The Fibonacci Sequence and Nature.

We all have those students.  You know, the ones who spend the period recreating the latest Manga figure, or want to recreate a picture they saw on the internet.  It doesn't matter what class they are in, they'd rather draw.

I have some who are extremely talented at recreating artwork but have no real interest in the regular academic subjects.  So how do we interest them in math when they only see the theoretical equations and not applications.

In fact, we read that nature is a real life application of the Fibonacci sequence but how does it get from a formula to the flowers.  If you remember, the Fibonacci sequence begins with 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc.  I understand the sequence mathematically but I do not understand how it translates to nature.

According to a Science article, sunflowers show certain parts of the Fibonacci sequence beautifully for either 34 and 55, or 55 and 89, or 89 and 144 depending on the size of the sunflowers.  These numbers represent the spirals of seeds from the center to the outer edge.  It doesn't matter if you go clockwise or counterclockwise, those are the two numbers which repeat.

For the past four years, the Museum of Science and Industry in Manchester, England has had members of the public growing sunflowers, taking pictures, counting seed spirals, and reporting back their results. After checking the results of over 650 flowers, scientists have discovered that one in five flowers produced either a non-Fibonacci sequence or one that is more complex than the usual.

Now for the actual mathematics of sunflower heads and other flowers.  The spiral is produced by a slight fractional turn. For instance if you choose a 90 degree turn it is 1/4th of a circle but in reality the seeds are placed in irrational fractions such as 2/3, 3/5, 5/8, 8/13, 13/21, etc.  Notice how the digits in these fractions relate to the Fibonacci sequence.  A relationship of how the Fibonacci sequence relates to the creations of sunflower spirals and other spirals in nature.

According to another article I read, the number of petals a flower contains is one from the Fibonacci sequence or if the flower is more complex, you can count spiraling petals either counterclockwise or clockwise and still discover numbers from the sequence.  In addition, certain flowers have a specific number of petals. This article shows how to count the spiraling pedals and has several pictures for students to practice on.

This article explains more about the number of petals and their placement within the flowers. For instance, the 8 petal rose has a center, three petals around that and five which surround that.  Each set of added petals are added in a new level.  As far as real flowers go, the marigold has 13 petals which the daisy has 21, 34, 55, or 89 petals arranged in levels.

This phenomenon is not just seen in flowers.  It can be seen in pine cones, the vegetable Romanesco, pineapples and so many more things.

Now how do you involve your artists?  Let them draw flowers any way they want and then compare their drawings with the real life photos to see if their's looks as "right" as those in the photos.  Introduce the Fibonacci Sequence to them mathematically, then discuss it with pictures and activities letting them see how prevalent it is in nature.  Let them redraw their flowers so they are done with the correct sequences to look more  correct.

Let me know what you think.  I'd love to hear.