The Golden Ratio

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|ImageName=The Golden Ratio
|ImageName=The Golden Ratio
|Image=Goldenratio.jpg
|Image=Goldenratio.jpg
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|ImageIntro=The '''golden number,''' often denoted by lowercase Greek letter "phi", is <br /><math>{\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399...</math>. <br />
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|ImageIntro=The '''golden number,''' often denoted by lowercase Greek letter "phi", is <br />
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:<math> \frac{a+b}{a} = \frac{a}{b} \equiv \varphi,</math>
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where the Greek letter [[Phi (letter)|phi]] (<math>\varphi</math>) represents the golden ratio. Its value is:
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<math>{\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399...</math>. <br />
The term '''golden ratio''' refers to any ratio which has the value phi. The image to the right illustrates dividing and subdividing a rectangle into the golden ratio. This page explores how the Golden Ratio can be observed and found in the arts, mathematics, and nature.
The term '''golden ratio''' refers to any ratio which has the value phi. The image to the right illustrates dividing and subdividing a rectangle into the golden ratio. This page explores how the Golden Ratio can be observed and found in the arts, mathematics, and nature.
|ImageDescElem===The Golden Ratio as an Irrational Number==
|ImageDescElem===The Golden Ratio as an Irrational Number==
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Another instance in which the Golden ratio appears is in Leonardo Da Vinci’s drawing of the Vitruvian Man. Da Vinci’s picture of man’s body fits the approximation of the golden ratio very closely. This picture is considered to a depiction of a perfectly proportioned human body. The Golden Ratio in this picture is the distance from the naval to the top of his head, divided by the distance from the soles of the feet to the top of the head.
Another instance in which the Golden ratio appears is in Leonardo Da Vinci’s drawing of the Vitruvian Man. Da Vinci’s picture of man’s body fits the approximation of the golden ratio very closely. This picture is considered to a depiction of a perfectly proportioned human body. The Golden Ratio in this picture is the distance from the naval to the top of his head, divided by the distance from the soles of the feet to the top of the head.
===Music===
===Music===
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Some people have argued that the Golden Ratio in music produces aesthetically pleasing sounds. Major sixth and the minor sixth chords are considered to be the most pleasing intervals, and subsequently, they are the intervals related to the Golden Ratio. The standard tuning tone used is an A and it vibrates at 440 vibrations per second. A major sixth interval below that would be a C, which has a frequency of 264 vibrations per second. The ratio of the two frequencies reduces to 5/3, which is the ratio of two Fibonacci numbers. Similarly, a minor sixth can be obtained from a high C, which has a frequency of 528 vibrations per second, and an E, which is 330 vibrations per second. This ratio reduces to 8/5, which is also a ratio of two Fibonacci numbers.
:[[Image:Chromaticscales.gif|350px]]
:[[Image:Chromaticscales.gif|350px]]
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The Golden Ratio manifests in the idea that music is proportionally balanced. However, it is an area of debate, because some scholars claim that these appearances are purely coincidental, or a result of number juggling from aficionados. However, many contemporary artists have intentionally used the Golden Ratio in their pieces. Composer, mathematician and teacher Joseph Schillinger believed that music could be based entirely on mathematical formulation. He developed a System of Musical Composition in which successive notes in a melody were followed by Fibonacci intervals when counted in half steps. Half steps are the smallest intervals possible, and it is the closest note that can be played higher or lower.
 
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For example, if C were the first note in a composition, the following note would be half a step higher, because one is a Fibonacci number. So, the second note would be a D flat. Schillinger would alternate between moving up and down in intervals. Thus, the third note would be two half steps (because two is the next Fibonacci number), down from the D flat, which would be B flat. The next Fibonacci number after two is three, so the next note would be three half steps higher than the last. Three half steps higher than B flat is E flat. Schillinger believed that these notes convey the same sense of harmony as the phyllotactic ratios found in leaves.
 
 +
The Golden Ratio manifests in the idea that music is proportionally balanced. However, it is an area of debate, because some scholars claim that these appearances are purely coincidental, or a result of number juggling from aficionados. However, many contemporary artists have intentionally used the Golden Ratio in their pieces. Composer, mathematician and teacher Joseph Schillinger believed that music could be based entirely on mathematical formulation. He developed a System of Musical Composition in which successive notes in a melody were followed by Fibonacci intervals when counted in half steps. Half steps are the smallest intervals possible, and it is the closest note that can be played higher or lower.
 +
For example, if C were the first note in a composition, the following note would be half a step higher, because one is a Fibonacci number. So, the second note would be a D flat. Schillinger would alternate between moving up and down in intervals. Thus, the third note would be two half steps (because two is the next Fibonacci number), down from the D flat, which would be B flat. The next Fibonacci number after two is three, so the next note would be three half steps higher than the last. Three half steps higher than B flat is E flat. Schillinger believed that these notes convey the same sense of harmony as the phyllotactic ratios found in leaves.
|ImageDesc===Fibonacci Numbers==
|ImageDesc===Fibonacci Numbers==
One of the greatest breakthroughs regarding the Golden Ratio came when its relation to Fibonacci numbers, also known as the Fibonacci sequence, was discovered. Fibonacci numbers can also be found in the arts, and nature. The Fibonacci sequence is the series of numbers,
One of the greatest breakthroughs regarding the Golden Ratio came when its relation to Fibonacci numbers, also known as the Fibonacci sequence, was discovered. Fibonacci numbers can also be found in the arts, and nature. The Fibonacci sequence is the series of numbers,
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610/377=1.618037 <br/>
610/377=1.618037 <br/>
987/610=1.618033 <br/>
987/610=1.618033 <br/>
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==The Golden Ratio in Nature==
==The Golden Ratio in Nature==
===Spirals & Phyllotaxis===
===Spirals & Phyllotaxis===
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[[Image:Sunflower head.jpeg|px200]]
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Spirals are abundant concerning phyllotaxis, which describes the way leaves are arranged on a plant stem. In 92 percent of Norway spruce cones, the spirals were found to appear in rows five and eight rows. They appeared in rows of four and seven in six percent, and four and six in four percent. In addition, the number of right-handed spirals appears to be equal to the number of left-handed spirals. The arrangements of the spirals in these spruce cones are found as the following pairs of rows: 2/3,3/5,5/8,8/13,13/21,21/34,34/55,55/89, and 89/144. These numbers are all numbers that belong to the Fibonacci Series.
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[[Image:Pinecone3.gif]]
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A similar phenomenon can be observed in sunflower heads. A sunflower head has both clockwise and counterclockwise spirals. The numbers of the spirals in a sunflower usually depend on the size of the sunflower. However, the ratios of the spirals that usually occur are 89/55,144/89, and even 233/144. Once again, all of the numbers in these ratios are consecutive Fibonacci numbers.
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|AuthorName=Joyce Han
|AuthorName=Joyce Han
|Field=Algebra
|Field=Algebra

Current revision

Image:inprogress.png
The Golden Ratio
Fields: Algebra and Geometry
Image Created By: Joyce Han

The Golden Ratio

The golden number, often denoted by lowercase Greek letter "phi", is
 \frac{a+b}{a} = \frac{a}{b} \equiv \varphi,

where the Greek letter phi (\varphi) represents the golden ratio. Its value is:

{\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399....
The term golden ratio refers to any ratio which has the value phi. The image to the right illustrates dividing and subdividing a rectangle into the golden ratio. This page explores how the Golden Ratio can be observed and found in the arts, mathematics, and nature.


Contents

Basic Description

==The Golden Ratio as an Irrational Number==

The Golden Ratio in the Arts

Parthenon

Image:ParthenonDIAG.gif

There is an abundance of artists who consciously used the Golden Ratio from as long ago as 400 BCE. Once such example is the Greek sculptor Phidias, who built the Parthenon. The exterior dimensions of the Parthenon form the golden rectangle, and the golden rectangle can also be found in the space between the columns.

Vitruvian Man

Another instance in which the Golden ratio appears is in Leonardo Da Vinci’s drawing of the Vitruvian Man. Da Vinci’s picture of man’s body fits the approximation of the golden ratio very closely. This picture is considered to a depiction of a perfectly proportioned human body. The Golden Ratio in this picture is the distance from the naval to the top of his head, divided by the distance from the soles of the feet to the top of the head.

Music

Some people have argued that the Golden Ratio in music produces aesthetically pleasing sounds. Major sixth and the minor sixth chords are considered to be the most pleasing intervals, and subsequently, they are the intervals related to the Golden Ratio. The standard tuning tone used is an A and it vibrates at 440 vibrations per second. A major sixth interval below that would be a C, which has a frequency of 264 vibrations per second. The ratio of the two frequencies reduces to 5/3, which is the ratio of two Fibonacci numbers. Similarly, a minor sixth can be obtained from a high C, which has a frequency of 528 vibrations per second, and an E, which is 330 vibrations per second. This ratio reduces to 8/5, which is also a ratio of two Fibonacci numbers.

The Golden Ratio manifests in the idea that music is proportionally balanced. However, it is an area of debate, because some scholars claim that these appearances are purely coincidental, or a result of number juggling from aficionados. However, many contemporary artists have intentionally used the Golden Ratio in their pieces. Composer, mathematician and teacher Joseph Schillinger believed that music could be based entirely on mathematical formulation. He developed a System of Musical Composition in which successive notes in a melody were followed by Fibonacci intervals when counted in half steps. Half steps are the smallest intervals possible, and it is the closest note that can be played higher or lower. For example, if C were the first note in a composition, the following note would be half a step higher, because one is a Fibonacci number. So, the second note would be a D flat. Schillinger would alternate between moving up and down in intervals. Thus, the third note would be two half steps (because two is the next Fibonacci number), down from the D flat, which would be B flat. The next Fibonacci number after two is three, so the next note would be three half steps higher than the last. Three half steps higher than B flat is E flat. Schillinger believed that these notes convey the same sense of harmony as the phyllotactic ratios found in leaves.

A More Mathematical Explanation

Fibonacci Numbers

One of the greatest breakthroughs regarding the Golden Ratio came when its rela [...]

Fibonacci Numbers

One of the greatest breakthroughs regarding the Golden Ratio came when its relation to Fibonacci numbers, also known as the Fibonacci sequence, was discovered. Fibonacci numbers can also be found in the arts, and nature. The Fibonacci sequence is the series of numbers, 0,1,1,2,3,5,8,13,21,34,55,89,144,233… The next term in the Fibonacci sequence, starting from the third, is determined by adding the previous two terms together. The Fibonacci sequence is related to the Golden Ratio because as the sequence grows, the ratio of consecutive terms gradually approaches the Golden Ratio. For example here are the ratios of the successive numbers in the Fibonacci sequence:
1/1=1.000000
2/1=2.000000
3/2=1.500000
5/3=1.666666
8/5=1.600000
13/8=1.625000
21/13=1.615385
34/21=1.619048
55/34=1.617647
89/55=1.618182
144/89=1.717978
233/144=1.618056
377/233=1.618026
610/377=1.618037
987/610=1.618033


The Golden Ratio in Nature

Spirals & Phyllotaxis

px200

Spirals are abundant concerning phyllotaxis, which describes the way leaves are arranged on a plant stem. In 92 percent of Norway spruce cones, the spirals were found to appear in rows five and eight rows. They appeared in rows of four and seven in six percent, and four and six in four percent. In addition, the number of right-handed spirals appears to be equal to the number of left-handed spirals. The arrangements of the spirals in these spruce cones are found as the following pairs of rows: 2/3,3/5,5/8,8/13,13/21,21/34,34/55,55/89, and 89/144. These numbers are all numbers that belong to the Fibonacci Series.

 Image:Pinecone3.gif

A similar phenomenon can be observed in sunflower heads. A sunflower head has both clockwise and counterclockwise spirals. The numbers of the spirals in a sunflower usually depend on the size of the sunflower. However, the ratios of the spirals that usually occur are 89/55,144/89, and even 233/144. Once again, all of the numbers in these ratios are consecutive Fibonacci numbers.




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