Appl. Math. 84 (1–3) (1998) 93−105.
26. T. Došlić, Mycielskians and matchings, Discuss. Math. Graph Theory 25
(3) (2005) 261−266.
27. T. Jensen, G. Royle, https://www. researchgate.net/ publication/227668205.
28. https://math.stackexchange.com/questions/1561029/.
29. C. L. Nagy, M. V. Diudea, Nano−Studio software, Babes−Bolyai
University, Cluj, 2009.
30. Wolfram Res., Inc., Mathematica, Version 10.4, Champaign, II.
31. M. Veith, P. König, A. Rammo, V. Huch, Cubane-like Li4H4 and
Li3H3Li(OH): stabilized in molecular adducts with alanes, Angew. Chem.
Int. Ed. 44 (2005) 5968−5971.
32. K. Chen, C. Sun, D. Xue, Morphology engineering of high performance
binary oxide electrodes, Phys. Chem. Chem. Phys. 17 (2015) 732−750.