Jabulani Polyhedron

"Jabulani polyhedron" is a term introduced here to refer to the polyhedron illustrated above which underlies the shape of the soccer ball used in the 2010 World Cup in South Africa. It can be constructed by truncation of a truncated tetrahedron by cutting the corners of its hexagons with oblique planes to form regular nonagons and the corners of its triangles to form regular hexagons (Kuchel 2012). It is implemented in the Wolfram Language as PolyhedronData["JabulaniPolyhedron"].

Here, 'Jabulani' is the name given to the ball by its manufacturer, Adidas, which means "celebrate' in the Zulu language. The ball was featured on the cover of Volume 96, Issue 536 of The Mathematical Gazette, accompanying the analysis of its underlying polyhedron by Kuchel (2012).

The Jabulani polyhedron has 36 vertices, 20 faces (consisting of 12 trapezoids, 4 regular hexagons, and 4 regular nonagons), and 54 edges (6 of which are short and 48 of which are long, with length ratio ). Here, the ratio of long to short edge length is given by

			(1)
			(2)

As seen in the net and further demonstrated above, the trapezoids are connected in pairs along their short edges, which provide the 6 short edges of the solid. If the short edges of the trapezoids are taken to have unity edge lengths, the trapezoids have base and side lengths and base angle

			(3)
			(4)

(Kuchel 2012).

The skeleton of the Jabulani polyhedron is a cubic Hamiltonian graph illustrated above in a number of embeddings. It is implemented in the Wolfram Language as GraphData["JabulaniSkeleton"].