Fibonacci series, Pi, Golden Ratio — Simple Relationships

Fibonacci numbers, the associated Golden Ratio and Pi appear abundantly in all phenomena of nature, from the very small to the very large. In this post, we deliver simple relationships between these three that allow their simple calculation, either exactly (Golden-Ratio and Fibonacci terms) or to high accuracy (Pi).

The start of the Fibonacci series (first seventeen terms) is:

{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …}.

A Fibonacci number is obtained by adding the last two terms preceding it in the series, for example, 55 is the sum of 21 and 34.

As the length of the Fibonacci series increases, the ratio between two consecutive Fibonacci numbers converge to the Golden Ratio. Simple exact expressions to calculate the Golden Ratio (denoted herewith Φ, capital of φ) and its reciprocal (denoted herewith φ, small Φ) are Eq. [1] and Eq. [2] (refer to PDF downloadable file below).

Employing Φ and φ, a simple formula for the k-th term in a Fibonacci series is given in Eq. [3]. Note that F(0)=0.

A formula that combines Fibonacci numbers and the Golden-Ratio (Phi = Φ =1.618…) delivers a compact expression for π (Eq. [4]).

For example, for n=3: F(2n+1)=F(7)=13.

Inserting in this equation the formula for a Fibonacci number in terms of the Golden-Ratio, as given earlier, we finally obtain a formula to calculate Pi in terms of the Golden-Ratio (Φ) and its reciprocal (ϕ) (Eq. [5]).

This formula delivers highly accurate values for π even for relatively small upper summation limit of n.

Below are values of π obtained for different upper summation values:

“Exact” Pi value (π): 3.141592654…

{Upper summation limit, calculated π}:

{{5, 3.141148432}, {6, 3.141739012}, {7, 3.141543509}, {8, 3.141609399}, {9, 3.141586881}, {10, 3.141594663}, {11, 3.141591949}, {12, 3.141592902}, {13, 3.141592565}, {14, 3.141592685}, {15, 3.141592642}}.

We realize that already for upper summation limit of 14 — exact value of Pi to seven decimal points is obtained‼

Playing Pi (π) and Phi (Φ, Golden Ratio) on the piano:

• Song from π! (reproduced 2015; with Sheet Music/HQ Download):

• Song from π! (original, 2011):

• What Phi (Golden Ratio) Sounds Like (reproduced 2012):

• What Phi (Golden Ratio) Sounds Like (original, 2011):

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Partial Source for this post: Castellanos D. Rapidly Converging Expansions with Fibonacci Coefficients 1986; Fibonacci Quarterly 24: 70-82.