Geometrical and Topological Dimensions of the Diamond

  • G. V. Zhizhin
  • Z. Khalaj
  • M. V. DiudeaEmail author
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 9)


The question of possible existence of molecules in spaces of higher dimensions, as a consequence of special distribution of matter, is addressed. The geometrical features of the adamantane molecule are examined in detail. It is shown that the adamantane molecule has the dimension 4. The connection ways of the adamantane molecules are investigated on the basis of their geometric properties. Topological properties of the diamond, a 3-periodic net of adamantane, and of a hyperdiamond, called diamond D5, are given in terms of Omega and Cluj polynomials.


Valence Bond Methane Hydrate Phosphorus Oxide Quasi Crystal Nonhomogeneous Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Adams GB, O’Keeffe M, Demkov AA, Sankey OF, Huang Y-M (1994) Wide − band − gap Si in open fourfold − coordinated clathrate structures. Phys Rev B 49:8048–8053CrossRefGoogle Scholar
  2. Aihara J (1976) A new definition of Dewar − type resonance energies. J Am Chem Soc 98:2750–2758CrossRefGoogle Scholar
  3. Ashrafi A, Ghorbani M, Jalali M (2008) The vertex PI and Szeged indices of an infinite family of fullerenes. J Theor Comput Chem 7:221–231CrossRefGoogle Scholar
  4. Ashrafi AR, Koorepazan − Moftakhar F, Diudea MV, Stefu M (2013) Chap. 18: Mathematics of D5 network. In: Diudea MV, Csaba CL (eds) Diamonds and related nanostructures. Springer, Dordrecht, pp 321–333Google Scholar
  5. Atwood W, Maykeylson P, Ritz S (2008) The window in the extreme universe. In the world of science. Sci Am 3:16–21Google Scholar
  6. Balaban AT (2013) Chap 1: Diamond hydrocarbons and related structures. In: Diudea MV, Csaba CL (eds) Diamonds and related nanostructures. Springer, Dordrecht, pp 1–28Google Scholar
  7. Bauschlicher CW, Liu Y, Ricca A, Mattioda AL, Allamandola LJ (2007) Electronic and vibrational spectroscopy of diamondoids and the interstellar infrared bands between 3.35 and 3.55 μm. Astrophys J 671:458–469CrossRefGoogle Scholar
  8. Benedek G, Colombo L (1996) Hollow diamonds from fullerenes. Mater Sci Forum 232:247–274CrossRefGoogle Scholar
  9. Böhme B, Guloy A, Tang Z, Schnelle W, Burkhardt U, Baitinger M, Yu G (2007) Oxidation of M4Si4 (M = Na, K) to clathrates by HCl or H2O. J Am Chem Soc 129:5348–5349CrossRefGoogle Scholar
  10. Burgess K, Quevedo F (2008) Large space travel on the “roller coaster”. In the world of science. Sci Am 3:22–31Google Scholar
  11. Dahl JE, Liu SG, Carlson RMK (2003) Isolation and structures of higher diamondoids, nanometer − sized diamond molecules. Science 229:96–99CrossRefGoogle Scholar
  12. Decarli PS, Jamieson JC (1961) Formation of diamond by explosive shock. Science 133:1821–1822CrossRefGoogle Scholar
  13. Diudea MV (1997) Cluj matrix invariants. J Chem Inf Comput Sci 37:300–305CrossRefGoogle Scholar
  14. Diudea MV (2006) Omega polynomial. Carpath J Math 22:43–47Google Scholar
  15. Diudea MV (2009) Cluj polynomials. J Math Chem 45:295–308CrossRefGoogle Scholar
  16. Diudea MV (2010a) Counting polynomials in partial cubes. In: Gutman I, Furtula B (eds) Novel molecular structure descriptors − theory and applications I. University of Kragujevac, Kragujevac, pp 191–215Google Scholar
  17. Diudea MV (2010b) Counting polynomials and related indices by edge cutting procedures. In: Gutman I, Furtula B (eds) Novel molecular structure descriptors − theory and applications II. University of Kragujevac, Kragujevac, pp 57–78Google Scholar
  18. Diudea MV (2013) Hyper − graphenes. Int J Chem Model 5:211–220Google Scholar
  19. Diudea MV, Klavžar S (2010) Omega polynomial revisited. Acta Chim Sloven 57:565–570Google Scholar
  20. Diudea MV, Nagy CL (eds) (2013) Diamond and related nanostructures, vol 6, Carbon materials: chemistry and physics. Springer, DordrechtGoogle Scholar
  21. Diudea MV, Szefler B (2012) Nanotube junctions and the genus of multi − tori. Phys Chem Chem Phys 14(22):8111–8115CrossRefGoogle Scholar
  22. Diudea MV, Gutman I, Jäntschi L (2002) Molecular topology. Nova, New YorkGoogle Scholar
  23. Diudea MV, Vizitiu AE, Janežič D (2007) Cluj and related polynomials applied in correlating studies. J Chem Inf Model 47:864–874CrossRefGoogle Scholar
  24. Diudea MV, Cigher S, John PE (2008) Omega and related counting polynomials. MATCH Commun Math Comput Chem 60:237–250Google Scholar
  25. Diudea MV, Ilić A, Ghorbani M, Ashrafi AR (2010a) Cluj and PIv polynomials. Croat Chem Acta 83:283–289Google Scholar
  26. Diudea MV, Dorosti N, Iranmanesh A (2010b) Cluj Cj polynomial and indices in a dendritic molecular graph. Studia Univ “Babes−Bolyai”Chemia 55(4):247–253Google Scholar
  27. Diudea MV, Nagy CL, Žigert P, Klavžar S (2010c) Cluj and related polynomials in tori. Studia Univ “Babes−Bolyai”Chemia 55(4):113–123Google Scholar
  28. Diudea MV, Ilić A, Medeleanu M (2011) Hyperdiamonds: a topological view. Iranian J Math Chem 2:7–29Google Scholar
  29. Diudea MV, Nagy CL, Bende A (2012) On diamond D5. Struct Chem 23:981–986CrossRefGoogle Scholar
  30. Dorosti N, Iranmanesh A, Diudea MV (2009) Computing the Cluj index of dendrimer nanostars. MATCH Commun Math Comput Chem 62(2):389–395Google Scholar
  31. Došlić T, Vukičević D (2007) Computing the bipartite edge frustration of fullerene graphs. Discret Appl Math 155:1294–1301CrossRefGoogle Scholar
  32. Dubrovinskaia N, Dub S, Dubrovinsky L (2006) Superior wear resistance of aggregated diamond nanorods. Nano Lett 6:824–826CrossRefGoogle Scholar
  33. Einstein A (1930) The problem of space, fields and ether in physics. Dia Koralle 5:486–487Google Scholar
  34. Euclid (2012) Beginnings. URSS, MoscowGoogle Scholar
  35. Fischer J, Baumgartner J, Marschner C (2005) Synthesis and structure of sila − adamantane. Science 310:825–830CrossRefGoogle Scholar
  36. Fisher ME, Pfeuty P (1972) Critical behavior of the anisotropic n–vector model. Phys Rev B 6:1889–1891CrossRefGoogle Scholar
  37. Greene B (2011) Theelegant universe. Superstrings, hidden dimensions and the quest for the ultimate theory. Librokom, MoscowGoogle Scholar
  38. Grunbaum B (1967) Convexpolytopes. Springer, LondonGoogle Scholar
  39. Guloy A, Ramlau R, Tang Z, Schnelle W, Baitinger M, Yu G (2006) A quest − free germanium clathrate. Nature 443:320–323CrossRefGoogle Scholar
  40. Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes of NY 27:9–15Google Scholar
  41. Gutman I, Klavžar S (1995) An algorithm for the calculation of the Szeged index of benzenoid hydrocarbons. J Chem Inf Comput Sci 35:1011–1014CrossRefGoogle Scholar
  42. Gutman I, Milun M, Trinajstić N (1977) Graph theory and molecular orbitals. 19. Nonparametric resonance energies of arbitrary conjugated systems. J Am Chem Soc 99:1692–1704CrossRefGoogle Scholar
  43. Harary F (1969) Graph theory. Addison − Wesley, ReadingGoogle Scholar
  44. Hosoya H (1988) On some counting polynomials in chemistry. Discret Appl Math 19:239–257CrossRefGoogle Scholar
  45. Hosoya H (1990) Clar’s aromatic sextet and sextet polynomial. Top Curr Chem 153:255–272CrossRefGoogle Scholar
  46. Hosoya H, Yamaguchi T (1975) Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons. Tetrahedron Lett 16(52):4659–4662CrossRefGoogle Scholar
  47. Janssen T, Chapuis G, De Boissieu M (2007) Aperiodic crystals. From modulated phases to quasicrystals. Oxford University Press, OxfordCrossRefGoogle Scholar
  48. John PE, Vizitiu AE, Cigher S, Diudea MV (2007) CI index in tubular nanostructures. MATCH Commun Math Comput Chem 57:479–484Google Scholar
  49. Kadanoff LP (1966) Scaling laws for Isingmodels near τc* Physics 2:263–272Google Scholar
  50. Khachatryan AK, Aloyan SG, May PW, Sargsyan R, Khachatryan VA, Baghdasaryan VS (2008) Graphite-to-diamond transformation induced by ultrasound cavitation. Diam Relat Mater 17:931–936CrossRefGoogle Scholar
  51. Khalaj Z, Ghoranneviss M (2012) Investigation of metallic nanoparticles produced by laser ablation method and their catalytic activity on CVD diamond growth. Studia Univ “Babes−Bolyai”Chemia 57(2):21–28Google Scholar
  52. Khalaj Z, Ghoranneviss M, Vaghri E, Saghaleini A, Diudea MV (2012) Deposition of DLC film on stainless steel substrates coated by Nickel using PECVD method. Acta Chim Slov 59:338–343Google Scholar
  53. Khalifeh M, Yousefi − Azari H, Ashrafi A (2008) A matrix method for computing Szeged and vertex PI indices of join and composition of graphs. Linear Algebra Appl 429:2702–2709CrossRefGoogle Scholar
  54. Klavžar S (2008) A bird’s eye view of the cut method and a survey of its applications in chemical graph theory. MATCH Commun Math Comput Chem 60:255–274Google Scholar
  55. Landa S, Machacek V (1933) Sur l’adamantane, nouvel hydrocarbure extait du naphte. Collection Czech Commun 5:1–5CrossRefGoogle Scholar
  56. Landau LD (1937) On the theory of phase transitions I. J Exp Theor Phys 7:19–38Google Scholar
  57. Lobachevsky NI (1835) Imaginary geometry. Sci Notes Kazan Univ 1:3–88Google Scholar
  58. Lorenz HP (1995) Investigation of TiN as an interlayer for diamond deposition on steel. Diam Relat Mater 4:1088–1092CrossRefGoogle Scholar
  59. Mansour T, Schork M (2009) The vertex PI index and Szeged index of bridge graphs. Discret Appl Math 157:1600–1606CrossRefGoogle Scholar
  60. Mathematical encyclopedia 4 (1984) Sov encyclopedia, MoscowGoogle Scholar
  61. Meier WM, Olson DH (1992) Atlas of zeolite structure types, 3rd edn. Butterworth − Heineman, LondonGoogle Scholar
  62. Merkle RC, Freitas RA Jr (2003) Theoretical analysis of a carbon-carbon dimer placement tool for diamond mechanosynthesis. J Nanosci Nanotechnol 3(4):319–324CrossRefGoogle Scholar
  63. Nagy CL, Diudea MV (2009) NANO–studio software program. Babes–Bolyai University, ClujGoogle Scholar
  64. Nagy CL, Diudea MV (2013) Chap 5: Diamond D5. In: Diudea MV, Csaba CL (eds) Diamonds and related nanostructures. Springer, Dordrecht, pp 91–106Google Scholar
  65. Osawa E (2007) Recent progress and perspectives in single − digit nano diamond. Diam Relat Mater 16:2018–2022CrossRefGoogle Scholar
  66. Osawa E (2008) Monodisperse single nano diamond particulates. Pure Appl Chem 80:1365–1379CrossRefGoogle Scholar
  67. Poincaré A (1895) Analysis situs. J de Ecole Polyt 1:1–121Google Scholar
  68. Poincaré A (1902) La science et Chypothe’se. Flammarion, ParisGoogle Scholar
  69. Riemann B (1868) On the hypotheses underlying geometry. Gëtt. Abhandlungen 13Google Scholar
  70. Saheli M, Diudea MV (2013) Chap. 10: Cluj and other polynomials of D6 and related networks. In: MV Diudea, CL Nagy (eds) Carbon materials: chemistry and physics, 6: Diamond and related nanostructures, Springer, Dordrecht, Heidelberg, New York, London, pp 191–204Google Scholar
  71. Schwarz U, Wosylus A, Böhme B, Baitinger M, Hanfland M, Yu G (2008) A 3D network of four − bonded germanium: a link between open and dense. Angew Chem Int Ed 47:6790–6793CrossRefGoogle Scholar
  72. Shafranovsky II (1964) Diamonds. Nauka, Moscow − LeningradGoogle Scholar
  73. Sharda T, Rahaman MM, Nukaya Y, Soga T, Jimbo T, Umeno M (2001) Structural and optical properties of diamond and nano–diamond films grown by microwave plasma chemical vapor deposition. Diam Relat Mater 10:561–467CrossRefGoogle Scholar
  74. Shevchenko VYA, Zhizhin GV, Mackay A (2013a) On the structure of quasicrystals in the space of higher dimension. News RAS Chem Ser 2:269–274Google Scholar
  75. Shevchenko VYA, Zhizhin GV, Mackay A (2013b) Chapter 17: On the structure of the quasicrystals in the high dimention space. In: Diamonds and related nanostructures. Springer, Dordrecht, pp 311–320Google Scholar
  76. Sourina O, Korolev N (2005) Design and analysis of a molecular tool for carbon transfer in mechanosynthesis. J Comput Theor Nanosci 2(4):492–498CrossRefGoogle Scholar
  77. Stefu M, Diudea MV (2005) CageVersatile_CVNET software program. Babes–Bolyai University, ClujGoogle Scholar
  78. Takano Y, Nagao M, Takenouchi T, Umezawa H, Sakaguchi I, Tachiki M, Kawarada H (2005) Superconductivity in polycrystalline diamond thin films. Diam Relat Mater 14:1936–1938CrossRefGoogle Scholar
  79. Tarasov D, Izotova E, Alisheva D, Akberova N, Freitas RA Jr (2011) Structural stability of clean, passivated, and partially dehydrogenated cuboid and octahedral nanodiamonds up to 2 nanometers in size. J Comput Theor Nanosci 8:147–167CrossRefGoogle Scholar
  80. Ursu O, Diudea MV (2005) TopoCluj software program. Babes–Bolyai University Cluj, ClujGoogle Scholar
  81. Williams OA, Douhéret O, Daenen M, Haenen K, Osawa E, Takahashi M (2007) Enhanced diamond nucleation on monodispersed nanocrystalline diamond. Chem Phys Lett 445:255–258CrossRefGoogle Scholar
  82. Wilson RG (1971a) Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys Rev B 4:3174–3183CrossRefGoogle Scholar
  83. Wilson RG (1971b) Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior. Phys Rev B 4:3184–3205CrossRefGoogle Scholar
  84. Yamazaki K, Furuichi K, Tsumura I, Takagi Y (2008) The large–sized diamond single–crystal synthesis by hot filament CVD. J Cryst Growth 310:1019–1022CrossRefGoogle Scholar
  85. Zhizhin GV (2014a) World 4D. Polytechnic Service, St. PetersburgGoogle Scholar
  86. Zhizhin GV (2014b) Disproportionate and fluctuating structure in space earthly reality. Biosphere 3:211–221Google Scholar
  87. Zhizhin GV (2014c) On higher dimension in nature. Biosphere 4:1–10Google Scholar
  88. Zwiebach B (2011) Initial course theory string. URSS, MoscowGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Member of “Skolkovo” OOO “Adamant”Saint-PetersburgRussia
  2. 2.Department of Physics, Shahr-e-Qods BranchIslamic Azad UniversityTehranIran
  3. 3.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania

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