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Phi and Pi III - 2014 - 127cm x 78cm. Engraved and painted sandstone. [not for sale]

Phi and Pi III engraved sandstone


This relationship between pi and phi was an accidental undergraduate discovery in 20021.

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.

Phi and Pi I engraved sandstone Phi and Pi II engraved sandstone
Phi and Pi I. 2014. 30cm x 30cm
Engraved and painted sandstone.
Phi and Pi II. 2014. 30cm x 42cm
Engraved and painted sandstone.


The series for pi converges fairly fast. Here are the values of the terms expressed geometrically as line segments:

Phi and Pi III equation

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/Φ2, 1/Φ3, 1/Φ4, 1/Φ5, etc....

Phi and Pi III pentagon 1


The edge lengths involved in the terms of the series for pi are the odd inverse powers - 1/Φ, 1/Φ3, 1/Φ5, 1/Φ7, etc....


Phi and Pi III equation 2

I rather like fractions and phi, but here are other versions, for those of you who don’t:

Phi and Pi III equation 2


1 MILNER, J. 2002. The Dodecahedron, the Icosahedron and the Golden Section. Unpublished paper. Sheffield Hallam University, UK.


© JIM MILNER 2018 • milner44@btinternet.com • 01226 763124