Abstract
A simple cut-and-patch method is presented for the construction and classification for fullerenes belonging to the octahedral point groups, O or \(O_h\). In order to satisfy the symmetry requirement of the octahedral group, suitable numbers of four- and eight-member rings, in addition to the hexagons and pentagons, have to be introduced. An index consisting of four integers is introduced to specify an octahedral fullerenes. However, to specify an octahedral fullerene uniquely, we also found certain symmetry rules for these indices. Based on the transformation properties under the symmetry operations that an octahedral fullerene belongs to, we can identify four structural types of octahedral fullerenes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adidas Telstar. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Accessed 17 July 2014
D. Kotschick, The topology and combinatorics of soccer balls. Am. Sci. 94, 350–357 (2005)
Adidas Brazuca. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Accessed 17 June 2014
S. Hong, T. Asai, Effect of panel shape of soccer ball on its flight characteristics. Sci. Rep. 4(5068), 1–7 (2014)
H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, \(\text{ C }_{60}\): Buckminsterfullerene. Nature 318, 162–163 (1985)
P.W. Fowler, D.E. Manolopoulos, An Atlas of Fullerenes (Dover Publications, New York, 2007)
M. Goldberg, A Class of multi-symmetric polyhedra. Tohoku Math. J. 43, 104–108 (1937)
D.L.D. Caspar, A. Klug, Physical principles in the construction of regular viruses. Cold Spring Harb. Symp. Quant. Biol. 27, 1–24 (1962)
D.E. Manolopoulos, P.W. Folwer, Molecular graphs, point groups, and fullerenes. J. Chem. Phys. 96, 7603–7614 (1992)
P.W. Fowler, J.E. Cremona, J.I. Steer, Systematics of bonding in non-icosahedral carbon clusters. Theor. Chim. Acta 73, 1–26 (1988)
S. Jendrol’, F. Kardoš, On octahedral fulleroids. Discret. Appl. Math. 155, 2181–2186 (2007)
P.W. Fowler, D.E. Manolopoulos, D.B. Redmond, R.P. Ryan, Possible symmetries of fullerene structures. Chem. Phys. Lett. 202, 371–378 (1993)
P.W. Fowler, K.M. Rogers, Spiral codes and Goldberg representations of icosahedral fullerenes and octahedral analogues. J. Chem. Inf. Comput. Sci. 41, 108–111 (2001)
A.-C. Tang, F.-Q. Huang, Electronic structures of octahedral fullerenes. Chem. Phys. Lett. 263, 733–741 (1996)
S. Compernolle, A. Ceulemans, \(\pi \) Electronic structure of octahedral trivalent cages consisting of hexagons and squares. Phys. Rev. B 71, 205407 (2005)
S. Compernolle, A. Ceulemans, Frontier orbitals of trivalent cages: (3,6) cages and (4,6) cages. J. Phys. Chem. A 109, 2689–2697 (2005)
B.I. Dunlap, R. Taylor, Octahedral \(\text{ C }_{48}\) and uniform strain. J. Phys. Chem. 98, 11018–11019 (1994)
H.-S. Wu, H.-J. Jiao, What is the most stable \(\text{ B }_{24}\text{ N }_{24}\) fullerene? Chem. Phys. Lett. 386, 369–372 (2004)
V. Barone, A. Koller, G.E. Scuseria, Theoretical nitrogen NMR chemical shifts in octahedral boron nitride cages. J. Phys. Chem. A 110, 10844–10847 (2006)
R.A. LaViolette, M.T. Benson, Density functional calculations of hypothetical neutral hollow octahedral molecules with a 48-atom framework: Hydrides and oxides of boron, carbon, nitrogen, aluminum, and silicon. J. Chem. Phys. 112, 9269–9275 (2000)
K.M. Rogers, P.W. Fowler, G. Seifert, Chemical versus steric frustration in boron nitride heterofullerene polyhedra. Chem. Phys. Lett. 332, 43–50 (2000)
R.R. Zope, T. Baruah, M.R. Pederson, B.I. Dunlap, Electronic structure, vibrational stability, infra-red, and Raman spectra of \(\text{ B }_{24}\text{ N }_{24}\) cages. Chem. Phys. Lett. 393, 300–304 (2004)
C. Chuang, Y.-C. Fan, B.-Y. Jin, Generalized classification scheme of toroidal and helical carbon nanotubes. J. Chem. Inf. Model. 49, 361–368 (2009)
C. Chuang, Y.-C. Fan, B.-Y. Jin, Dual Space Approach to the Classification of Toroidal Carbon Nanotubes. J. Chem. Inf. Model. 49, 1679–1686 (2009)
C. Chuang, B.-Y. Jin, Systematics of high-genus fullerenes. J. Chem. Inf. Model. 49, 1664–1668 (2009)
C. Chuang, Y.-C. Fan, B.-Y. Jin, Comments on structural types of toroidal carbon nanotubes. J. Chin. Chem. Soc. 60, 949–954 (2013)
C. Chuang, Y.-C. Fan, B.-Y. Jin, Systematics of Toroidal Helically-coiled Carbon Nanotubes, High-genus Fullerenes, and Other Exotic Graphitic Materials. Procedia. Eng. 14, 2373–2385 (2011)
C. Chuang, B.-Y. Jin, Classification of hypothetical doubly and triply periodic porous graphitic structures by tilings of neck-like units. J. Math. Chem 47, 1077–1084 (2010)
S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry (Springer, New York, 1991)
D. Tománek, Guide Through the Nanocarbon Jungle: Buckyballs, Nanotubes, Graphene and Beyond, IOP Concise Physics series (Morgan & Claypool Publishers, San Rafael, 2014)
K. Delp, B. Thurston, Playing With Surfaces: Spheres, Monkey Pants, and Zippergons. in Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture 2011, pp. 1–8
A. Šiber, Shapes and energies of giant icosahedral fullerenes. Euro. Phys. J. B 53, 395–400 (2006)
E.U. Yong, D.R. Nelson, L. Mahadevan, Elastic platonic shells. Phys. Rev. Lett. 111(177801), 1–5 (2013)
Acknowledgements
The research was supported by the Ministry of Science and Technology, Taiwan. B.-Y. Jin thanks Center for Quantum Science and Engineering, and Center of Theoretical Sciences of National Taiwan University for partial financial supports. We also wish to thank Chern Chuang and Prof. Yuan-Chung Cheng for useful discussions and comments on this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fan, YJ., Jin, BY. From the “Brazuca” ball to octahedral fullerenes: their construction and classification. J Math Chem 55, 873–886 (2017). https://doi.org/10.1007/s10910-016-0719-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-016-0719-3