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The information for this page has been taken from an excellent site for Fibonacci (Except for YouTube video clips) http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/



Benford's Law and initial digits

Are there any patterns in the initial digits of Fibonacci numbers?
What are the chances of a Fibonacci number beginning with "1", say? or "5"? We might be forgiven for thinking that they probably are all the same - each digit is equally likely to start a randomly chosen Fibonacci number. You only need to look at the Table of the First 100 Fibonacci numbers or use Fibonacci Calculator to see that this is not so. Fibonacci numbers seem far more likely to start with "1" than any other number. The next most popular digit is "2" and "9" is the least probable!

This law is called Benford's Law and appears in many tables of statistics. Other examples are a table of populations of countries, or lengths of rivers. About one-third of countries have a population size which begins with the digit "1" and very few have a population size beginning with "9".

Here is a table of the initial digits as produced by the Fibonacci Calculator:

Initial digit frequencies of fib(i) for i from 1 to 100: 
    Digit:    1   2   3   4   5   6   7   8   9
Frequency:   30  18  13   9   8   6   5   7   4  100 values
  Percent:   30  18  13   9   8   6   5   7   4
What are the frequencies for the first 1000 Fibonacci numbers or the first 10,000? Are they settling down to fixed values (percentages)? Use the Fibonacci Calculator to collect the statistics. According to Benford's Law, large numbers of items lead to the following statistics for starting figures for the Fibonacci numbers as well as some natural phenomena
Digit:  1  2  3  4  5  6  7  8  9
Percentage: 30 18 13 10  8  7  6  5  5



Honeybees and Family trees

There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee").
First, some unusual facts about honeybees such as: not all of them have two parents!
bee iconIn a colony of honeybees there is one special female called the queen.
bee icon There are many worker bees who are female too but unlike the queen bee, they produce no eggs.
bee icon There are some drone bees who are male and do no work.
Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father!
bee icon All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home (a hive) in search of a place to build a new nest.
Bee Tree Key

So female bees have 2 parents, a male and a female whereas male bees have just one parent, a female.

Here we follow the convention of Family Trees that parents appear above their children, so the latest generations are at the bottom and the higher up we go, the older people are. Such trees show all the ancestors (predecessors, forebears, antecedents) of the person at the bottom of the diagram. We would get quite a different tree if we listed all the descendants (progeny, offspring) of a person as we did in the rabbit problem, where we showed all the descendants of the original pair.

Bee Tree Let's look at the family tree of a male drone bee.

  1. He had 1 parent, a female.
  2. He has 2 grand-parents, since his mother had two parents, a male and a female.
  3. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.
  4. How many great-great-grand parents did he have?

Again we see the Fibonacci numbers :

                                       great-     great,great   gt,gt,gt
                           grand-      grand-     grand         grand
Number of       parents:   parents:    parents:   parents:      parents:
of a MALE bee:    1           2           3          5             8
of a FEMALE bee:  2           3           5          8            13



Fibonacci Rectangles and Shell Spirals

fibspiral fibspiral We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).

We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.

fibspiral2.GIF Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangment of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the center are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.

Cundy and Rollett (Mathematical Models, second edition 1961, page 70) say that this spiral occurs in snail-shells and flower-heads referring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6 "The Equiangular Spiral". Here Thompson is talking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phi expansion factor.

Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers provide boyancy in the water. Click on the picture to enlarge it in a new window. Draw a line from the center out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it. This shows that the shell has grown by a factor of the golden ratio in one turn.
On the poster shown here, this factor varies from 1.6 to 1.9 and may be due to the shell not being cut exactly along a central plane to produce the cross-section.



A Fibonacci Number Trick

Here is a little trick you can perform on friends which seems to show that you have amazing mathematical powers. We explain how it works after showing you the trick.

Alice and Bill

Here is Alice performing the trick on Bill:
Alice: Choose any two numbers you like, Bill, but not too big as you're going to have to do some adding yourself. Write them as if you are going to add them up and I'll, of course, be looking the other way!
Bill: OK, I've done that.
Bill chooses 16 and 21 and writes them one under the other: 16
Alice: Now add the first to the second and write the sum underneath to make the third entry in the column.
Bill: I don't think I'll need my calculator just yet.... Ok, I've done that.
Bill writes down 37 (=16+21) under the other two: 16
Alice: Right, now add up the second and your new number and again write their sum underneath. Keep on doing this, adding the number you have just written to the number before it and putting the new sum underneath. Stop when you have 10 numbers written down and draw a line under the tenth.
There is a sound of lots of buttons being tapped on Bill's calculator!
Bill: OK, the ten numbers are ready.
Bills column now looks like this:   16
Alice: Now I'll turn round and look at your numbers and write the sum of all ten numbers straight away!
She turns round and almost immediately writes underneath: 2728.
Bill taps away again on his calculator and is amazed that Alice got it right in so short a time [gasp!]

So how did Alice do it?

The sum of all ten numbers is just eleven times the fourth number from the bottom. Also, Alice knows the quick method of multiplying a number by eleven. The fourth number from the bottom is 248, and there is the quick and easy method of multiplying numbers by 11 that you can easily do in your head:
Starting at the right, just copy the last digit of the number as the last digit of your product. Here the last digit of 248 is 8 so the product also ends with 8 which Alice writes down: ...
Now, continuing in 248, keep adding up from the right each number and its neighbour, in pairs, writing down their sum as you go. If ever you get a sum bigger than 10, then write down the units digit of the sum and remember to carry anything over into your next pair to add.
Here the pairs of 248 are (from the right) 4+8 and then 2+4. So, next to the 8 Alice thinks "4+8=12" so she writes 2 and remembers there is an extra one to add on to the next pair:
Then 2+4 is 6, adding the one carried makes 7, so she writes 7 on the left of those digits already written down: ...
Finally copy down the left hand digit (plus any carry). Alice sees that the left digit is 2 which, because there is nothing being carried from the previous pair, becomes the left-hand digit of the sum.

The final sum is therefore 2728 = 11 x 248 .


Why does it work?

You can see how it works using algebra and by starting with A and B as the two numbers that Bill chooses.
What does he write next? Just A+B in algebraic form.
The next sum is B added to A+B which is A+2B.
The other numbers in the column are 2A+3B, 3A+5B, ... up to 21A+34B.
  A    B
  A +  B
  A + 2B
 2A + 3B
 3A + 5B
 5A + 8B
 8A +13B
13A +21B
21A +34B
55A +88B
If you add these up you find the total sum of all ten is 55A+88B.
Now look at the fourth number up from the bottom. What is it?
How is it related to the final sum of 55A+88B?

So the trick works by a special property of adding up exactly ten numbers from a Fibonacci-like sequence and will work for any two starting values A and B!

Perhaps you noticed that the multiples of A and B were the Fibonacci numbers? This is part of a more general pattern which is the first investigation of several to spot new patterns in the Fibonacci sequence in the next section.


The Fibonacci Numbers in Pascal's Triangle


A simple definition of Phi

What number squared is equal to itself plus one?


A bit of history...

Euclid, the Greek mathematician of about 300BC, wrote the Elements which is a collection of 13 books on Geometry (written in Greek originally). It was the most important mathematical work until this century, when Geometry began to take a lower place on school syllabuses, but it has had a major influence on mathematics.

It starts from basic definitions called axioms or "postulates" (self-evident starting points). An example is the fifth axiom that

there is only one line parallel to another line through a given point.
From these Euclid develops more results (called propositions) about geometry which he proves based purely on the axioms and previously proved propositions using logic alone. The propositions involve constructing geometric figures using a straight edge and compasses only so that we can only draw straight lines and circles.
bisect a line For instance, Book 1, Proposition 10 to find the exact centre of any line AB
  1. Put your compass point on one end of the line at point A.
  2. Open the compasses to the other end of the line, B, and draw the circle.
  3. Draw another circle in the same way with centre at the other end of the line.
  4. This gives two points where the two red circles cross and, if we join these points, we have a (green) straight line at 90 degrees to the original line which goes through its exact centre.

In Book 6, Proposition 30, Euclid shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line". He describes this geometrically.

                <-------- 1 --------->
                A            G       B
                       g        1–g               
Euclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (ie is the same as the ratio AG/AB). We can see that this is indeed the golden section point if we let the line AB have unit length and AG have length g (so that GB is then just 1–g) then the definition means that
GB = AG i.e. using the lengths of the sections 1-g = g
-- -- -- --
AG AB g 1
which we rearrange to get 1 – g = g2.
Notice that earlier we defined Phi2 as Phi+1 and here we have g2 = 1–g or g2+g=1.
We can solve this in the same way that we found Phi above:
g =  –1 +sqrt5   or g =  –1 – sqrt5  



So there are two numbers which when added to their squares give 1. For our geometrical problem, g is a positive number so the first value is the one we want. This is our friend phi also equal to Phi–1 (and the other value is merely –Phi).



The Value of Phi

Phi has the value  sqrt5 + 1   and phi is  sqrt5 – 1  .



Both have identical fractional parts after the decimal point. Both are also irrational which means that
  • They cannot be written as M/N for any whole numbers M and N;
  • their decimal fraction parts have no pattern in their digits, that is, they never end up repeating a fixed cycle of digits

Here is the decimal value of Phi to places grouped in blocks of 5 decimal digits. The value of phi is the same but begins with 0·6.. instead of 1·6.. .

Phi is 1·61803398874...

Phi and the Fibonacci numbers

graph of y=Phi x The graph on the right shows a line whose gradient is Phi, that is the line

y = Phi x = 1·6180339.. x

Since Phi is not the ratio of any two integers, the graph will never go through any points of the form (i,j) where i and j are whole numbers - apart from one trivial exception - can you spot it?
So we can ask
What are the nearest integer-coordinate points to the Phi line?
Let's start at the origin and work up the line.
The first is (0,0) of course, so here ARE two integers i=0 and j=0 making the point (i,j) exactly on the line! In fact ANY line y=kx will go through the origin, so that is why we will ignore this point as a "trivial exception" (as mathematicians like to put it).
The next point close to the line looks like (0,1) although (1,2) is nearer still. The next nearest seems even closer: (2,3) and (3,5) even closer again. So far our sequence of "integer coordinate points close to the Phi line" is as follows: (0,1), (1,2), (2,3), (3,5)
What is the next closest point? and the next? Surprised? The coordinates are successive Fibonacci numbers!

Let's call these the Fibonacci points. Notice that the ratio y/x for each Fibonacci point (x,y) gets closer and closer to Phi=1·618... but the interesting point that we see on this graph is that

the Fibonacci points are the closest points to the Phi line.



Can we write Phi as a fraction?
The answer is "No!" and there is a surprisingly simple proof of this. Here it is. [with thanks to Prof Shigeki Matsumoto of Konan University, Japan]
First we suppose that Phi can be written as a fraction and then show this leads to a contradiction, so we are forced to the conclusion Phi cannot be written as a fraction:
Suppose Phi = a/b and that this fraction is in its lowest terms which means that:
  • a and b are whole numbers
  • a and b are the smallest whole numbers to represent the fraction for Phi
  • Since Phi > 1 then a > b
But we know from the alternative definitions of Phi and phi that
Phi = 1/phi and phi = Phi – 1 which we put in the fraction for Phi:
Phi = 1/(Phi –1). Now we substitute a/b for Phi:
a/b = 1/(a/b –1) = b/(a – b)
But here we have another fraction for Phi that has a smaller numerator, b since a > b
which is a contradiction because we said we had chosen a and b were chosen to be the smallest whole numbers.
So we have a logical impossibility if we assume Phi can be written as a proper fraction
and the only possibility that logic allows is that Phi cannot be written as a proper fraction - Phi is irrational.




Another definition of Phi

We defined Phi to be (one of the two values given by)
Phi2 = Phi+1

Suppose we divide both sides of this equation by Phi:
Phi = 1 + 1/Phi

Here is another definition of Phi - that number which is 1 more than its reciprocal
(the reciprocal of a number is 1 over it so that, for example, the reciprocal of 2 is 1/2 and the reciprocal of 9 is 1/9).

Phi as a continued fraction

Look again at the last equation:
Phi = 1 + 1/Phi
This means that wherever we see "Phi" we can substitute (1 + 1/Phi).
But we see Phi on the right hand side, so lets substitute it in there!
Phi = 1 + 1/(1 + 1/Phi)
In fact, we can do this again and again and get:
    Phi = 1 +       1         = 1 + 1/( 1 + 1/( 1 + 1/( 1 +.. ))) 
               1 +     1    
                   1 +   1  
                       1 + ..
This unusual expression is called a continued fraction since we continue to form fractions underneath fractions underneath fractions.


Fibonacci and the Share market?






 From The Canberra Times

Explores the relationship between health and wealth with some very interesting conclusions..... not what you would expect