| Where Recherche duTemps Perdu
---- meets Kirchliche Dogmatik
Most of this was written originally on Saturday evening, but needed some refinement and a lot of uploading of formulas, plus adding enough cute stuff to give it appeal to a general public. Hope you like the Staypuft Marshmallow man gif I concocted.
The weather continuous to be quite nice, and definitely appropriate for summer. We are heading into the high nineties again today. A couple of weeks ago I procured my annual season pass to the Beulah Park Pool, here in Smalltown, USA, and I’m making good use of it, every day if I at all possible. It’s funny (so to speak): on the first day I went out there this year, I felt like I had lost an enormous amount of strength in my arms and legs, and I wasn’t surprised. I couldn’t even manage to swim one entire lane without taking a break. However, by this past Friday, after only about a week, I managed four, and as of yesterday and today, I’m up to six already. That’s definitely also been a surprise, and obviously a good one, for which I’m grateful. On most days, after swimming one set, I usually take a break and visit the water slide once or twice to satisfy my need for speed. Then I do another set, casually and without trying to go to any limit.
Well, let’s get back to the numbers.
There are several mathematical surprises connected to the Fibonacci series in its own right, apart from its involvement with ϕ. For example, even though ϕ is definitely an irrational number, it does manifest a certain amount of regularity, such as certain digits recurring at precise intervals. As intriguing as those things are, I need to refer you to Livio for details. If I don’t observe some limits, this series would still be going a year from now, and—who knows—even the spammers might be scared away by then. So I’m going to continue to focus on ϕ as much as I can, and talk some more specifically about the relationship between the Fibonacci series and ϕ.
I’m going to introduce a new feature in this entry. The post would become meaningless if you or I just skipped all of the details, but I think I’ll keep more readers interested if I show them what they can skip without missing out on all of the content. So, I will introduce the problem for the day, but then give you the option of clicking on a link that will take you right to the conclusion without going through all of the steps to take you there. You will see it shortly below as a green rectangle.
There is a formula known as “Binet’s Equation,” named after Jacques Phillipe Binet (1786-1856), who was actually not the first person to discover it, but apparently created more interest in it than other famous mathematicians before him. Now, I’m going to trot out Binet’s equation, and you may just be horrified when you first see it. It looks like a huge monster, perhaps reminiscent of one such as the StayPuft Marshmallow Man ® from “Ghost Busters.” We’re going to describe Binet’s equation, and pare it down until it becomes one simple calculation. Then we’ll use it to find out what the 12th Fibonacci number is.
[Added on Monday: When I was working on this on Saturday night I had no guarantee that it would actually work. To be more specific, I assumed that the formula was right, but I did not know if I was interpreting and using it correctly. If it didn’t work out, I had no one to ask, at least immediately, what I had done wrong. Thus, my emotional reaction at the end was totally authentic. I couldn’t believe that I had brought of both stages: understanding the formula (a good first step to clarifying it for my readers) and applying it in order to get the right answer. ]
Here goes. This is Binet’s Equation:
Intimidated? You should be—but only until we’ve taken it apart and seen the simplicity that’s behind the apparent complexity.
Seriously, I’m going to go extremely slowly in clarifying this equation and highlighting all steps. Maybe you will realize that working with such formulas one does have to be careful, but it does not have to be “*cough* laborious *cough*, if I may tease John of LRT one more time (all in good fun, I hope he understands if he should ever read it). One of the nicest compliments I ever received on my writing was with regard to my commentary on 1 and 2 Chronicles, when the editor at B&H told me: “Win, you have made Chronicles come alive.” I don’t know if it’s a comparable challenge, but I shall try to make Binet’s formula come alive as well.
“Fn” stands for which Fibonacci number in the chain you wish to calculate, i.e. the “nth” number. This is totally of your own choosing. For now, we’ll try to use the formula to figure out the 12th Fibonacci number, a number we can easily verify by just adding up the Fibonaccis. Maybe later, then, we’ll try our hands at calculating a much larger number, perhaps F95, but first we’ll have to see if we make it through this trial.
The expression “for n ϵ ℤ” reminds us that n has to be an integers. The Fibonacci numbers per se are not irrational, and are inhabitants of ℤ. The irrationality of phi and squareroot of 5 comes into play insofar our answer may need rounding up or down to keep the solution within the ℤ circle as well. From here on out, we're going to dispense with that little tail.
Wow! What are we going to do with that messy clutter highlighted in red? Actually, you may know already what we’re looking at here once I let it stand by itself. It’s the positive root of our good old equation for ϕ:
So, we can substitute ϕ in for that formula, and life is beginning to look a lot simpler already. We now know that, when the time comes for actual calculations, we can insert 1.61803 as the value of ϕ.
A similar thing is true for the next packet that I’ve highlighted in red.
This is the negative solution for phi’s equation, and you may or may not recall that it’s value is the same as the negative conjugate of ϕ, but terminology aside, you may remember that it amounts to -0.61803. Rather than writing “-1/ϕ”, which would mean that we're building up fractions again, I’ll use an expression that’s equivalent, but a little easier on the eyes, ϕ-1.
Let’s not forget that, when we eventually subtract that negative number, we can substitute a “plus” sign for the two negatives—if we ever have to do so. The last thing I can point out before we actually install some numbers is that both ϕ and its negative reciprocal will be raised to the power of n.
Now we’re looking at the formula that we want to instantiate.
I know; I know. You’re probably still not buying into my claim towards simplicity. Okay, I have one more trick up my sleeve to try to persuade you, and it's a good one. There’s a corner we may cut without doing ourselves any serious harm (just don’t run with the scissors, please). That whole –(-ϕn) business is going to get extremely tiny, yea infinitesimal, so quickly that it can be pretty much treated as negligible for our practical purpose. I’m allowed to say that because this formula does have a practical purpose, namely to find a specific number in the Fibonacci series. (Alright, different definitions of “practical” perhaps.) If we give ourselves permission to leave off the negative part, we can actually calculate the “nth” Fibonacci number of the series with the simple formula: One multiplication (ϕ raised to the nth power) followed by one division, dividing the product by the square root of five.
We set out to find the 12th Fibonacci number, and now we’re at the point where we can put some numbers into that simplified formula. If our result comes out really skewed we’ll have to work a little harder. Let’s substitute 12 for n:
If you’re using a sophisticated program, you can just plug in the symbols. I don’t expect that too many of us carry a value for ϕ12 in our heads, but Wolfram/Alpha has given me an answer of 321.987… . We can use a value of 2.236 for √5. So we can take that number and divide it by the number that Wolfram has given us. The crucial moment is drawing nigh:
We do the division and get …
Let me tell you how nervous I am right now. As I said above, I have not done this before with all of the appropriate details. Before starting to write down anything formal, I gave it a rough once-over trying just to see if it might work, but now I’m really wondering. I’m sure you’re in suspense, too, about whether I worked it out so as to get a plausible result. Of course, it’s possible that it didn’t. I am fallible and prone to small mistakes that generate drastic changes in the outcome. That reminds me of a time when …
Okay, okay, I’ll stop dawdling. It’s taking me a bit to get past this approach/avoidance dilemma. Let’s take it to Wolfram then and have it do the last division for us. The expected answer was 144. Here is the result of applying the truncated form of Binet’s equation:
I can’t believe it. I really, honestly did not expect anything nearly as close to 144. I’m stunned and almost emotional. This was a long and winding road. Thank you to those of you who stayed with me through the entire length of this entry. I know it was demanding. But isn’t that result a whole can of high-octane awesome-sauce‼ Back in my childhood days in Gymnasium, the math teacher, Frau Dr. von Borke, had us memorize the multiplication table for up to 20. Thus I realize that 144 is not only the 12th Fibonacci number, but also 122. We can’t derive a rule from that fact, but we can see another piece of that beautiful mosaic of numbers. When God built in numbers onto the universe he created, he not only gave it regularity, but he infused into it a beauty that we ignore to our own loss.
It’s too bad that it has become almost fashionable these days to promote yourself as someone who doesn’t get along with numbers, wearing that self-deprivation almost like a badge of machismo. Look at what you’re missing!
I wish all of my readers a day filled with beauty on many different levels.