Monday, June 6, 2011

The Mathematics of Nature: Fractals

Our garden is beginning to bust at the seams. The roses are wonderful, as they always are, and the peonies are gargantuan this year. Coreopsis and asters, honeysuckle and coralbells are all in bloom, and so many things are right behind them: hydrangea, lamb's-ear, pincushion flower,  to name a few. Even the eggplants and potatoes don't want to miss out on the action, with their pretty flowers.



We have been examining the geometry of living things for our math studies. And we wanted to share a fun math exploration that you can do, right in your own backyard. And we owe it all to Leonardo di Pisa.


Leonardo di Pisa lived in Italy in the 12th century. His father's name was Bonacci, so he was nicknamed, "Fibonacci," which means "Son of Bonacci." He was a mathematician who was fascinated with patterns in numbers, especially as they could be observed in the natural world. He discovered that there was a special set of whole numbers which described just about every living thing in the world. This set of numbers has been called the "Fibonacci Series."

The Fibonacci Series is easy to calculate: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc., with each new number in the series being the sum of the last two. We stopped at 89 -- what do you think the next 4 numbers in the series would be?

Fibonacci discovered that, if you closely examine things in nature, most things existed in these numbers. To test this out, we dissected some of our flowers, to look for Fibonacci numbers in our garden.

First, we started with something simple: Yellow wood-sorrel (Oxalis stricta) (lots of people mistake it for clover, but it has yellow flowers, and tastes sour like lemons if you nibble a leaf). When we dissected the leaves into leaflets, we found these numbers in the Fibonacci series:

  • 3 leaflets per leaf
  • 2 lobes on each leaflet
I wonder what we would have discovered if we had counted the number of leaves on the whole plant. Do you think it would have been a Fibonacci number?

Does this work with more complicated flowers? Well, we next tried a flower from Lanceleaf tickseed (Coreopsis lanceolata). Being a member of the Composite (daisy) family of plants, the "flowers" are really clusters of many flowers. The "petals" are ray flowers, whose real petals are fused together; in the center are the disk flowers, making the "eye" of the daisy flower head. So there was a lot to look at!


We found lots of Fibonacci numbers here:
  • 8 ray flowers ("petals") per flower head
  • 3 points on each ray flower
  • 8 sepals in each calyx (i.e., the green bracts behind the flower head)
We didn't count the disk flowers -- what do you think we would have found out?

Feeling very confident, we began to eye those roses that I told you about previously. That's a lot of counting to do...








Imagine our surprise, to find that there really weren't as many petals on a rose as it appears ...
  • 21 petals
  • 5 sepals in the calyx 
You can find Fibonacci numbers in things besides flowers and plant parts. We decided to look for other living things and explore them for these special numbers...



Sea scallops:
  • 2 shells on a scallop
  • 8 ridges on each shell
  • 2 "wings" on the "hinge"
  • 1 muscle hinging the two shells together



 
Our dog, Lucky, was watching this process, so we decided to use him as a test subject, too:

  • 2 each (eyes, ears, nostrils, front legs, back legs)
  • 1 tails
He got bored with the activity after awhile, and went off to chew on his Nylabone...
Where is the "3"? What about "5?"  Leonardo di Vinci did a study of the human body (The Vitruvian Man), and found all kinds of Fibonacci numbers:

  • There are 3 parts of each arm (hand, lower arm, upper arm)
  • There are 3 parts of each leg (foot, lower leg, upper leg)
  • There are 5 "points" to the body (head, arms, legs)
  • These also are in a special proportion, called the Golden Ratio (we'll talk about that later).
Questions for YOU!
  1. What natural objects are in your yard that you can explore for Fibonacci numbers?
  2. Why do you think so many natural objects exist in these numbers?
  3. How high can you extend the Fibonacci series? Does it ever end? Why do you say this?
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As we slide into June, here are some resources for keeping reading alive and thriving, in the dog days of summer:

-Why not enroll your kids in a summer reading program? Check out "Summer Means... Summer Reading Programs" for links to 10 of the bigger programs available nationwide.


-For more ideas to use as follow up activities to summer reading, check out "Keep Kids Reading All Summer Long," by Scholastic.
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To more activities for math and art, using number patterns in nature, check out my article, "Fractals in Nature: Where Geometry Meets Nature Study" -- I've added new links and activities!

Thinking of doing some online summer activities with your kids? Check out CurrClick for free and for fee online courses and downloadable resources -- a ton of stuff is available!