Fibonacci Numbers

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was how many pairs will there be in one year? When attempting to solve this problem, a pattern is detected:

Figure 1: Recognizing the pattern of the "rabbit problem".

If we were to keep going month by month, the sequence formed would be 1,1,2,3,5,8,13,21 and so on. From here we notice that each new term is the sum of the previous two terms. The set of numbers is defined as the Fibonacci sequence. Mathematically speaking, this sequence is represented as:

The Fibonacci sequence has a plethora of applications in art and in nature. One frequent finding in nature involves the use of an even more powerful result of the Fibonacci sequence: phi and the golden ratio. The following is an example of what I will later discuss: the golden spiral.

Figure 2: The arrangement of the whorls on a pine cone follows a sequence of Fibonacci numbers.

The following example is just one of the numerous examples of the fascination applications found within the Fibonacci sequence in nature. Now, we turn to one of the most fundamental concepts of the Fibonacci sequence: the golden ratio.

Consider the ratio of the Fibonacci numbers (1,1,2,3,5,8,...)

As, the sequence progresses, we notice that the sequence seems to converge and approach a number. The question is what exactly is that number?

Answer:

One of the most interesting and frequent applications of phi is that of the golden rectangle. The golden rectangle is created in a way such that the area of the rectangle is phi (meaning that the length/width is one and the length/width is phi). Though ancient Greeks were unaware of the Fibonacci sequence, they were much aware of the golden rectangle. As a matter of fact, the golden rectangle is widely seen in many Greek architecture and art. The following rectangle is considered to be the most pleasing to the eye:

Figure 3: A Golden Rectangle

Another interesting application which uses both Fibonacci numbers and phi is the golden spiral. The spiral itself is a