The Brazuca and octahedral fullerenes

Yuan-Jia and I submitted a manuscipt entitled, From the "Brazuca" ball to Octahedral Fullerenes: Their Construction and Classification, to the physics preprint archive, arXiv, recently. Basically, we observed that the symmetry of the Brazuca ball used in the FIFA World Cup held now in Brazil is exactly that of an octahedral fullerene, which Yuan-Jia has been thinking about for more than a year. Therefore, we modified the original manuscript a little bit to point out this peculiar connection. Hopefully, researchers can know that not only the original the Adidas Telstar, a truncated icosahedron, has a microscopic analog, namely the famous C₆₀, the current Brazuca ball could also have microscopic correspondence, in principle.

The Adidas Telstar vs a C₆₀ molecule

The Adidas Brazuca vs an octahedral fullerene

Here I give a list of symmetry groups for all official match footballs (soccer balls) adopted by the FIFA since 1970:

Year	Country	Name of the official match ball	Point group
1970	Mexico	The Adidas Telstar	Ih
1974	West Germany	The Adidas Telstar Durlast	Ih
1978	Argentina	The Adidas Tango Durlast	Ih
1982	Spain	The Adidas Tango Espana	Ih
1986	Mexico	The Adidas Azteca	Ih
1990	Italy	The Adidas Etrvsco	Ih
1994	USA	The Adidas Questra	Ih
1998	France	The Adidas Tricolore	Ih
2002	Korea Japan	The Adidas Fevernova	Ih with T pattern
2006	Germany	The Adidas Teamgeist	Th
2010	South Africa	The Adidas Jabulani	Td
2014	Brazil	The Adidas Brazuca	O

It is interesting to note that it took FIFA 28 years to move away from the icosahedral symmetry to the tetrahedral symmetry, and then another 12 years to come to the last of three Platonic symmetry groups, namely the octahedral group. Another peculiar difference between the Brazuca ball and the previous soccer balls is the lack of inversion and mirror symmetries in the Brazuca ball, meaning that the Brazuca ball is chiral. This should lead to nonvanishing coupling between translational and rotational (spinning) motions, I suspect.