If you open the icosahedron, you can easily see that the shortest line between the geometrical centres of two opposite faces is the straight line. This is the geodesic line, which you would obtain in absence of friction if you stretched between the two points an elastic band. If you establish the Cartesian axes as shown in the figure, the coordinates of the two points are (0, 1/3) and (5/2, 2/3).

Therefore, the solution is given by:

d = sqrt((5/2)

^{ 2}+ (1/3)^{2}) = sqrt(226) / 6 = 2.505549396395485.
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