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A new way to look at Fibonacci Numbers
[Nanook, the Eagle scout is talking with Bill, the supervisor of the Electronics shop]
"So, you want to know about some of the other discoveries I’ve made? How about one in mathematics?”
“Sure. I’m pretty good in math. What’s it about?”
"Have you ever heard of Fibonacci numbers?”
“Yeah, sure. One of the most famous number sequences in history. It describes the branching in trees, the arrangement of leaves on stems, the fruiting of pineapples, the flowering arrangement of artichokes and sunflowers and pine cones, the curve of sea shells and the breeding of bees. So, what have you done? Turn them inside out?”
"Hmmm… I never thought about what that would mean. So no, not inside out. What I did was figure out a new way to generate the series.”
“So? Why is that important?”
“First, let’s go through the standard description so we can be sure we’re both talking about the same thing. The Fibonacci number sequence is actually a pretty simple sequence. Here’s how they taught us to create it in school.
First you start out with the number 1. That’s the first number in the sequence. Then you add a second 1 to the sequence. Now you have two numbers in the sequence: 1, 1. To get the third number in the sequence, you add the two numbers preceding it. So, the third number in the sequence is 2. The first 3 numbers in the sequence are therefore 1,1,2. To get the next number in the sequence, and in fact, any number in the sequence, you simply add the previous two numbers. So the last two numbers so far are 1 and 2. The sum is 3. So the sequence is now 1,1,2,3. The next number is the sum of 2 and 3 which equals 5. We now have 1,1,2,3,5. The first 10 numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Each number is always the sum of the previous two numbers. In mathematical form this can be written: F(n) = F(n-1) + F(n-2) where n is the position count in the sequence. Agreed?”
“So far, so good. Now what?”
"Well, what got me thinking about this was something my chemistry professor said. He said there are always a lot of equations that can be found that ALMOST fit a set of data. What’s important, when you are asked to help someone find an equation to fit some data, is to understand the problem well enough to pick an equation that represents the true “nature” of the data. So, when I came across the Fibonacci sequence a few days later, I wondered what the underlying principle was. I went to the library. I was amazed. There were a lot of books that showed how the sequence matched things in nature, like the spiral of the Chambered Nautilus, but almost every book honestly admitted that no one knew why? So I thought back to what my professor said. His point was, a lot of data in society will match interesting patterns of numbers. But if you don’t have the CORRECT formula, then you can’t predict what will happen when you extend the data. Furthermore, when you have the wrong formula, the characteristics of the formula are not right for the true science going on. So you don’t really know what’s going on and the formula doesn’t help you understand it. It may even mislead you.
I decided to try to figure out WHY the Fibonacci series fit all these things in nature. What it didn’t take me long to figure out was that this was a much bigger task than I had time for.”
Bill broke up laughing.
“I guess it never occurred to you that some of the greatest mathematicians of all time spent their whole life trying to figure it out - and still couldn’t do it.”
“Actually, I did think about that. I knew I wasn’t the first one to have this idea. So, I pretty quickly gave up on it. But as I now understand myself, some second channel in my brain sometime keeps going when my conscious brain heads off to do other things. One day, I woke up with some interesting additions to the theory.
First off, I realized that you could add 0 to the beginning of the sequence without ruining it. This seems kind of trivial, but I was always bothered that to start the traditional series, you had to pull the first 1 and a second 1 out of the air. I guess starting with a 0 and 1 just seemed more logical to me. So, then I had: 0, 1, 1, 2, 3 etc.”
“Very interesting. Zero isn’t at all a trivial number. With all the things that the Roman’s knew, they didn’t consider Zero a number.”
“Of course, once you add 0 to the sequence, you’re going to ask what’s on the other side of 0. “
“Yeah right. Maybe something like that would seem obvious to YOU.”
“Using the standard method, the series that results looks like this: 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5 . So, the magnitudes of the numbers before 0 are the same as after 0, but they alternate in sign.”
“Yeah? So what does that mean?”
“I don’t know! I don’t have a clue what use this would be. But I never saw that before and it just seemed interesting.
There was another thing that I discovered, however, that seemed to be much more important. I found another way to generate the sequence.
Let’s start out with this part of the sequence: 0, 1, 1, 2, 3, 5, 8. Take the second number in the sequence and double it. Then add the first number. This gives 1 times 2 equals 2 plus 0 equals 2 which is the fourth number in the sequence. Now let’s move up one and try the method again. Take the third number and double it. Then add the second. This gives 1 times 2 equals 2 plus 1 equals 3 which is the fifth number in the sequence. Move up once more. Take the fourth number and double it. Then add the third. This gives 2 times 2 equals 4 plus 1 equals 5 which is the sixth number in the sequence. Move up once more. Take the fifth number and double it. Then add the fourth. This gives 3 time 2 equals 6 plus 2 equals 8 which is the seventh number in the sequence. In equation form this is F(n) = 2 X F(n-2) + F(n-3).” I wrote it out for him.
“Hang on now. I was interested in this for awhile, so I know a little bit about it. Don’t you mean F(n) = 2XF(n-1) + F(n-2), which is called the Pell sequence?
"No. I later came across that as well. My approach is different because it skips a number in the sequence.”
“OK. So? This is going to be good.”
“Here’s why this seemed so important to me. A living process like cells or bacteria grow by dividing. One element splits into two. Two become four etc. This is a simple geometric progression. But what about more complex processes in which after reproduction, members age but no longer reproduce. After aging, they die. Or in some cases, after being born, there is a dormant period before reproduction can occur. The equation I discovered, which produced the same Fibonacci sequence, seemed important because there was a doubling process in it. That’s what my professor was trying to tell me. This new way of generating the sequence could help people fit it more logically to what we observe in nature. Here’s how the new way could fit a natural process in a logical way.
Using the sequence above, start with a single female ( the second term ). In the first cycle, she produces an offspring. How many total members are there? Using my formula, we get 2, the fourth term. But the child is not yet productive, so how many are productive? Still 1 (the third term ). In the next cycle, that single female ( the third term in the sequence this time ) also produces one child. How many total members now? Using my formula, we get 3 ( the fifth term in the series). The new child is not productive, but now the previous one is. Total productive females equal 2 ( the fourth term in the sequence). Those two females have children. Total number of members is 5 ( sixth term in the series ). The prior children now become productive. How many productive members? 3 ( the fifth term in the series. )”
“OK. I can’t actually visualize it without drawing it out on paper. But I believe you. This could be a really important discovery.”