Fibonacci in Nature

This flower has 13 petals.
13 is a Fibonacci number!
Photo credit:
www.constructingtheuniverse.com

The Fibonacci sequence is interesting from a math standpoint...if you are into math. But if the only place to find Fibonacci was in a math class, it wouldn't be nearly as interesting as it inevitably is. So let's take a walk outside and see what's out there.

This trillium flower has three petals.
Photo credit: britton.disted.camosun.bc.ca

Where I live, it's still winter. Cold, dark, no snow surprisingly, but definitely no spring flowers yet. Still, it won't be long before our flowers are blooming in full color and I wanted to show you some examples of what I like to call 'Fibonacci flowers.' 

There are a variety of places you can see Fibonacci numbers in flowers. The simplest is in the number of petals on a bloom. Starting with the number three, you can find countless examples of flowers with three, five, eight, thirteen, and even twenty-one petals, as you will see in the photos that follow.

This flower has five petals.
Photo credit: www.bio.tamu.edu

This beautiful pink clematis has eight petals.
Photo credit: alexorbit.com

This star daisy has thirteen petals.
Photo credit: murderousmaths.com.uk

This daisy has twenty-one petals.
Photo credit: britton.disted.camosun.bc.ca

So does this one!
Photo credit: psophis.blogspot.com

Why these numbers? No one knows. It's just one of those oddities of nature that SO many flowers bloom with a Fibonacci number of petals, and not four, or seven, or twelve, for example. In all fairness, there are certainly flowers with petals that number something other than a Fibonacci number. But as winter gives way to spring, take a look around and see what you find. You just might be surprised to find Fibonacci numbers all around you!

Strange and Wonderful

Behold, the beginning of the Fibonacci sequence:

n        1     2     3     4     5     6     7     8     9     10     11     12     13     14     15 ...

F(n)   1     1     2     3     5     8    13    21    34   55    89    144    233   377   610 ...

Photo credit:
www.theoremoftheday.org

Now, here's something you may not know. Pick any two consecutive Fibonacci numbers.... let's take numbers 5 and 6. You will find the fifth Fibonacci number listed as F(5), and it's actually the number five! F(6) is 8.

Here's where it gets interesting. If you square both numbers, you will see that 5² = 25, and 8² = 64. Now add those up and you get 25 + 64 = 89. This is another Fibonacci number! But wait- there's more. WHICH Fibonacci number is it? It's the 11th....as in... 5+6 = 11. Hmm...let's try that one again.

Let's pick two consecutive Fibonacci numbers. We'll pick Fibonacci number 7 (13) and Fibonacci number 8 (21). Now, square those numbers. 13² = 169, and 21² = 441. Add those together (169 + 441) and you get ... 610 - another Fibonacci number! And which one is it? Remember, we chose F(7) and F(8).... 7 + 8 = 15, and the 15th Fibonacci number is.... 610!

What does this look like mathematically? Well, perhaps it's more than you came here for, but here goes:

F(n)² + F(n+1)² = F(n + [n+1])

When you square and then sum consecutive Fibonacci numbers, the answer you get is the Fibonacci number at the location that is the sum of the two original locations.

Whew!

Wanna see if you 'get' it? The 29th Fibonacci number is 514,229.

A Most Unusual Finding in Fibonacci - Video (9:48)

As much as I wish this was my own work, I can't claim to have even known about this phenomenon until I watched this video. If you are really into Fibonacci by now, then this is the video for you to watch! But if you consider yourself more of a casual fan of the sequence, this is still going to blow you away.

This video was produced in the UK, and is one of a vast collection of interesting mathematical tidbits. One click and you might be hooked. This one stands out to me because it talks about the Fibonacci sequence itself, and some of the hidden wonders inside it. We will spend a great deal of time in the coming weeks and months exploring places where Fibonacci numbers can be found, but I thought I would take a brief side-trip and let you look at Fibonacci numbers that lie...inside the Fibonacci sequence.

I would love to hear your thoughts or comments about this video. Maybe it will spark something in you. Maybe it will be too much for you at this part of your journey, and that's alright. But at the very least, I hope you find it entertaining and informative. And if it causes you to seek these folks' other videos, be sure to watch the one on McDonald's and the impossible quest to purchase exactly 43 Chicken McNuggets.

How Honeybees Grow

Photo credit:
http://saadine117.blogspot.com

Bees. To some they represent the very worst part of any summer day. Me? I'm quite allergic to bee stings, but I no longer discriminate between the different types of bees in my life. I run from them all.

But let's give bees credit. They give us honey, which is delicious. But more than that, as I understand, bees are the key to transferring pollen between flowers. And in pollinating plants and flowers, they represent a key piece in our country's agricultural puzzle. They can also teach us about the Fibonacci sequence.

You see, honey bees reproduce according to an unusual set of constraints. When the queen bee mates with a male, their offspring is a female. These bees are typically known as 'worker bees'. But what makes them interesting is that they always have exactly two parents. Interesting? Don't all animals have two parents? NO!

When a queen bee lays an egg that is not fertilized, that egg becomes a 'drone' bee. They are males and do no work (insert joke about males here). So to simplify, a male (drone) bee has only one parent. But a female (worker) bee always has two parents. If you were to build a family tree for a single drone (male) bee, it would look something like this:

Photo credit:
www.askipedia.com

Do you see what I see? Look on the right side of this image....Fibonacci numbers! How do they work?

The first male bee is just one bee (Generation 1). That male must have a SINGLE parent, and that parent must be a female (Generation 2). But we know that the female must have come from two parents (Generation 3). Now it gets interesting. The female in generation 3 came from two parents, but the male in that generation only had one parent (a female). So in Generation 4, there would be three bees.

Moving on, you can see how each bee in Generation 4 came about. The two females each had two parents, but the male had only one. Therefore, in Generation 5, there will now be five bees. And finally, you can see how in Generation 6, there are going to be eight bees.

1, 1, 2, 3, 5, 8, ...  Fibonacci numbers.

This is where most people being to realize that this sequence is something more than just a mathematical oddity. The rabbit problem was a little contrived, don't you think? But bees are bees. And in the coming posts, we will be exploring some very unusual and unexpected examples of objects in nature that somehow show up as Fibonacci numbers. Stick around - it's going to be an interesting ride!

Rabbits Galore!

Photo credit:
http://theinsanium.blogspot.com

This is the image I posted in the last entry. Now I want to go through it with you so you can see how exactly this pattern leads to the Fibonacci sequence.

Recall that the premise of Fibonacci's question was this:

1. You begin with one male rabbit and one female rabbit. These rabbits have just been born.
2. A rabbit will reach sexual maturity after one month.
3. The gestation period of a rabbit is one month.
4. Once it has reached sexual maturity, a female rabbit will give birth every month.
5. A female rabbit will always give birth to one male rabbit and one female rabbit.
6. Rabbits never die.

Ok, some of these may be a bit of a stretch, but let's just go with it. What do we have in Month #0 (the very start)?

Month 0: There is ONE pair of rabbits.

Month 1: The two rabbits have mated, but are not yet ready to give birth. Therefore, there is still just the ONE pair.

Month 2: Babies! But note Rule #5 - the female always gives birth to a pair of rabbits, so now we have TWO pairs.

Month 3: The original pair gives birth again (see #4), but the newest pair is now only able to mate. They will need to wait one more month before giving birth. So now we have THREE pairs.

Month 4: Finally, BOTH pairs of rabbits give birth and we now have FIVE pairs.

Month 5: Here is where it gets a little tricky, but by now, we have three pairs that are able to give birth and do so, leaving us with a total of EIGHT pairs of rabbits.

1, 1, 2, 3, 5, 8.... So develops the Fibonacci sequence. In my next post, we'll look at honeybees, and how they build their colonies. The process is quite different, but a) the result is the same, and b) unlike these hypothetical rabbits, the bees are real.

Who was this Fibonacci guy anyway?

Leonard Pisano (1170-1250),
also known as Fibonacci.
Photo credit:
http://html.rincondelvago.com/fibonacci.html

You may already know of a famous Italian named Leonardo. He was an artist, inventor, mathematician, scientist, and more. Leonardo was from a town called Vinci in the Tuscany region. For that reason, we commonly know him as Leonardo da Vinci, or Leonardo from Vinci.

However, you may not know about the other famous Leonardo, who was also from Tuscany. You see, he was born and, later in life, lived in Pisa (the town with that leaning tower). So he was originally known as Leonardo Pisano (as in 'Leonardo, the guy from Pisa'). He also went by the name Leonardo Bigollo (wanderer, good-for-nothing).

But you see, as a child, Leonardo was not as well known as his father, Guglielmo Bonacci. Bonacci was a diplomat who represented Pisa in business dealings in North Africa. Leonardo was trained in accounting by African teachers, and didn't return to Pisa until he was around 30 years of age.

In Italian, the word 'figlio' (FEEL-yo) means 'son'. So Leonardo, who was the son of Bonacci, was known as "Figlio di Bonacci", the son of Bonacci. Over time, it shortened down to Fibonacci.

Fibonacci's most famous work was the Liber Abaci, or The Book of Calculation. In that book, he posed the following question:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

One can see that this problem elicits the sequence 1, 1, 2, 3, 5, 8 ,13, 21, 34, .... 

The Fibonacci Sequence as seen in a
growing rabbit population.
Photo Credit: http://theinsanium.blogspot.com/2011/01/
fibonaccis-rabbits.html

I doubt old Leonardo could have predicted just how many places his numbers would turn up over the next 800 years. Over the next several posts, I will be exploring the unusual, the unexpected, and the truly unbelievable places you might run into a Fibonacci number. We will be looking at the fields of botany, zoology, marine biology, design, architecture, and even the stock market.

Stay tuned.

Welcome to Author Tom Robinson's Blog!

1, 1, 2, 3, 5, 8, 13, 21, 34, .... ?

Welcome to my author blog! This is my first entry and tonight I am celebrating the signing of my latest contract. 

Coming soon to a bookstore near you : Fibonacci Zoo, from Arbordale Publishing.

In this blog, I will be discussing the MANY different examples and applications of Fibonacci numbers, a little of the history behind the sequence, and even a little bit about Fibonacci himself. 

The Fibonacci numbers are made according to a very simple rule. Start with the numbers 1 and 1 (some versions start with 0, but my book will begin with 1, because a zoo with zero animals in it just isn't that interesting!) and then form the next number by ADDING up the two previous numbers. 

So after 1 and 1, the next number will be 1+1 =2. After that, 1+2 = 3, 2+3 = 5, and so on!

Over the next several months, I will be posting examples for YOU to see Fibonacci numbers in action. You will find them in the fields of botany, zoology and marine biology, and in the careers of graphics, design, and architecture. 

For now, however, I want to simply say a big thank you to Katie Hall,  my editor at Sylvan Dell, and to encourage you to check out my other books currently for sale:

Everything Kids Science Experiments Book - #61 on Amazon's Best Seller List, as of December 7, 2013

Everything Kids Magical Experiments Book

Forcing Out: A Guide To Better Physics Fitness