Forum Outposts

The Geometry Forum Newsletter

Winter 1995, page 3

Ask Dr. Math!, continued

Look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), and notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero. Again, we would want the limit of this sequence to be 1/0, but looking at the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative infinity. So which would we assign to 1/0? Negative infinity or positive infinity? Instead of just assigning one of these willy nilly, we say that infinity isn't a number, and that 1/0 is indeterminate.

I hope this helps. -- Ken "Dr." Math

Dear Dr. Math:

My name is Erica Anderson and I need to know examples of where the Fibonacci sequence is found in nature and how that relates to the Golden Mean. This is for Pre Calculus class. Thanks for your help.

Hey Erica,

I love this problem. The Fibonacci sequence happens all the time in nature, so much it is amazing. I am having trouble not just sitting here and listing all the occurrences that I can think of, but I will try to resist sending you a seven-page e-mail message. A few examples:

Think about a pine cone. Have you ever noticed that the petals spiral up in two directions? Well, the number of petals it takes to get once around is almost always a Fibonacci number.
Why are four leaf clovers so rare? (Four isn't a Fibonacci number.)
Many flowers also demonstrate Fibonacci sequences--check a few and count their petals.
Other things to explore are the rotations of the pods of pussywillows, sunflower blooms, pineapples...

Now, about the Golden Mean. The Golden Mean is the ratio of successive Fibonacci numbers. The series
1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21 55/34 89/55 144/89
has as its limit the Golden Ratio.

Hope that helps. -- Ethan, Doctor On Call

Dear Erica,

I'll add on, including three of my favorite examples of where the Fibonacci sequence is found:

As Ethan said, pine cones, pineapples, and sunflowers are a few examples of where the numbers of spirals are Fibonacci number--and you can actually count the spirals in two directions, one tighter than the other (three for a pineapple). Now is a great time to get pinecones! (I like to mark my starting spiral with a marker or whiteout so I can keep track of what I have counted.) Pine cones are usually 5 and 8 or 8 and 13, pineapples 13, 21 and 34 (I think), and sun-flowers 34 and 55 or 55 and 89 (also, I think)--try it.

You can also draw a logarithmic spiral like the one in a nautilus shell, where all the angles are equal, based on the Fibonacci sequence. Using graph paper, draw two 1x1 squares next to each other. Draw a 2x2 square next to them and then spiral round so that the 3x3 is touching a 1x1 and the 2x2. The 5x5 shares a side with the 2x2 and 3x3, and it continues on. When you have run out of paper, you can draw in the spiral itself, connecting one corner of each square to the opposite corner with a curve. It make take a try or two before you figure out how the squares line up against one another, but it's worth it--the result is wonderful!

A third favorite example has to do with the Golden Ratio, which Ethan described. Many artists have used the Golden Ratio in their work. If you can, check out the Time/Life Science Library book on Mathematics. It has examples of how the Parthenon fits into a golden rectangle, how DaVinci used these proportions (knowingly or not), and how more modern artists such as Seurat and Mondrian used them quite knowingly.

Have fun!

-- Jane Stavis, Westtown School

If you use a web browser like Mosaic or Netscape, you can read previous questions and responses at
http://olmo.swarthmore.edu/dr-math/dr-math.html

Picture caption:
And you thought the Internet was safe...
Steve Weimar and Annie Fetter, Ask Dr. Math coaches and workshop leaders, shown here at the close of the Forum's 1994 Summer Institute, where one group of teachers presented their final project, "Geometry Under Construction," bringing hardhats and tools as props and persuading Steve and Annie to pose for this picture.

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Sarah Seastone
30 December 1994