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/
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
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 4What 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
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
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.
Let's look at the family tree of a male drone bee.
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
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
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.
Alice and BillHere is Alice performing the trick on Bill:
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:
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.
If you add these up you find the total sum of all ten is 55A+88B.A B A + B A + 2B 2A + 3B 3A + 5B 5A + 8B 8A +13B 13A +21B 21A +34B -------- 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.
What number squared is equal to itself plus one?
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.
For instance, Book 1, Proposition 10 to find the exact centre of any line AB
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 1gEuclid 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 1g) then the definition means that
Notice that earlier we defined Phi2 as Phi+1 and here we have g2 = 1g or g2+g=1.
We can solve this in the same way that we found Phi above:
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 Phi1 (and the other value is merely Phi).
Both have identical fractional parts after the decimal point. Both are also irrational which means that
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.. .
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:
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.
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 fractionLook again at the last equation:
Phi = 1 + 1/PhiThis 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
Explores the relationship between health and wealth with some very interesting conclusions..... not what you would expect