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.
Using a piece of graph paper, draw a spiral using the Fibonacci series. Starting in the center of the page, draw a 1 X 1 square, next to it draw another 1 X 1 square,
After, draw 2 X 2 squares touching the last two squares,
Then continue to add on squares until the graph paper is filled. To finish the spiral draw arcs (quarter circles) in each square starting in the center and working outward.
Do you notice any similarity to the spiral you have drawn and the image of the shell?
Fibonacci SeriesActivity 2
Take the Fibonacci sequence listed below and divide each pair of number and record the results in the...
...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 Goldenratio
In order to define the goldenratio 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 goldenratio.
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...
...What is the GoldenRatio?
Most people are familiar with the number Pi because it can be found in so many different math problems and equations. There is, however, another irrational number like Pi. This number isn¡¦t as well known as Pi however. This number is called Phi. This number is also called the goldenratio. The goldenratio is equal to the square root of five plus one, divided by two. If you work this out it comes out as 1.618033988749895. This is also the only number that if squared, is equal to itself plus one. Mathematically speaking, Phi^2 = Phi + 1. Also if you find the reciprocal of Phi, it is equal to itself minus one, Phi^1 = Phi ¡V 1.
The GoldenRatio is the basis for many things in nature. Even ones fingers use the GoldenRatio. First measure the length of the longest finger bone. Then measure the shorter one next to it. Finally if you divide the longer one by the shorter one, you should get a number that is close to 1.168 which is really close to the GoldenRatio. Most parts of the human body are proportional to the GoldenRatio.
The GoldenRatio can even be traced back into the times of the Romans and Pyramids. For example, the Great Pyramid of Giza, which was built in 2560 BC, is one of the earliest ways 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...
...Fibonacci sequence in arithmetic sequence
The Fibonacci sequence is a series of numbers in which each number is the sum of the previous two. It starts with 0 and 1, which equals 1. Then 1 plus 2 equals 3, 2 plus 3 equals 5, and so on.
n mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation
With seed values[1]
The Fibonacci numbers are represented practically everywhere. In the petals on a flower, or the arrangement of leaves along a stem, you will find this sequence of numbers. The petals on most flowers display one of the Fibonacci numbers. The numbers also appear in certain parts of sea shell formations. Parts of the human body also reveal these ratios, including the five fingers, and a thumb on each hand. Fibonacci also can be seen in a piano that produces harmony through a beautiful music. A piano has one keyboard with five black keys (sharps and flats) arranged in groups of two and three, and eight white keys (whole tones) for the 13 chromatic musical octaves.
The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.
IN NATURE
A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.[52] This has the form
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie...
...wayfaring.info
Source: www.wayfaring.info
BASED ON GOLDENRATIO:
A golden rectangle is a rectangle whose side lengths are in the goldenratio, onetophi, that is, approximately 1:1.618. A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is, with the same proportions as the first. Square removal can be repeated infinitely, which leads to an approximation of the golden or Fibonacci spiral.
Source: www.archgeom.blogspot.com
Source: www.archgeom.blogspot.com
Taj Complex is basically constructed by using the unit called angulams, where each angulam is measures 1.763 cm. and 12 angulams mean 1 Vitasti. Units of measure mentioned in the Arthasastra.
Taj Complex is basically constructed by using the unit called angulams, where each angulam is measures 1.763 cm. and 12 angulams mean 1 Vitasti. Units of measure mentioned in the Arthasastra.
The plan of Taj Mahal is an example of a perfect square specially plan of the mausoleum of Taj complex. The mausoleum’s design is basically the overlapping of golden rectangles. Here, red, white, yellow and blue are the individual golden rectangles. Based on the traditional method of calculating dimension of each rectangle in the figure 180v is the length of all sides which assumed as the perfect square that means 1,...
...GOLDENRATIO maths project
Index
Serial no. chapter
1  Introduction 
2  History 
3  In nature 
4  In human body 
5  In architecture 
6  In art 
7  In day to day life 
8  SIGNIFICANCE 
ACKNOWLEDGEMENT
I would like to express my special thanks of gratitude to my teacher sonali durgam on the topic goldenratio, which also helped me in doing a lot of Research and I came to know about so many new things. I am really thankful to her.
Secondly I would also like to thank my parents and friends who helped me a lot in finishing this project’s information finding work.
I am making this project not only for marks but to also increase my knowledge.
THANKS AGAIN TO ALL WHO HELPED ME
Introduction
The goldenratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes .
The designations "phi" (for the goldenratio conjugate ) and "Phi" (for the larger quantity ) are sometimes also used (Knott), although this usage is not necessarily recommended.
The term "golden section" (in German, goldener Schnitt or der goldene Schnitt) seems to first have...
...The GoldenRatio
The goldenratio is a number used in mathematics, art, architecture, nature, and architecture. Also known as, the divine proportion, golden mean, or golden section it expresses the relationship that the sum of two quantities is to the larger quantity as is the larger is to the smaller. It is also a number often encountered when taking the ratios of differences in different geometric figures.
Represented mathematically as approximately 1.618033989, and by the Greek letter Phi, the number tends to show up frequently in geometrical shapes. For example, the goldenratio is the basis for the construction of a pentagram. This shape looks like a regular star; five straight lines form a star with five points. The pentagon within the star in the center is proportional to the points of the star by a ratio of 1: 1.618.
The goldenratio appeared so much in Geometry, as stated above with the pentagram example, that it intrigued the Ancient Greeks. They studied the ratio for most of the same reasons mathematicians study it today. They found it to have unique and interesting properties. It is said that the Parthenon, among other Greek architecture have many proportions approximate to the goldenratio. Other classical buildings and structures have been...
...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...
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