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Sunday, January 26, 2014

Fibonacci in Nature

This flower has 13 petals.
13 is a Fibonacci number!
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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.
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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.
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This beautiful pink clematis has eight petals.
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This star daisy has thirteen petals.
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This daisy has twenty-one petals.
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So does this one!
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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!

Thursday, January 9, 2014

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 ...

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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 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.


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

Sunday, January 5, 2014

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

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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:

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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!