Pi & Fibonacci NumbersPi Logo

Recall that the Fibonacci sequence is defined by F(1) = 1, F(2) = 1, F(3) = 2, F(n) + F(n+1) = F(n+2). The following relation involving the Fibonacci numbers was proven by Ko Hayashi [3].

(*) arctan(1/F(2i)) = arctan(1/F(2i+1)) + arctan(1/F(2i+2))

The connection to pi is that arctan(1) = pi/4. Thus pi can be expressed in terms of Fibonacci numbers

The first three cases have been demonstrated geometrically using the Geometer's Sketchpad. (These are not really interactive sketches.)

Geometry of arctan(1) = arctan (1/2) + arctan(1/3)

Geometry of arctan(1/3) = arctan(1/5) + arctan(1/8)

Geometry of arctan(1/3) = arctan(1/5) + arctan(1/8)

Geometry of arctan(1/8) = arctan(1/13) + arctan(1/21)

Geometry of arctan(1/8) = arctan(1/13) + arctan(1/21)

One can represent pi as the sum of an arbitrary number of terms involving Fibonacci numbers by continuing in this manner. The repeated application of equation (*) yields

pi = 4*arctan(1/F(2n+2)) + 4*SUM{i=1...n}[arctan(1/F(2i+1))] , for any natural number n.

Each sum starting with n =1, is an exact representation of pi. One can look at four times the summation to be a partial sum with

pi = 4*arctan(1/F(2n+2))

as the error term for pi. The sequence of these partial sums converges to pi also. I.e.,

pi = 4*SUM{i=1...infinity}[arctan(1/F(2i+1))].

Using the formula for the tangent of the sum of two angles, these relationships can be easily verified.

Return to Historical Overview of Pi

Go to A Slice of Pi Home Page


Pi Logo

Created: March 1996 ---- Last Modified: July 6, 1997

Copyright © 1996-1997 Michael D. Huberty, Ko Hayashi & Chia Vang