Large Icosahedral Clusters

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


Large multi-shell clusters, of icosahedral symmetry, are derived, by operations on maps, from seeds like Bergman’s C45 and a larger C125 clusters. The most used operations were medial and truncation, eventually followed by dualization. The clusters were characterized by figure count while their topological symmetry was described in terms of ring signature and centrality index. Clusters composed of dodecahedral shapes and having dodecahedral topology, both spongy and filled, were designed by an original procedure. Rhomb decorated clusters were realized by medial operation followed by dualization, or by p4 operation. Attention was focused on the cluster C152 that includes the smallest rhombic Rh3 substructure (the skeleton of a real molecule, named [1,1,1]-propellane), which is not a polyhedron but a tile; a new class of structures, called “propellanes” was thus discovered. An atlas section illustrates the discussed multi-shell polyhedral clusters.

Supplementary material


  1. Ammann R, Grünbaum B, Shephard GC (1992) Aperiodic tiles. Discret Comp Geom 8:1–25CrossRefGoogle Scholar
  2. Bergman G, Waugh JLT, Pauling L (1952) Crystal structure of the intermetallic compound Mg32(Al, Zn)49 and related phases. Nature 169:1057–1058CrossRefGoogle Scholar
  3. 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
  4. de Boisieu M, Dubois JM, Audier M, Dubost B (1991) Atomic structure of the icosahedral AlLiCu quasicrystal. J Phys Condens Matter 3:1–25CrossRefGoogle Scholar
  5. Coxeter HSM (1973) Regular polytopes, 3rd edn. Dover, New York, NYGoogle Scholar
  6. Diudea MV (1994) Layer matrices in molecular graphs. J Chem Inf Comput Sci 34:1064–1071CrossRefGoogle Scholar
  7. Diudea MV (2013) Quasicrystals, between spongy and full space filling. In: Diudea MV, Nagy CL (eds) Carbon materials: chemistry and physics, vol 6. Diamond and related nanostructures. Springer, Dordrecht, Chap. 19, pp 333–383Google Scholar
  8. Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem A 42(6):1283–1294Google Scholar
  9. Duneau M, Gratias D (2002) Covering clusters in icosahedral quasicrystals. In: Coverings of discrete quasiperiodic sets, vol 180 of the series Springer tracts in modern physics, pp 23–62Google Scholar
  10. Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comment Acad Sci Imp Petropolitanae 4:109–160Google Scholar
  11. Frank FC, Kasper JS (1958) Complex alloy structures regarded as sphere packings. Definitions and basic principles. Acta Crystallogr 11:184–190CrossRefGoogle Scholar
  12. Gardner M (2001) Packing spheres. Ch10 in The colossal book of mathematics: classic puzzles, paradoxes, and problems. WW Norton, New York, NY, pp 128–136Google Scholar
  13. Grünbaum B, Shephard GC (1987) Tilings and patterns. W. H. Freeman, New York, NYGoogle Scholar
  14. Hales TC (1992) The sphere packing problem. J Comput Appl Math 44:41–76CrossRefGoogle Scholar
  15. Hales TC (2005) A proof of the Kepler conjecture. Ann Math 162:1065–1185CrossRefGoogle Scholar
  16. Heath TL (1981) A history of Greek mathematics. Dover, New York, NYGoogle Scholar
  17. Koltover VK (2007) Endohedral fullerenes: from chemical physics to nanotechnology and nanomedicine. In: Lang M (ed) Progress in fullerene research. Nova Science Publishers, New York, NY, pp 199–233Google Scholar
  18. Nagy CL, Diudea MV (2009) Nano studio software. “Babes-Bolyai” Univ, ClujGoogle Scholar
  19. Nagy CL, Diudea MV (2017) Ring signature index. MATCH Commun Math Comput Chem 77(2):479–492Google Scholar
  20. Papacostea C (1930–1935) P1 aton. Opere, 1–11 (Roum. translation), Casa Şcoalelor, BucureştiGoogle Scholar
  21. Penrose R (1974) The role of aesthetics in pure and applied mathematical research. Bull Inst Math Appl 10:266–271Google Scholar
  22. Popov AA, Yang S, Dunsch L (2013) Endohedral fullerenes. Chem Rev 113(8):5989–6113CrossRefGoogle Scholar
  23. Reddy BV, Khanna SN, Dunlap BI (1993) Giant magnetic moments in 4d clusters. Phys Rev Lett 70(21):3323–3326CrossRefGoogle Scholar
  24. Samson S (1965) The crystal structure of the phase β of Mg2Al3. Acta Crystallogr 19:401–413CrossRefGoogle Scholar
  25. Samson S (1972) Complex cubic A6B compounds. II. The crystal structure of Mg6Pd. Acta Crystallogr B 28:936–945CrossRefGoogle Scholar
  26. Saunders M, Jiménez-Vázquez HA, Cross RJ, Poreda RJ (1993) Stable compounds of helium and neon. He@C60 and Ne@C60. Science 259(5100):1428–1430CrossRefGoogle Scholar
  27. Schläfli L (1901) Theorie der vielfachen Kontinuität Zürcher und Furrer, Zürich (Reprinted in:Ludwig Schläfli, 1814–1895, Gesammelte Mathematische Abhandlungen, Band 1:167–387, Verlag Birkhäuser, Basel, 1950)Google Scholar
  28. Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Combin Theory Ser A 40(2):305–330CrossRefGoogle Scholar
  29. Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Cryst A 70:203–216CrossRefGoogle Scholar
  30. Steinitz E (1922) Polyeder und Raumeinteilungen, Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries) (IIIAB12), pp 1–139, Abgeschlossen am 31 August 1916Google Scholar
  31. Taylor AE (1928) A commentary on Plato’s Timaeus. Clarendon, OxfordGoogle 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

Personalised recommendations