The golden ratio 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 golden ratio 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 golden ratio 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 golden ratio. Other classical buildings and structures have been said to have been modeled off of the golden ratio, so it can be argued that the artists and architects who designed such things were aware of the golden ratio and used it to create an aesthetically pleasing design. However, it can be argued that such designers used their basic sense of good proportion. On the other hand, such analyses can always be questioned on the ground that the designer chooses the points from which the measurements are made and that these choices affect the proportions observed.

Personally, I think that the golden ratio is very interesting, but it does not have enough solid backing to be looked upon as a perfect proportionality between different things other than that of...

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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 ; 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, 1571-1630
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 Fibonacci...

...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...

...Golden Patterns
It is a common misconception to believe that mathematics is only found in books, written on paper, expressed as variables, functions, shapes and alphanumerical characters or as a real world application of a mathematical problem found in real life now worded and written into text. As a student progresses through grade school a student can’t help but feel no connection between the world’s nature and mathematical work, this concept however could not be further from the truth. As the student journeys through college and higher division mathematics, the student finds that mathematical patterns and sequences can be found naturally and without the need of human influence and that furthermore, patterns influence nature heavily. Patterns have always been of interest in mathematics, after all it can be said that mathematics is the science of patterns. In nature there are many simple yet elegant patterns present. Let’s consider the patterns composed by the scales on a pineapple, or the ones found on an acorn, is there a possible way to model these patterns mathematically? We can most certainly answer this question with a “yes” given that patterns have always been of interest as stated above. Furthermore, the amount of research and time that has gone into finding mathematical representations for these patterns is phenomenal and I will be using just a few of these works to give some insight on some of the most important patterns in mathematics, that is,...