Fibonacci and Phi

by Nikhat Parveen, UGA

An interesting pattern evolves from the ratios of consecutive Fibonacci numbers. The graph below plots the quotients of the first 13 Fibonacci numbers and their antecedents.

Fig 1.

The limit of this sequence is called the golden mean or golden section. Φ = (1 + sqrt (5))/2, approximately 1.618034.

This value is
known as the
Golden
Ratio. Like the
irrational number* Pi*, the digits of* Phi*, go on forever without
repeating.

Since the
decimal continues to infinity, the most accurate way to represent *Phi* (Φ)
is to calculate it algebraically and express it as the root of a quadratic
equation.

We now see
that *Phi* is actually the __positive__ root of the quadratic equation.
The negative root is -*phi*, such that * phi* =
0.6180339887499

__PHI FACTS! __

A fun fact about Φ is that its approximate value is 1.618...and the approximate value of the reciprocal of Φ is .618...The term "Phi" was not used until the 1900s when American mathematician Mark Barr used the Greek letter Φ (Phi) to designate this proportion.

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