Pi, Phi and Fibonacci Numbers

Phi ( Φ ) and pi (p) and Fibonacci numbers can be related in several ways:

You might be surprised to learn that the two most famous numbers in the history of mathematics, phi ( Φ ) and pi (p) and are exactly related to each other by a rational fraction, even though both are irrational numbers.  Roger Logan, in his paper entitled THE MAGNIFICENT PERFECT SQUARE © 2001, introduces the concept that both Φ2 and p are composite numbers sharing a common irrational factor.  When put in the form of a fraction,  Φ2/p, this common irrational factor cancels out, leaving a rational fraction comprised of two perfect squares (1, 2, 4, 9, 16, ...).

The composite number when substituted for the Greek letter is shown to satisfy and clarify Euler's famous equation giving proof that and are truly composite numbers by reason that things equal to the same thing are equal to each other. Eulers equation is not only satisfied, but it is clarified as to the relationship of the numbers e, pi, i, and -1.

Mr.. Logan's paper is intentionally expository and shows the mean proportional relationship between and is easily expressed by his " structure", using right triangles, the Pythagorean theorem, the laws governing mean proportionals, and high school - level mathematics (calculus is not required). In addition to the math, Mr. Logan provides the reader with a narrative giving a brief review of the Number System and short history about the number Pi along with an interactive Glossary as an aid for the reader.

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:

 

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.

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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

 

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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.