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!

Queen Anne's Lace and Other Pretty Things

It seems like we went from winter to summer in the blink of an eye. We have had a stretch of weather with high, high humidity and temperatures in the high 80's for about a week, and have found ourselves in 90+ degree weather for the holiday weekend. But gardening still calls!

I came upon some free vegetable plants. I mean lots of them. My husband saw them all and inquired, "Where ARE you gonna put all those?" To which I replied, "Everywhere." And we started today!

I always have loved the gardens where roses mingled with some pole beans, and nasturtiums rambled between pepper plants. So I plopped some tomatoes in between the knock-out roses (you know, the ones my son and I sat in a few weeks ago), and found some "extra" space in front of the rose bed that became part of the flower bed "accidentally." Here's how you  do that:

 Yes, I simply moved the brick border out and filled in the gap with soil and woodchips. My husband didn't notice the first time I did it, but he caught me this time and accused me of taking over the whole yard. I said, "But look how pretty the garden is," and he concurred. You can follow along with our gardening adventures by checking our online family journal, The Urban Farmer.

I bought a couple of yellow trumpet creepers and planted them along a trellis where I currently hang my bird feeders. It's really a bad place for them, traffic wise -- I hit my head exactly three times on them while I worked in the yard today. Malik thought that was funny, and I guess it really was, by the third time. But it's the best place to hang it for viewing from the back of the house. So I practice ducking. Except today.

Part of one flower bed has become invaded by Queen Anne's Lace. I let it grow there, because I just love them, and they remind me of the banks of them growing along the rail bed near my home, growing up. So I let them tumble into the lawn, and my husband weed whacks around them. When they get too unruly, I root some out. But I don't get too hung up on them growing in the garden.

We began the Queen Anne's Lace Year-long study last week, fighting an incredible batch of mosquitoes until we were forced to take a sample in to study inside (it makes a pretty centerpiece, too).  Here are our sketches.

Malik was particularly fascinated by the arrangement of the leaves and leaflets of the plant, since we are studying fractals (repeating patterns in nature -- see my new article on Fractals in Nature for more information and photos). He noticed that the leaflets looked like little versions of the whole leaf, so they must be fractals, and said, "See, I really AM learning stuff in homeschool!"

As the school year winds down, check out these updates to two of my previous articles:

"Engaging Books for Boys" is a list of book series that covers boys from grades K through 6, with links to other books that will keep boys reading all summer. Check this out in my Literacy 101 article.

I also added some additional activities and links to the "Sandy Beach" excursion in my Road Trip! 2010-2011 article -- Aloha!

Remember: We live in the land of the free BECAUSE of the brave. Have a safe Memorial Day weekend.