Chapbook

[sasha]

Nothin' Up My Sleeve...

There's a well-known series of numbers called the Fibonacci sequence, named for Leonardo Bonacci, aka "Fibonacci", a 13th century mathematician who studied it. A sequence is simply a list of numbers generated from some arithmetic rule - in this case, each member of the list is the sum of the previous two, starting with 0 and 1. Adding these together gives the third: 0 + 1 = 1. Adding this to the 1 just before it gives us the fourth, 2. Add this to the 1 preceding it and we get 3... which when added to the 2 gives 5; and so on.

You get the idea. Skipping the intermediate details, the sequence goes

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... ad infinitum.

This sequence is full of unexpected patterns & surprises, and has been delighting numerophiles for centuries. You'll find it in just about every mathematician's toy box. I was playing with it one rainy afternoon several years ago and discovered something that left me so baffled & amazed I thought I'd made a mistake. But I hadn't.

These numbers make a few surprise appearances in nature - the arrangement of leaves in plants, & the structure of chambered nautilus shells - but their greatest appeal lies in what happens when we calculate the ratio between adjacent pairs. The first pair is the one we started with: 0 and 1. The ratio of 0 to 1 is just 0 (0/1 = 0). The next pair is 1,1; 1/1 = 1. Now that we're past the starting gate, things start to get interesting:

third pair: 1/2 = 0.5
fourth pair: 2/3 = 0.66666
fifth pair: 3/5 = 0.6

followed by

5/8 = 0.625
8/13 = 0.61538...
13/21 = 0.61904...
21/34 = 0.61764...
34/55 = 0.61818...
55/89 = 0.61797...
...

It almost looks like the ratio is closing in on a value somewhere around 0.617 or 0.618 - and in fact, that's exactly what it's doing. That number is called the Golden Ratio (or sometimes the Golden Section) - rounded to 6 decimal places, its value is 0.618034, and was so revered by the ancient Greeks that they gave it a name, Φ (phi). They considered it - among other things - the most visually pleasing aspect ratio for a rectangular shape - the most ideal, most beautifully rectangly rectangle (something I've kept in mind when building shelves, as has my daughter when designing garden layouts.) The aesthetic appeal we associate with this number can't be overstated.

On that long-ago rainy day, I was curious about the path of convergence these ratios follow on their way to Φ, and had coded a spreadsheet to do the calculations. Progressive ratios seemed alternately too large and too small, but each time closer than before - a typical alternating path to convergence, like a bouncing ball gradually coming to a stop. Interesting.

This is where I stepped off the trail & found myself in the woods. It occurred to me that the Fibonacci sequence was only one of an infinite number of such sequences that could be generated by the same rule - all I had to do was start things off with a different pair of numbers to generate an entirely different sequence, whose pairs would converge to a different ratio - if they converged at all. So I replaced the two starting values with something else - I don't remember what. Let's just say 37 and -5.

As I'd known I would, I got an entirely different string of numbers: 32, 27, 54, 83... but I was more interested in where their succesive ratios would lead. So I ran off a few dozen, and saw that, as before, their successive ratios rapidly converged...

... to 0.618034 - the Golden Ratio.

This was so unexpected I was sure I'd made a mistake somewhere. I double-checked my coding, and found no errors - unsurprising given the calculations' simplicity. I tried doing a few by hand - and got the same result as the spreadsheet. I tried changing the starting values again, this time to non-integers, like π and √2. But no matter where these sequences started, the ratio of their consecutive values always converged to 0.618034. I was as baffled and awed as when I was 13 or 14 and quite accidentally discovered the parabola, hiding in the zig-zag path between the tiles of the shower stall in our house.

The story should end here - at an amazingly counterintuitive truth about numerical sequences I'd accidentally stumbled across on a rainy day. But a mystery without resolution is like an unscratched itch, & I had to go scratch it. If seeing how a magic trick is done spoils it for you, consider this The End. But if you're not averse to a little off-roading, buckle up & I'll try to explain just why all Fibonaccies lead to the Golden Ratio. It's going to involve some algebra and a bit of pre-calculus hand-waving, but I hope it doesn't diminish the magic and beauty of the result.



*********************************************************************

Still here? OK the question is: Why does the ratio between consecutive elements of ANY Fibonacci sequence - not just THE Fibonacci sequence - converge to the same value, regardless of where they start? To answer this, we need to cast the sequence in purely symbolic terms. We need to translate the statement "each number in the sequence is the sum of the previous two" into math-speak:

To do this, we need to give names to each of the components in this definition. So let's call the nth term in the sequence Fn - where n can mean ANY counting number. We'll call the one just before it - the (n-1)th - Fn1 (1st previous), and the one before that Fn2 (2nd previous). Now we can express the rule for generating the sequence in math-speak as

1) Fn= Fn1 + Fn2. But for this to be true, there have to be at least two members of the series already in existence before we can calculate a third. It doesn't tell us anything more about them, so they can be whatever we like. Fibonacci just chose 0 and 1 because - well, why not.


We want to find out why the ratio of Fn1 to Fn converges to 0.618034, as n gets very large. So let's call that ratio R, recognizing that R will be different for each pair of numbers. We'll also assume that n is fairly large, putting us some distance downstream from the starting pair.

2) R = Fn1/Fn.

Returning to equation 1): let's divide Fn by Fn1. We can do that, as long as we do the same to what it's equal to, namely (Fn1 + Fn2). (If I have the same number of coins in each pocket and remove half of them from each, or add three to each, the number in my left pocket still equals the number in my right - though not the same number as before.) So let's divide both sides by Fn1:

3) Fn/Fn1 = (Fn1 + Fn2)/Fn1.

This is almost equation 2), which tells is what Fn1/Fn is equal to. In fact, Fn/Fn1 is just ____1____    or 1/R.
                                                                                                                                      (Fn/Fn1)

Dividing the sum of two numbers by a third number is the same as dividing each number individually by the 3rd, and adding the two results. [ Try it yourself with (6 + 4)/2. ] Doing this to the right-hand side turns equation 3) into

4) 1/R = (Fn1/Fn1) + (Fn2/Fn1).

Any number divided by itself is just 1; so 4) becomes

5) 1/R = 1 + Fn2/Fn1.

But Fn2 and Fn1 are consecutive "Fibonacci" numbers too, the pair just before Fn; so as long as we're far enough along in the sequence, their R-ratios won't be too different. In other words, when n is very large,  Fn2/ Fn1 is ALMOST the same as Fn1/Fn - they're "close enough" to consider the same R. (This is the hand-wavy part.)

6) 1/R = 1 + R.

We can get rid of that fraction by multiplying it by R, as long as we do the same to 1+R:

7) R(1/R) = R*(1 + R)

The R's on the left cancel each other out; and the R outside the parentheses on the right multiplies each term inside them for the same reason equation 3) became equation 4):

8] 1 = R + R*R.

Finally, we can subtract 1 from both sides and rearrange the terms a bit to give us

9) R² + R -1 = 0.

This is a particular class of equation known as a quadratic, whose solutions are well-known; so without taking that side trail, I'll just cut to the chase and say that there are two values of R that satisfy it: (√5 - 1)/2, and -(√5 + 1)/2... equal to 0.618034 and -1.618034 respectively.

This is true only because each term in the series is the sum of the two preceding it - it says nothing about where such a series needs to start. Though each string of numbers is different, they ALL have the property that the further out you go, each one gets closer and closer to 0.618034 times the next.

I never saw this one coming - after all these years, Nature can still pull a fast one on me!