Over 40 years ago, I made a significant breakthrough related to the famous Fibonacci sequence. While this math equation is world renown as a description for so many natural phenomenon, I have never been able to find any explanation for how it explains the natural mechanisms it is associated with other than describing their geometry.
On August 21, 2020, I saw an article about a Boston College graduate, Professor Lisa Piccirillo (now at MIT) about how she solved the problem called the “Conway knot”. The article says, through help from others at U. Texas, she was able to get her discovery published by the Annals of Mathematics. Since I don’t have a PhD stamp from anywhere, I can’t even get an open door at any of these publications. For what it’s worth, here is the content of a letter I wrote to Prof. Piccirillo about my breakthrough hoping she might have a lead to getting it some higher exposure.
In high school, the way we are taught to construct the series is: first start with the number 1. Then add a second 1 to the sequence. To get the third number in the sequence, you add the two numbers preceding it, etc. This process produces the well known form: F(n) = F(n-1) + F(n-2). What got me thinking about this was something one of my chemistry professors said, (paraphrased) “there are always many equations 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 model that represents the true “nature” of the data. I wondered, what natural processes produced this algorithm. A quick visit to the library moved me to the next step. 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. This led to the following:
First I realized that 0 could be added to the beginning of the sequence without ruining it. This seemed kind of trivial, but I was always bothered that to start the traditional series, the first 1 and a second 1 just got pulled out of the air. Starting with a 0 and 1 just seemed more logical to me. That gives: 0, 1, 1, 2, 3 etc.
Once you add 0 to the sequence, of course, you’re going to ask what’s on the other side of 0. Using the standard method, the series that results looks like: 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5 . So, the magnitudes of the numbers in the negative direction are the same as those on the positive side, but they alternate in sign. This was fun, but I never found any use for it, and never saw any reference to it either.
The real breakthrough followed that.
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.
Move up one and try the method again. Take the third number and double it; add the second. That equals 3 which is the fifth number in the sequence.
This produces a completely different equation form for the same sequence: F(n) = 2 X F(n-2) + F(n-3). (NB: this is not the Pell sequence: F(n) = 2 x F(n-1) + F(n-2))
The importance of my new sequence was that it both doubles a number in the sequence, but also skips a number in the sequence. This is important because these are life-form processes! Cells double. There is a delay process while they mature, they double again, the earlier ones die or become non-productive.
If you can help, this new interpretation might help someone else.
Good luck with your effort. . .
There are plenty more breakthrough concepts like this on my website. And, of course, all the social breakthroughs in my new books:
