This relationship between pi and phi was an accidental undergraduate discovery in 2002^{1}.
I was studying the geometry of the dodecahedron in terms of the golden section and found that the dihedral angle of the dodecahedron was 2ArcTanΦ, where phi (Φ) = ½ + ½√5.
Checking my result with published results, I found out that the dihedral angle was also equal to π – ArcTan2. The infinite series for pi in terms of phi was derived from this relationship.

The series for pi converges fairly fast. Here are the values of the terms expressed geometrically as line segments:
This detail from Pi & Phi III illustrates the orders of magnitude of the inverse powers of phi (Φ) in terms of the side lengths of the pentagons. If the fractal itself has an edge length of 1, then the different generations of pentagons within it have edge lengths of 1/Φ, 1/Φ^{2}, 1/Φ^{3}, 1/Φ^{4}, 1/Φ^{5}, etc....
The edge lengths involved in the terms of the series for pi are the odd inverse powers  1/Φ, 1/Φ^{3}, 1/Φ^{5}, 1/Φ^{7}, etc....
I rather like fractions and phi, but here are other versions, for those of you who don’t:
^{1} MILNER, J. 2002. The Dodecahedron, the Icosahedron and the Golden Section. Unpublished paper. Sheffield Hallam University, UK.
