### An explanation of the “magic” property of the Fibonacci series

#### Alejandro Serrano

A well extended myth confers magic properties on Fibonacci numbers. These numbers make up the so-called Fibonacci series, which starts with two ones and subsequent numbers are calculating by adding the two previous ones. Therefore, the first ten numbers of the series are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. An intriguing property of this series is that the ratio between two consecutive numbers is closer and closer to the so-called golden ratio (the golden ratio is usually denoted by φ and its value is (1 + √ 5)/ 2 = 1.618033989… ) For instance, dividing the 20^{th} by the 19^{th} element of the series yields a number with eight identical digits: 6,765 / 4,181 = 1.61803396…,

The magic nature of Fibonacci numbers stem from this outstanding property, since the golden ratio is described as one of the most mysterious numbers in nature (see Mario Livio’s cite below.)

*“Some of the greatest mathematical minds of all ages, from **Pythagoras** and **Euclid** in **ancient Greece**, through the medieval Italian mathematician **Leonardo of Pisa** and the Renaissance astronomer **Johannes Kepler**, to present-day scientific figures such as Oxford physicist **Roger Penrose**, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. *

(Mario Livio. *The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number*, p.6

It is believed that the Pythagoreans used a 5-pointed star, called a pentagram, as a secret symbol to identify themselves. The pentagram was chosen because it hides the golden ratio everywhere.

The golden ratio is also related to beauty in nature. From sunflowers to shells, the golden ratio is supposed to confer beauty on the objects that contain it. Moreover, several dimensions in human creations are also driven by the golden number. For instance, many of the proportions of the Parthenon in Athens exhibit the golden ratio. A contemporary example is the (golden) ratio between the length and the height of a credit card.

The truth of the matter is that this “magic” property of the Fibonacci numbers can be explained with a bit of math. In fact, the numbers in the Fibonacci series can be calculated using the following expression

For instance, if n=4:

The eigenvalues of the matrix on the right-hand side of this expression turn out to be (1 + √ 5)/ 2 and (1 – √ 5)/ 2 , i.e., φ and -0.61803… respectively. Note that φ > 1 and 0 < -0.61803… < 1. That is why, after a number of iterations, the impact of the second eigenvalue on the vector on the letf-hand side is negligible . Therefore, for n sufficiently large, only the first eigenvalue, φ, is relevant when calculating the following element of the series, and then

which explains the supposed-to-be “magic” nature of the Fibonacci numbers.