"Mathematicians Solve The Topological Mystery Behind The “Brazuca” World Cup Football Just in time for the World Cup final"

I just noticed that there is a page from Medium - Everyone’s stories and ideas,
Mathematicians Solve The Topological Mystery Behind The “Brazuca” World Cup Football Just in time for the World Cup final,
which reported the manuscript, From the "Brazuca" ball to Octahedral Fullerenes: Their Construction and Classification, Yuan-Chia Fan and I submitted to the arXiv at the end of last month.

In fact, Yuan-Jia has been working on the general structural rules for this peculiar family of generalized fullerenes with octahedral symmetry for a period of time. When the FIFA World Cup started in the mid June, I noticed quite accidentally that the official soccer ball, the “Brazuca”, for the FIFA World Cup held in Brazil this year has exactly the same octahedral symmetry. Then, we rewrote the draft a little bit and changed the title to its current form. However, the manuscript was first rejected by the Journal of Chemical Information and Modelling because the paper is not of general interest to the readers of the journal. And then it was rejected again by the Journal Physical Chemistry A because the editor thought this paper falls outside the bounds of the journal.

Instead of finding another journal, I decided to put it in the print archive, arXiv, at the end of June. So people might find it interesting during the World Cup season.

The evolution of the official Adidas Soccer Balls since 1970.

In addition to the page on Medium - Everyone’s stories and ideas, I also noticed three more pages which refer to the Yuan-Jia's work on the octahedral fullerenes and their connection to the Brazuca:

• Physics World: Molecular footballs, knotty headphones and a naming contest for exoplanets.
• Brazuca, la scienza a servizio del calcio (Italian)
• Il Brazuca e i ‘misteri topologici’ dietro i palloni dei mondiali (Italian)

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.

Carbon cube

Using the same combination of an octagon together with four pentagons for a face as described in the carbon cuboctahedron, one can make the following bead model of the carbon cube. Each corner of a cube is mimicked by three fused pentagons. In principle, we can use the same strategy to create a whole series of fullerenes with the shape of a truncated cube.

Carbon cuboctahedron

Most of cage-like fullerenes belong to either icosahedral or tetrahedral groups. But if we remove the restriction of using only pentagons and hexagons, we can create cage-like fullerenes with octahedral symmetry (or cubical shape) too. Of course, all P-type TPMSs, which are extended systems, posted before have the same symmetry. But I have never made cage-like fullerenes with cubical shape before. I found a few examples of cage-like fullerenes with cubical shape in an interesting book with the title, Periodic Nanostructures, by M. V. Diudea and C. L. Nagy recently. In this structure, each face contains an octahedron surrounded by four pentagons, which give a topological charge of 2. There are six faces in a cube, so the total topological charge is 12 as required by the Euler theorem. It is also easy to see that the eight vertices of this molecule are covered by flat coronenes. Therefore, the molecule looks like a cuboctahedron (立方八面體).

It is not hard to see that one can grow eight carbon nanotubes along eight vertices of the cube. The result will be a Schoen's I-WP surface I described before. If one inserts six tubes along the six faces, one get a single unit cell of the P-surface. Or one can also terminate the CNTs to get a dendritic fullerene with a cubic-shape core.