Pi, Phi and Fibonacci
Numbers
Phi ( Φ ) and pi (p)
and Fibonacci numbers can be related in several ways:
 The
Pi-Phi Product and its derivation through limits
The product of phi and pi,
1.618033988... X
3.141592654..., or 5.083203692,
is found in golden geometries:
|
Golden Circle
|
Golden Ellipse
|
 |
 |
Circumference =
p
Φ
|
Area =
p
Φ
|
Ed Oberg and Jay A. Johnson have developed a unique
expression for the pi-phi product (p
Φ)
as a function of the number
2 and an expression they call "The Biwabik Sum,"
a function of phi, the set of all
odd numbers and the set
of all Fibonacci numbers, as follows:
| p
Φ
=
22
{1 |
+ [ (2/3)
/ (F1+F2Φ)
+ (1/5) /
(F3+F4Φ)
- (1/7) /
(F5+F6Φ)
] |
| |
- [
(2/9) /
(F7+F8Φ)
+ (1/11) /
(F9+F10Φ)
- (1/13) /
(F11+F12Φ)
] |
| |
+ [ (2/15)
/ (F13+F14Φ)
+ (1/17) /
(F15+F16Φ)
- (1/19) /
(F17+F18Φ)
] |
| |
- … } |
| |
= 5.083203692.... |
 |
This
relationship was derived after Oberg noticed an interesting
relationship between pi and phi while contemplating geometric questions
related to the location of the King and Queen’s burial chambers in the Great
Pyramid, Cheops, of Giza,
Egypt, the design of which is based on phi.You can
access the complete paper published by
Ed Oberg and Jay A. Johnson,
The Pi-Phi Product, in Word, or the Pi-Phi Product
in Excel to see their formulation illustrated numerically. |
Trigonometric functions
relating phi ( Φ ) 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:
Note: Above formulas expressed in radians, not
degrees
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.)
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