Chiral Multi-tori

  • Mircea Vasile Diudea
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 10)


Chirality is one of the basic characteristics of biological structures; chirality is a symmetry property. Multi-tori are complex structures consisting of more than one torus, embedded in negatively curved surfaces. Design of multi-tori may be achieved by operations on maps. The “monomers” used to design the chiral multi-tori discussed in this chapter are snubs of Platonic solids. A snub polyhedron s(P) is achieved by dualizing the p5(P) transform: s(P) = d(p5(P)); since p5-operation is prochiral, all the consecutive structures will be chiral; high genus structures of rank k = 3 were thus obtained. The genus and rank (or space dimension) of a structure are parameters of complexity.

Topological symmetry of the structures herein discussed was evaluated by ring signature and centrality index and confirmed by symmetry calculation using the adjacency matrix permutations. C60-related chiral tori were also designed and their symmetry evaluated. An atlas section illustrates the discussed chiral multi-tori.

Supplementary material


  1. Babić D, Klein DJ, Schmalz TG (2001) Curvature matching and strain relief in bucky-tori: usage of sp3-hybridization and nonhexagonal rings. J Mol Graph Model 19:223–231Google Scholar
  2. Barborini E, Piseri P, Milani P, Benedek G, Ducati C, Robertson J (2002) Negatively curved spongy carbon. Appl Phys Lett 81:3359–3361CrossRefGoogle Scholar
  3. Benedek G, Vahedi-Tafreshi H, Barborini E, Piseri P, Milani P, Ducati C, Robertson J (2003) The structure of negatively curved spongy carbon. Diam Relat Mater 12:768–773CrossRefGoogle Scholar
  4. Blatov VA, O’Keeffe M, Proserpio DM (2010) Vertex-, face-, point-, Schläfli-, and Delaney-symbols in nets, polyhedra and tilings: recommended terminology. Cryst Eng Comm 12:44–48CrossRefGoogle Scholar
  5. Buckley F (1979) Self-centered graph with given radius. Congr Numer 23:211–215Google Scholar
  6. Buckley F (1989) Self-centered graphs. Ann NY Acad Sci 576:71–78CrossRefGoogle Scholar
  7. Diudea MV (1994) Layer matrices in molecular graphs. J Chem Inf Comput Sci 34:1064–1071CrossRefGoogle Scholar
  8. Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, DordrechtCrossRefGoogle Scholar
  9. Diudea MV, Petitjean M (2008) Symmetry in multi-tori. Symmetry Cult Sci 19(4):285–305Google Scholar
  10. Diudea MV, Petitjean M (2016) Chiral multitori as snub derivatives. Rev Roum Chim 61(4–5):329–337Google Scholar
  11. Diudea MV, Rosenfeld VR (2017) The truncation of a cage graph. J Math Chem 55(4):1014–1020. doi: 10.1007/s10910-016-0716-6 CrossRefGoogle Scholar
  12. Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem A 42(6):1283–1294Google Scholar
  13. Euler L (1752–1753) Elementa doctrinae solidorum-Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita. Novi Comment Acad Sc Imp Petropol 4:109–160Google Scholar
  14. Harary F (1969) Graph theory. Addison-Wesley, ReadingCrossRefGoogle Scholar
  15. Hargittai M, Hargittai I (2010) Symmetry through the eyes of a chemist. Springer, DordrechtGoogle Scholar
  16. Higuchi Y (2001) Combinatorial curvature for planar graphs. J Graph Theory 38:220–229CrossRefGoogle Scholar
  17. Janakiraman TN, Ramanujan J (1992) On self-centered graphs. Math Soc 7:83–92Google Scholar
  18. Kelvin L (1904) Baltimore lectures on molecular dynamics and the wave theory of light, Appendix H, Sect. 22, footnote p. 619. C.J. Clay and Sons, Cambridge University Press Warehouse, LondonGoogle Scholar
  19. Klein DJ (2002) Topo-combinatoric categorization of quasi-local graphitic defects. Phys Chem Chem Phys 4:2099–2110CrossRefGoogle Scholar
  20. Klein DJ, Liu X (1994) Elemental carbon isomerism. Int J Quantum Chem S28:501–523CrossRefGoogle Scholar
  21. Lenosky T, Gonze X, Teter M, Elser V (1992) Energetics of negatively curved graphitic carbon. Nature 355:333–335CrossRefGoogle Scholar
  22. Lijnen E, Ceulemans A (2005) The symmetry of Dyck graph: group structure and molecular realization. J Chem Inf Model 45(6):1719–1726CrossRefGoogle Scholar
  23. Mackay AL, Terrones H (1991) Diamond from graphite. Nature 352:762–762CrossRefGoogle Scholar
  24. Meier WM, Olson DH (1992) Atlas of zeolite structure types, 3rd edn. Butterworth-Heineman, LondonGoogle Scholar
  25. Nagy CL, Diudea MV (2009) Nano studio software. Babes-Bolyai University, ClujGoogle Scholar
  26. Nagy CL, Diudea MV (2017) Ring signature index. MATCH Commun Math Comput Chem 77(2):479–492Google Scholar
  27. Nazeer W, Kang SM, Nazeer S, Munir M, Kousar J, Sehar A, Kwun YC (2016) On center, periphery and average eccentricity for the convex polytopes. Symmetry 8:145–168CrossRefGoogle Scholar
  28. Negami S, Xu GH (1986) Locally geodesic cycles in 2-self-centered graphs. Discret Math 58:263–268CrossRefGoogle Scholar
  29. Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Combin Theory A 40(2):305–330CrossRefGoogle Scholar
  30. Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Crystallographica A 70:203–216CrossRefGoogle Scholar
  31. Terrones H, Mackay AL (1997) From C60 to negatively curved graphite. Prog Crystal Growth Charact 34:25–36CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Mircea Vasile Diudea
    • 1
  1. 1.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania

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