Trigonometric functions
relating phi ( 
) and pi (p)
Divide a 360° circle into 5 sections of 72° each and
you get the five points of a
pentagon, whose dimensions are all based on phi relationships.
Accordingly, it shouldn't be too
surprising that phi, pi and 5 (a Fibonacci number) can be related through trigonometry:
Or, a much simpler way involving, contributed by Dale Lohr:
Pi = 5 arccos (.5 Phi)
Note: The angle of .5 Phi is 36
degrees, of which there are 10 in a circle or 5 of in pi radians.
Note: Above formulas expressed in radians, not
degrees
Alex Williams, MD, points out that
you can use the Phi and Fives relationship to
express pi as follows:
5arccos((((5^(0.5))*0.5)+0.5)*0.5) = pi
Robert Everest discovered that you can express Phi as a function of Pi and
the numbers 1, 2, 3 and 5 of the Fibonacci series:
Phi = 1 - 2 cos ( 3 Pi / 5)
Pi squared (p2)
and 987
Pi squared (p2)
is 9.8696..., which, if you round to 9.87 and ignore the decimals, is 987, the 17th number of the Fibonacci series.
(Contributed by William Erman.)
More on the relationship of Phi squared
and Pi
If you're looking for other interesting ways to relate
pi and phi, 6/5 *
Phi^2 = 3.1416, which approximates pi. (Contributed by
Steve Lautizar.)
- 