A soccer ball (also known as a "truncated icosahedron") is equivalent to a (1,1) icosadeltahedron. To see this, draw the "dual" set of vertices, edges, and faces (new vertex in the center of each old face and new edges connecting new vertices associated to old faces that shared an old edge). The links in purple show a (1,1) path between five-fold vertices (in blue).

Some other figures used in class: