KCC Biology Department BioLab 2 Exercise 1

BioLab 2 Exercise 1
Fibonacci Numbers in Biology

Objectives:

To examine the life of the mathematician Leonardo of Pisa, also known as Fibonacci.

To discover what Fibonacci numbers are, and how to use them.

To utilize the Internet to examine and analyze Fibonacci numbers as applied to Biological organisms.

To submit answers in writing to your instructor based on your observations and calculations.

Materials:

A computer.
Internet access through an Internet service provider (ISP), and a browser such as Netscape Navigator.®
A calculator.
Pen and paper.

Background:

Leonardo of Pisa (1180-1250) was better known as Fibonacci, which literally means "son of Bonaccio". Born in Pisa, Italy, Fibonacci was an international merchant, following in his father's footsteps. In addition to Pisa, Leonardo also lived in North Africa and Constantinople and traveled extensively. Among the many places he traveled to are Sicily, Egypt, Syria, Greece, Provence and other Mediterranean locations. Along his travels, he encountered Greek and Roman culture and Arab mathematical ideas. On a trip to Pisa, the emperor Frederick II met Fibonacci and appointed him a member of his imperial entourage.
     Fibonacci authored several memorable books, Liber Quadratorum (1225), Flos (1225), Practica geometriae (1220) and Liber Abaci (1202), which literally means "book of the abacus", a misleading title since it is not about the abacus. Liber abacus is, however, about numbers. Fibonacci is probably best known for the sequence of numbers that bears his name, The Fibonacci Numbers. This sequence arose from a problem that appeared in Liber Abaci. Amir Aczel, in his book, Fermat's Last Theorem, and Carl B. Boyer, in his book, A History of Mathematics, state that the problem is proposed as follows:

     "How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?"

It is from this problem that the Fibonacci sequence was created. In the Fibonacci sequence, each term after the first is the sum of the immediately preceding two terms, as follows:

     1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,...

For example, the third term,2, is the sum of 1 plus 1, the first two terms. Three is the sum of 1 plus 2. Five is the sum of 2 plus 3. Eight is the sum of 3 plus 5, and so on.
This sequence has some interesting properties. For example, any two successive terms are relatively prime. This means that the only common factor each number in the pair has is one. In another example, the ratio of two successive numbers in the sequence approaches the Golden Section, which is:

(sqrt(5)-1)/2
The ratios are:

1/1,1/2,2/3,3/5,5/8,8/13,13/21,21/34,34/55,55/89,89/144,144/233,...
Fibonacci was probably the most original and most capable mathematician of Medieval Christian times. However, much of his work was too advanced for his contemporaries to understand. For more on Fibonacci and pictures of this great mathematician, see the Additional Links below.

Fibonacci Numbers in Biology:

The Fibonacci Sequence appears in many places in nature:

Leaves on the branch of a plant grow at distances from one another corresponding to Fibonacci numbers.
In flowers, the number of petals is either 3,5,8,13,21,34,55 or 89, as the table below illustrates:
     Flower         Number of flower petals
     Lilies                   3
     Buttercups         5
     Delphiniums       8 (often)
     Marigolds         13
     Asters               21
     Daisies             35 (often)
In Sunflowers, the seeds in the head of the flower are arranged in 2 sets of spirals. One spiral winds clockwise; the other winds counter-clockwise. In the clockwise direction, there are often 34 spirals. In the counter-clockwise direction, there are often 55 spirals. The numbers of spirals could also be 55 and 89, respectively. These are all Fibonacci numbers. Further on, we will study a sunflower in detail.
The pattern in the spirals of shells, such as a chambered Nautilus, follow the Fibonacci sequence. We will explore this pattern in an exercise that follows.

References for Background:

Aczel, D. Amir, Fermat's Last Theorem Unlocking the Secret of an Ancient Mathematical Problem, 1996, Dell Publishing, Bantam Doubleday Dell Publishing Group, Inc., New York.
Boyer, Carl B., A History of Mathematics, 1968, John Wiley & Sons, Inc., New York.

Methods:

Bee-line Puzzle
Our First challenge will focus on bee hives. Go to the web site:
Making a bee-line with Fibonacci numbers
Solve the puzzle and submit your answer to your instructor. Then return to this page to continue.
Fibonacci Numbers and Nature
There is a wonderful web site called "Fibonacci Numbers and Nature". In a moment you will go to this web site to try your hand at several biology problems involving Fibonacci numbers. Once you arrive at the web site, try each of the following:
1. Fibonacci's Rabbits
  This is a problem in which you will work through Leonardo's original rabbit problem.
2. Honeybees, Fibonacci numbers and Family trees
  Here you will work on family trees involving Fibonacci numbers.
3. The Fibonacci Rectangles and Shell Spirals
  As stated earlier, the patterns in the spirals of shells involve Fibonacci numbers. This exercise makes the patterns come to life in a vivid animation.
4. Fibonacci Numbers and Branching Plants
  This section depicts how the branching of plants follows a Fibonacci sequence.
5. Petals on Flowers
  Earlier we were introduced to the fact that petals on flowers follow Fibonacci numbers. This section will elaborate on this topic.
6. Seed heads
  Here you will see how seed arrangements follow the Fibonacci sequence.
7. Pine cones
  The spirals in pine cones follow a Fibonacci sequence. This section will show you how.
8. Leaf arrangements
  Leaf arrangements on a plant also follow the Fibonacci sequence. Work through this section to see how.
9. Vegetables and Fruit
  Using different fruits and vegetables, you will seed how they are loaded with Fibonacci numbers.
Work through each of the above exercises. Submit your results to your instructor. Return to this page when you have completed the sections. Now go to Fibonacci Numbers and Nature
Dr. Stavroulakis's Sunflowers
In this exercise, we will make use of the photographs shown below:

These photographs of sunflowers were taken by Dr. Anthea Stavroulakis, Ph.D., of the Department of Biological Sciences at Kingsborough Community College, CUNY, Brooklyn, NY. The detail in the photos clearly shows the arrangement of the seeds. Study the photos carefully. Can you determine the Fibonacci numbers of both the clockwise and counter-clockwise spiral patterns of the seeds? For additional help with this sunflower problem, go to:
The Life and Numbers of Fibonacci
Submit your answers to your instructor.

Additional Links:

Computer and Internet Ethics

Do not provide personal information online, nor strike up conversations with strangers in online chat rooms.
Do not violate copyright laws through illegal duplication of software, or through utilization of the original online work of others without proper citation, or permission,
Keep any and all passwords confidential.
When on the KCC or other Website networks, do not attempt accessing restricted areas which require codes or passwords. In addition to not being the right thing to do, you could be arrested.
When using your home computer, always scan newly downloaded software for viruses utilizing the antiviral programs that are part of your computer's utilities.

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