Anatolia College
Mathematics HL investigation

The Fibonacci sequence
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence can be defined as the following recursive function: Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below: F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way: F0 F1 F2 F3 F4 F5 F6 F7
0 1 1 2 3 5 8 13
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression: x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52 Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence: φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the golden ratio we discussed before. Therefore: φ=x+1x φ=1+52+11+52 φ=1+52
Thus φ2=1+522 φ2=3+52 and F2φ+F1=1+52+1=3+52
Therefore:
φ2=F2φ+F1 We can simplify other powers of φ the same way, thus:
φ3=2+5 and φ4=35+72
In order to from a conjecture connecting φn, Fn and Fn1 we can apply the relationship we found for f2 to the other powers of f: F3φ+F2= 2+5 and F4φ+F3=35+72
By examining the last two relationships we can deduce that: φn=Fnφ+Fn1
We can prove the...
...
The Golden Number
1.61803 39887 49894 84820 is by no means a number of memorization. However, it is a recognizable one. Never will you find a combination of numbers that is more significant than this one. This ratio is known as the Golden Number, or the GoldenRatio. This mystery number has been used throughout different aspects of life, such as art, architecture, and of course, mathematics. One may wonder where theGoldenRatio came from? Who thought to discover it? When was it discovered? And how has it been used throughout time? The Goldenratio has been used throughout different aspects of life after being discovered during the ancient times.
About two to three thousand years ago, the GoldenRatio was first recognized and made use by the ancient mathematicians in Egypt. The goldenratio was introduced by its frequent use in geometry. An ancient mathematician, sculptor, and architect named Phidias, who used the goldenratio to make sculptures, discovered it. He lived from sometime around 490 to 430 BC. None of his original works exist, however he was highly spoken of by ancient writers who gave him high praise. Hegias of Athens, Agelades of Argos, and Polygnotus of Thasos were said to have trained him.
Although not much is known about Phidias’s life, he is...
...Introduction: The Fibonacci Series
The Fibonacci Series is a sequence of numbers first created by Leonardo Fibonacci (fibonachee) in 1202. It is a deceptively simple series, but its ramifications and
applications are nearly limitless. It has
fascinated and perplexed mathematicians
for over 700 years, and nearly everyone
who has worked with it has added a new
piece to the Fibonacci puzzle, a new tidbit
of information about the series and how it
works. Fibonacci mathematics is a
constantly expanding branch of number
theory, with more and more people being
Yellow flower with 8 petals, a Fibonacci
drawn into the complex subtleties of
Number.
Fibonacci's legacy.
The first two numbers in the series are one and one. To obtain each number of the
series, you simply add the two numbers that came before it. In other words, each number
of the series is the sum of the two numbers preceding it.
Note: Historically, some mathematicians have considered zero to be a Fibonacci
number, placing it before the first 1 in the series. It is known as the zeroth Fibonacci
number, and has no real practical merit. We will not consider zero to be a Fibonacci
number in our discussion of the series.
http://library.thinkquest.org/27890/mainIndex.html
Series:
(0,) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
EXAMPLE IN NATURE...
...The Discovery of the FibonacciSequence
A man named Leonardo Pisano, who was known by his nickname, "Fibonacci", and named the series after himself, first discovered the Fibonaccisequence around 1200 A.D. The Fibonaccisequence is a sequence in which each term is the sum of the 2 numbers preceding it. The first 10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbers are obviously recursive.
Fibonacci was born around 1170 in Italy, and he died around 1240 in Italy, but the exact dates of his birth and death are not known. He played an important role in reviving ancient mathematics and made significant contributions of his own. Even though he was born in Italy, he was educated in North Africa where his father held a diplomatic post. He published a book called Liber abaci, in 1202, after his return to Italy and it was in this book that the Fibonacci numbers were first discussed. It was based on bits of Arithmetic and Algebra that Fibonacci had accumulated during his travels with his father. Liber abaci introduced the HinduArabic placevalued decimal system and the use of Arabic numerals into Europe. Though people were interested, this book was somewhat controversial because it contradicted some of the foremost Roman and Grecian Mathematicians of the time, and even proved many...
...Goldenratio ; The Definition of Beauty
“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” Johannes Kepler, 15711630
The goldenratio is present in everyday Life. The golden proportion is the ratio of the shorter length to the longer length which equals the ratio of the longer length to the sum of both lengths. It can be expressed algebraicay like :
This ratio has always been considered most pleasing to the eye. It was named the goldenratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias.
The GoldenRatio is also known as the golden section, golden mean or golden rectangle. The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains, a smaller square can be removed and so on, resulting in a spiral pattern. It is a unique and important shape in mathematics which also appears in nature, music, and is often used in art and architecture.
Our human eye „sees“ the golden rectangle as a beautiful geometric form.
Leonardo...
...The GoldenRatio
By : Kaavya.K
In mathematics and the arts, two quantities are in the goldenratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The goldenratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for thegoldenratio are the golden section and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number, and mean of Phidias. The goldenratio is often denoted by the Greek letter phi, usually lower case (φ).
[pic]
The golden section is a line segment divided according to the goldenratio: The total length a + b is to the longer segment a as a is to the shorter segment b.
The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:
[pic]
This equation has one positive solution in the algebraic irrational number
[pic]
At least since the Renaissance, many artists and architects have proportioned their works to approximate the...
...The GoldenRatio
Body, art, music, architecture, nature – all connected by a simple irrational number – the GoldenRatio. According to Posamentier & Lehmann in their work The
(Fabulous) Fibonacci Numbers, there is reason to believe that the letter φ (phi) was
used because it is the first letter of the name of the celebrated Greek sculptor Phidias (490430 BCE). He produced the famous statue of Zeus in the Temple of Olympia and supervised the construction of the Parthenon in Athens Greece (Posamentier & Lehmann, 2007). In constructing this masterpiece building, Phidias used the GoldenRatio to create a masterpiece of work.
Figure 1: This is a model of Zeus in the Temple of Olympia. The red lines show the use of the GoldenRatio. (www.scarletcanvas.com/)
Phidias brought about the beginning of the one of the most universally recognized form of proportion and style used throughout history (Posamentier & Lehmann, 2007). The irrational number Phi, also known as the GoldenRatio, has had tremendous importance. To properly understand this mathematical concept, it is important to explore the definition, history, and the relations to architecture, art, music and the Fibonaccisequence.
Figure 2: This model shows the line segments in the Golden...
...GoldenRatio
In mathematics, two quantities are in the goldenratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b,
Where the Greek letter phi (φ) represents the goldenratio. Its value is:
Thegoldenratio is also called the golden section (Latin: sectio aurea) or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.
Some twentiethcentury artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the goldenratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the goldenratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). Mathematicians since Euclid have studied the properties of the goldenratio, including its appearance in the dimensions of a regular pentagon and in a golden...
...
The Fibonaccisequence
The Fibonaccisequence is a series of numbers developed by Leonardo Fibonacci as a means of solving a practical problem. The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. Suppose a newly born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair, one male, one female, every month from the second month on. The question that Fibonacci posed was how many pairs will there be in one year?
At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The Fibonaccisequence is the series of numbers, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… When squares are made with the widths, you get a nice spiral. If you look closely at the...
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