posted by @ulaulaman about #Brazuca #geometry #WorldCup2014 #Brazil2014
Brazuca is the ball of the World Cup 2014. The particular pattern of its surface is a consequence of the Pogorelov's theorem about convex polyhedron:
A domain is convex if the segment joining any two of its points is completely contained within the field.
Now consider two convex domains in the plane whose boundaries are the same length.(1)
Now we can create a solid using the two previous domains: we must simlply connect every point of one boundary with a point of the other boundary, obtaining a convex polyhedron, like showed by Pogorelov in 1970s.
The object you have built consists of two developable surfaces glued together on edge.
Instead of using two domains, you can, for example, start from six convex domains as the "square faces" of a cube. On the edges of each of these areas, you choose four points, as the vertices of the "square". We assume that the four "corners" that you have chosen are like the vertices, that is to say that the domains have angles in these points.(1)