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




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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