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This creates a shape made of 20 hexagons with 12 pentagons at the vertices of the icosahedron.In other words, this cut-and-paste procedure produces C60, the molecular analogue of the Telstar.Yuan-Jia and Bih-Yaw try several showing that a remarkable variety of shapes with octahedral symmetry can be formed in this way.However, the climax of their paper comes when they finally discover a shape that has both octahedral symmetry and physically feasible bonds for a fullerene.

But this World Cup was also significant for topological reasons.It turns out that Adidas has addressed this shortcoming with its current World Cup ball: the Brazuca now being used in Brazil.The Adidas Brazuca is made from six panels each with a four-leaf clover shape that knit together like a jigsaw to form a sphere.An important part of this problem is that pentagons cannot be adjacent in fullerenes so anything that forms this kind of structure is ruled out.So for example, Yuan-Jia and Bih-Yaw construct a fullerene with icosahedral symmetry by cutting 20 equilateral triangles out of graphene and then pasting them onto the triangular faces of an icosahedron.

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The adjacent carbon hexagons then knit together to form a polygon with the same symmetry.

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