Maths: Fibonacci and phi

The Fibonacci series has some very interesting properties that you can track down with a web search. This will deal with just a few of them. For example, the ratio of any two successive terms gets closer and closer to phi, the “Golden Mean”. Then again, the numbers of seed whorls in a sunflower counting clockwise and counter-clockwise, will always be two successive Fibonacci terms, if you count up through complete turns.

Start with a rectangle f x 1 (phi x 1) and draw squares like the ones here. Their sizes are in the Fibonacci series, and quarter-circles in the squares make a beautiful spiral. 

An ammonite fossil from Morocco. Nature got there first!

Or if you count the leaves going up a stem, then as you go, you will slowly rotate around the stem, after you have made as many turns as one of the Fibonacci numbers, the leaves you have counted will sum to another Fibonacci number. Why? Do a web search!

* Some of the numbers in the Fibonacci sequence above are composite (divisible by other numbers), and some of them are prime. Is there any deep pattern that determines what a particular Fibonacci number is divisible by, if anything?

* What is the maximum number of Fibonacci numbers you need to add together to reach any integer? Is this always a maximum? How would you prove it? (For example, 6 = 1 + 5, 7 = 1 + 1 + 5, and so on).

Squares and Fibonacci numbers

The square of any Fibonacci number above 1 can be expressed as the product of its neighbour Fibonacci numbers, plus or minus 1: is there a hidden pattern here?

2² = 1 x 3 +1
3² = 2 x 5 – 1

5² = 3 x 8 + 1
8² = 5 x 13 -1
13² = 8 x 21 + 1

Other ways of getting a value for F (phi)

Here is a formula that returns a value of phi:

(phi) = (5 + √5) / (5 - √5)

Get your calculator and try it out, then if you can, open a spreadsheet program.

First, these two clips from screenshots show you how to write the code for the Fibonacci series.

The spreadsheet codes for the Fibonacci series.

To save yourself time, type the code into cell C1, then use the mouse to highlight cells to the right. Once they are highlighted, holding down the Control key and tapping R is the same as Edit – Fill – Right. The main thing is that this is what you see:

The spreadsheet results for the Fibonacci series.

Then I added this code in row B:

Going from Fibonacci to phi.

And I got this result:

Going from Fibonacci to phi.

A friend mentioned in passing that the value of F (phi) to three decimal places is 1.618, and that you could generate a “pseudo-Fibonacci series”, beginning with the number 16 and 18, and that this series would also settle in to a close approximation to the Golden Mean. I tested this with a spreadsheet, and you can as well. Here is the code (and remember to use Control-R!)

Going from phi to Fibonacci.

Here is the result:

Going from phi to Fibonacci.

Anybody writing a book called Playwiths is going to play with things, so I tried copying the two rows and inserting my favourite number (1728) and the name of my favourite novel (1984) as start points. Here is what I got: There is something mysterious going on here: I can smell a pattern!

Going from 1729 or 1984 to phi.

Speaking of patterns, Joseph Lagrange tested the last digit of successive Fibonacci terms, and found that after sixty terms, the pattern of last digits repeated. Logically, the last two terms in the set have to be one and zero, but here is the actual pattern:

1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9…

Finding this may seem hard, because the 60^th term in the series is more than 1.5 trillion, but you can do it easily with a spreadsheet, using this small snippet below as a model. The code just takes the sum, and if it is more than 9, it takes 10 off it, but otherwise, it just provides the sum.

How to get the last digits of the Fibonacci series.

Post script

It was late at night when I wrote that the calculation must have been hard for Lagrange to do, but as my head hit the pillow, I realised that maybe he didn’t need to do the additions at all. Here is the end of the pattern, and the start of the next sequence in bold blue:

9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8.

My way to find the whole series is to know that 1, 0 must be the last two, and working back before that, the previous number must be 9, and 2 had to come before that, and so on!

I got up the next morning, and I deduced the whole sequence, running backwards, but I rather think that if Lagrange ever heard of spreadsheets, he would have killed for even Visicalc or Lotus 123. (If you are less than 70, you will probably need to look those up.)

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