The Golden Ratio
The golden ratio is a unique number approximately equal to 1.6180339887498948482. The Greek letter Phi (Φ) is used to refer to this ratio. The exact value for the golden ratio is the following:

`
A popular example of the application of the golden ratio is the Golden Rectangle. Interestingly enough, many artists and architects have proportioned their works to apply the golden ratio in the form of the golden rectangle. A golden rectangle is a rectangle where the ratio of the longer side (length) to the shorter side (width) is the golden ratio If one side of a golden rectangle is N ft. long, the other side will be approximately equal to N(1.62) or N(Φ). One interesting attribute about the golden rectangle is that if you cut a square off it so that what remains is a rectangle, the remaining rectangle will also have the length to width properties of the golden ratio, therefore making it another golden rectangle. What happens is that if you keep cutting squares off, each time you get a smaller and smaller golden rectangle. Leonardo Da Vinci, the famous mathematician and artist from the Renaissance, featured the golden ratio in many of his paintings. For example, lets take a look at the world famous "Mona Lisa". If a rectangle were drawn around her face, the measurements would be that of a Golden Rectangle.

...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: Natures Beautiful Proportion
At first glance of the title, many may wonder: What is the GoldenRatio? There are many names the GoldenRatio has been called including the Golden Angle, the Golden Section, the Divine Proportion, the Golden Cut, the Golden Number et cetera, but what is it and how is it useful for society today? One may have heard of the number π (Pi 3.14159265…) but less common is π’s cousin Φ (Phi 1.61803399…). Both Φ and π are irrational numbers, meaning they are numbers that cannot be expressed as a ratio of two whole numbers as well as the fact that they are never-ending, never-repeating numbers. The GoldenRatio is the ratio of 1:Φ (1.61803399…). The GoldenRatio is a surprising ratio that is based on the research of many composite mathematicians spanning over 2300 years, and it is found in many areas of everyday life including art, architecture, beauty and nature.
Euclid, a Greek mathematician who taught in Alexandria around 300 B.C., was one of the first to discover and record the bases for the GoldenRatio back 2300 years ago. Euclid’s discovery was that if one takes a line and divide it into two unequal sections in such a way that the longer section is the same...

...has never been deciphered. Until, the discovery of a natural ratio, that changed the game of beauty in all aspects. “Many of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kelper, to present day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties, called the goldenratio”(Livio, 2007). This can be located in thousands of natural and manmade objects, and is believed to hold the key to the secret of beauty. Still, no one knows who first discovered the goldenratio, but it is known that the Egyptians used it in the creation of the Great Pyramids at Giza. In addition, Phidias applied the goldenratio to the design of the Parthenon (Livio, 2007). These two marvelous structures were the first known cases of where the goldenratio had been used. The goldenratio started to make its mark on the world when it was first applied to the arts.
Modern artists do not receive the fame or glory from their artwork, and some of the artists are forgotten. However, the artist of the renaissance period are still praised and remembered well after their time, for their work. Their works of art whether it be a...

...rederick smith
The GoldenRatio
March 31 2011
1. The introduction:
Hello my name is Frederick Smith, I will be speaking you about a fascinating thing that is in everything, it’s a part of you, it created you & its not just in you, its all around you. Its also in all plants and in all animals. Take for example an octopus has eight tentacles hence the name “octo’~pus, each one of its tentacles has the exact number of suckers on it and each tentacle is the same length pretty amazing right... (Pause for a break…) and the intricate design on a butterfly. One wing as the exact pattern as the other side in the exact spot adjacent to its counterpart (the other wing) It is exactly the same on one side as it is on the other? Or How does a seashell create a perfect spiral? so how does all this happen… (Another pause…)
2. Thesis statement
In nature there is something not visible bi the untrained eye. It happens because there is something in nature called the goldenratio. (Say softly & clearly…). Think of goldenratio as natures secret un~seen Architect! Although I am not a fan of mathematics, it’s in everything around you
(Pause for break, let them think about it for a second)
Have any of you heard about the goldenratio before?
Other names frequently used for the goldenratio are the golden section and...

...previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence can be defined as the following recursive function:
Fn=un-1+ un-2
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=F1-F0F2=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 x2-x-1=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 goldenratio 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...

...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
The theory of the Italian mathematician Leonardo Pisano is extremely present today. While he was trying to sort out the number of rabbits that mated in a year, he discovered a series of numbers, that are profoundly consistent in man, nature & animals. This discovery was extraordinary, but he also found that the ratio always resulted in 1.618. Although it is called differently, this ratio is often called „the goldenratio“. It's marked with the Greek letter phi. It's just amazing how we've used it to create beauty in art & architecture, today you may find the goldenratio in everydays objects such as tables, couches, doors,posters, books and etc.
Because it is very pleasing to the eye, the goldenratio is used alot in art. Leonardo da Vinci used the goldenratio in many paintings including The Vetruvian Man"(The Man in Action)" The Annuncation, The Mona Lisa, St. Jerome, Micahelangelo in Holy Family, Raphael in Crucifixion, Rembrandt in the self-portrait by and other art works. The goldenratio was especially used in the Renaissance and by the greeks and the romans. Various important proportions of Michelangelo’s amazing sculpture, David, are carved in the GoldenRatio...

...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...