Fibonacci Everywhere?

Handsome Fibonacci?

If you were even taught about the Fibonacci number sequence in school, you probably don’t remember much about it. As with other higher levels of math, many aren’t sure how Fibonacci could possibly be relevant to their real lives so why should they even attempt to remember him or his sequence?  In reality, Fibonacci numbers are something you come across practically every day.  Even so, let’s go back and start at the beginning.

The Fibonacci number sequence is named after Leonardo of Pisa (1175-1240), who was known as Fibonacci. (I love to say that name because it sounds like I know a foreign language.)  In mathematics, Fibonacci numbers are this sequence of numbers:

As you can see, it is a pattern, (all math is based on patterns).  Can you figure out the number that follows 89?  Okay, let's pretend I waited for at least 60 seconds before giving you the answer….144.  By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.  For those who are still having difficulty (like my daughter who is sitting here), it is like this.

  

The next number is found by adding up the two numbers that precede it.

F The 8 is found by adding the two numbers before it (3 + 5)

F Similarly, the 13 is found by adding the two numbers before it (5 + 8),

F And the 21 is (8 + 13), and so on!

It is that simple!   For those who just love patterns, here is a longer list:

 

Can you figure out the next few numbers?

The Fibonacci sequence can be written as a "Rule “which is:   x_n = x_n-1 + x_n-2   The terms are numbered from 0 forwards as seen in the chart below.  x_n is the term number n.   x_n-1  is the previous term (n-1) and x_n-2   is the term before that (n-2)



Sometimes scientists and mathematicians enjoy studying patterns and relationships because they are interesting, but frequently it's because they help to solve practical problems. Number patterns are regularly studied in connection to the world we live in so we can better understand it. As mathematical connections are uncovered, math ideas are developed to help us be aware of the relationship between math and the natural world.

As stated previously, we come across Fibonacci numbers almost every day in real life.  For instance, many numbers in the Fibonacci sequence can be linked to ordinary things we see around us such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the frutlets of a pineapple. In addition, numerous claims of Fibonacci numbers are found in common sources such as the spirals of shells or the curve of waves. 

Fibonacci numbers can also be seen in the arrangement of seeds on sunflower heads, the state flower of Kansas (where I live). If you look at the seed arrangement in the center, you will observe what looks like spiral patterns curving left and right or clockwise and counter clockwise. Incredibly, if you count these spirals, the total will be a Fibonacci number. Divide the sunflower spirals into those pointed left and right, and you'll get two consecutive Fibonacci numbers. ­

Many other plants in nature also illustrate this sequence.  For instance, buttercups have 5 petals; lilies have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 while daisies can be found with 34 or 55 or even 89 petals.

Pine cones clearly show the Fibonacci Spirals. On the right is a picture of an ordinary pine cone seen from its base where the stalk attaches to the tree. Can you see the two sets of spirals going left and right? How many are in each set?

 Here are two questions to think about:  

1)      How might knowing this number pattern be useful?

2)      What kinds of things can the numbers in the Fibonacci sequence represent?

I want to close this discussion with a cartoon.  It is written by Bill Amend for his cartoon strip Fox Trot which appeared in the newspaper on February 8, 2009.  Just think!  Now that you know something about Fibonacci numbers, you can understand the humor in the cartoon.

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