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 golden ratio. The golden ratio 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 Golden Ratio is the basis for many things in nature. Even ones fingers use the Golden Ratio. 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 Golden Ratio. Most parts of the human body are proportional to the Golden Ratio.

The Golden Ratio 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 Golden Ratio was used. The length of each side of the base is 756 feet while the height of the Pyramid when build was 481 feet. If you divide 756 by 481, you would get 1.5717 which is very close to the Golden Ratio.

Another good example of the Golden Ratio is in Athens Greece. The Parthenon, which was build during 440 BC, uses the Golden Ratio also. The spaces in between the columns are proportional to the Golden Ratio. This shows that the golden ratio has been used for a very long time.

Since the Golden Ratio is an irrational number, it cannot be written as a regular fraction. You could however, get a very close estimate. One of the easiest ways is using the Fibonacci numbers. The Fibonacci numbers is a sequence of...

...
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...

...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 twentieth-century 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...

...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...

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