Natures Poetry: The Fibonacci Sequence

March 30th, 2014

sunflower

Math has a reputation for being abstract, and therefore, irrelevant to most peoples everyday experience. While math certainly can be appreciated purely for its beauty, it can also be used to describe the physical universe in exquisitely accurate detail. Math could perhaps be thought of as natures poetry. One great example of this is the Fibonacci sequence.

The Fibonacci Sequence

You may be familiar with the following number sequence (a.k.a. the Fibonacci series).

  0, 1, 1, 2, 3, 5, 8, 13, 21, 34

Each number in the sequence except for the first two, 0 and 1, is the sum of the two numbers preceding it.

For example:

    0 + 1 = 1,

    1 + 1 = 2,

    1 + 2 = 3,

    2 + 3 = 5,

    3 + 5 = 8,

    5 + 8 = 13,

and so on.

This famous sequence is named after the Italian mathematician, Leonardo Pisano Bogollo (1170-1250). He is more commonly known by his nickname, Fibonacci. In addition to studying the famous number sequence, he is also known for popularizing the modern decimal system (digits 0 through 9) over the cumbersome Roman numeral system.

Examples in Nature

Fibonacci numbers are apparent in the number of leaves, seeds, and petals of many plants (e.g. sunflower seeds, artichoke leaves). The phenomenon is easy to see in pinecones. Simply count the number of vertical rows of seeds starting at the base and ending at the top of the cone. Whether you count clockwise or counterclockwise spirals, the sum of rows will (almost always) be a Fibonacci number. You can do the same thing for seedpods with more seeds (e.g. sunflower center). Count the number of spiraling rows and you will get a Fibonacci number.

Pincone Spirals

Pseudo Example

Before writing this post, I assumed that the nautilus shell was a perfect example of the Fibonacci shell in nature due to its similar appearance to the golden spiral (see A Spiral-Shaped Progression). Rather than make assumptions, I decided to test my hypothesis.

nautilus_shell_spiral

While trying to superimpose the image of a golden spiral over a picture of a nautilus shell, I noticed that the spiral of the shell did not match up with the golden spiral. The shell’s spiral wouldn’t fit no matter how I scaled or rotated the golden rectangle diagram. By doing more research online, I learned the nautilus spiral follows a different logarithmic spiral than the golden spiral.

A Spiral-Shaped Progression

The golden spiral diagram shows how the Fibonacci sequence is related to the golden ratio, a special number that describes aesthetically pleasing proportions. Hence, picture frames and TV monitors often exhibit the golden ratio (i.e. when the length-to-width ratio is about 1.6).

Golden Spiral

In the image above, notice that each consecutive rectangle edge is growing in a Fibonacci pattern, starting from the center square in the middle of the spiral and moving outward in a counterclockwise direction. With each turn in the spiral, a new rectangle is created. Interestingly, as these rectangles get bigger, their proportions more closely approximate the golden ratio. We can observe this approximation with the first nine terms in the sequence.

    Rectangle 11:     length/width = 1/1 = 1

    Rectangle 12:     length/width = 2/1 = 2

    Rectangle 23:     length/width = 3/2 = 1.5

    Rectangle 35:     length/width = 5/3 = 1.6666

    Rectangle 58:     length/width = 8/5 = 1.6

    Rectangle 813:     length/width = 13/8 = 1.625

    Rectangle 1321:     length/width = 21/13 = 1.61538

It gets even more dramatic with larger length and width values.

    Rectangle 14489:     length/width = 144/89 = 1.61797

    Rectangle 41816765:     length/width = 6765/4181 = 1.61803

So as the number of Fibonacci numbers approaches infinity, the length-to-width ratios of the rectangles approach the golden ratio, or phi, which is (1 + sqrt(5))/2.

Defining the Fibonacci Sequence

Notice that the first two Fibonacci numbers, 0 and 1, are not the sum of two distinct nonnegative integers. Therefore, they are considered to be the fundamental building blocks for all other Fibonacci numbers. Below is a definition for obtaining all the Fibonacci numbers, starting with 0. (This definition is easily translatable into computer code if you want to make a program for creating Fibonacci numbers).

Let N be a positive integer. Then the Nth term in the Fibonacci sequence can be calculated as follows:

    i. Case N = 1: Nth term = 1st (first) term = 0

    ii. Case N = 2: Nth term = 2nd (second) term = 1

    iii. Case N > 1: Nth term = (N-1)th term + (N-2)th term

Putting it in Code

Below is a recursive function for computing the nth term of the Fibonacci sequence, using the C++ programming language.

// Computes the nth term in the fibonacci sequence
int fibonacci(int n)
{
if (n <= 1) return 0;
else if (n == 2) return 1;
else return fibonacci(n-1) + fibonacci(n-2);
}

Here is an iterative version of the function above.

// Computes the nth term in the fibonacci sequence
int fibonacci_2(int n)
{
    if (n <= 1) return 0;
    else if (n == 2) return 1;
    else
    {
        int sum = 0, term_A = 1, term_B = 0;
        for (int i = 2; i <= n; i++)
        {
            sum = term_A + term_B;
            term_A = term_B;
            term_B = sum;
        }
        return sum;
    }
}

Sources:


Categories: mathsciencetutorials

Previous Post: