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The Golden Ratio - Math Images

The Golden Ratio

From Math Images

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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 />
|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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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. The result is [[Field:Fractals|fractal-like]]. This page explores real world applications for the golden ratio, common misconceptions about the golden ratio, and multiple derivations of the golden number.
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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.
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|ImageDescElem===The Golden Ratio as an Irrational Number==
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==The Golden Ratio in the Arts==
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|ImageDescElem=[[Image:Monalisa01.jpg|Does the Mona Lisa exhibit the golden ratio?|thumb|400px|right]]The golden number, approximately 1.618, is called golden because many geometric figures involving this ratio are often said to possess special beauty. Be that true or not, the ratio has many beautiful and surprising mathematical properties. The Greeks were aware of the golden ratio, but did not consider it particularly significant with respect to aesthetics. It was not called the "divine" proportion until the 15th century, and was not called "golden" ratio until the 18th century. <ref>[http://en.wikipedia.org/wiki/Golden_ratio "Golden ratio"], Retrieved on 20 June 2012.</ref>
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===Parthenon===
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<br /> [[Image:Finalpyramid1.jpg|Markowsky has determined the above dimensions to be incorrect.|thumb|400px|left]]
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:[[Image:ParthenonDIAG.gif]]
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Since then, it has been claimed that the golden ratio is the most aesthetically pleasing ratio, and claimed that this ratio has appeared in architecture and art throughout history. Among the most common such claims are that the Parthenon and Leonardo Da Vinci's Mona Lisa use the golden ratio. Even more esoteric claims propose that the golden ratio can be found in the human facial structure, the behavior of the stock market, and the Great Pyramids. <br />
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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.
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However, such claims have been criticized in scholarly journals as wishful thinking or sloppy mathematical analysis. Additionally, there is no solid evidence that supports the claim that the golden rectangle is the most aesthetically pleasing rectangle.<ref>[http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf "Misconceptions about the Golden Ratio"], Retrieved on 24 June 2012.</ref>
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===Vitruvian Man===
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:[[Image:Vitruvian man mixed.jpeg]]
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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.
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===Music===
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===Misconceptions about the Golden Ratio===
 
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Many rumors and misconceptions surround the golden ratio. There have been many claims that the golden ratio appears in art and architecture. In reality, many of these claims involve warped images and large margins of error. One claim is that the Great Pyramids exhibit the golden ratio in their construction. This belief is illustrated below.
 
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In his paper, ''Misconceptions about the Golden Ratio,'' George Markowsky disputes this claim, arguing that the dimensions assumed in the picture are not anywhere close to being correct. Another belief is that a series of [[The Golden Ratio#Jump2|golden rectangles]] appears in the ''Mona Lisa''.
 
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However, the placing of the golden rectangles seems arbitrary. Markowsky also disputes the belief that the human body exhibits the golden ratio. To read more, [http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf click here!]
 
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====''What do you think?''====
 
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George Markowsky argues that, like the ''Mona Lisa,'' the Parthenon does not exhibit a series of golden rectangles (discussed below). Do you think the Parthenon was designed with the golden ratio in mind or is the image below simply a stretch of the imagination?
 
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:[[Image:Golden ratio parthenon.jpg|300px]]<ref>[http://lotsasplainin.blogspot.com/2008/01/wednesday-math-vol-8-phi-golden-ratio.html "Parthenon"], Retrieved on 16 May 2012.</ref>
 
==A Geometric Representation==
==A Geometric Representation==
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==Fibonacci Numbers==
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==The Golden Ratio in Nature==
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===Spirals & Phyllotaxis===
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Revision as of 00:53, 12 December 2012


The Golden Ratio
Fields: Algebra and Geometry
Image Created By: azavez1
Website: The Math Forum

The Golden Ratio

The golden number, often denoted by lowercase Greek letter "phi", 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

Image:Vitruvian man mixed.jpeg

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

A Geometric Representation

The Golden Ratio in a Line Segment

Image:Animation2.gif


The golden number can be defined using a line segment divided into two sections of lengths a and b. If a and b are appropriately chosen, the ratio of a to b is the same as the ratio of a + b to a and both ratios are equal to \varphi. The line segment above (left) exhibits the golden proportion. The line segments above (right) are also examples of the golden ratio. In each case,

\frac{{\color{Red}\mathrm{red}}+\color{Blue}\mathrm{blue}}{{\color{Blue}\mathrm{blue}} }= \frac{{\color{Blue}\mathrm{blue}} }{{\color{Red}\mathrm{red}} }= \varphi .

The Golden Rectangle

A golden rectangle is any rectangle where the ratio between the sides is equal to phi. When the sides lengths are proportioned in the golden ratio, the rectangle is said to possess the golden proportions. A golden rectangle has sides of length \varphi \times r and 1 \times r where r can be any constant. Remarkably, when a square with side length equal to the shorter side of the rectangle is cut off from one side of the golden rectangle, the remaining rectangle also exhibits the golden proportions. This continuing pattern is visible in the golden rectangle below.

Image:Coloredfinalrectangle1.jpg


Triangles

The golden number, \varphi, is used to construct the golden triangle, an isoceles triangle that has legs of length \varphi \times r and base length of 1 \times r where r can be any constant. It is above and to the left. Similarly, the golden gnomon has base \varphi \times r and legs of length 1 \times r. It is shown above and to the right. These triangles can be used to form regular pentagons (pictured above) and pentagrams.

The pentgram below, generated by the golden triangle and the golden gnomon, has many side lengths proportioned in the golden ratio.

Image:Star1.jpg
\frac{{\color{SkyBlue}\mathrm{blue}} }{{\color{Red}\mathrm{red}} } = \frac{{\color{Red}\mathrm{red}} }{{\color{Green}\mathrm{green}} } = \frac{{\color{Green}\mathrm{green}} }{{\color{Magenta}\mathrm{pink}} } = \varphi .

These triangles can be used to form fractals and are one of the only ways to tile a plane using pentagonal symmetry. Pentagonal symmetry is best explained through example. Below, we have two fractal examples of pentagonal symmetry. Images that exhibit pentagonal symmetry have five symmetry axes. This means that we can draw five lines from the image's center, and all resulting divisions are identical.


A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra, Geometry

Fibonacci Numbers

The Golden Ratio in Nature

===Spirals & Phyllotaxis===

Fibonacci Numbers

The Golden Ratio in Nature

Spirals & Phyllotaxis




Teaching Materials

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References


Future Directions for this Page

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http://www.metaphorical.net/note/on/golden_ratio http://www.mathopenref.com/rectanglegolden.html




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