The Golden Ratio
From Math Images
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- | 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 | + | 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 φ. The line segment above (left) exhibits the golden proportion. The line segments above (right) are also examples of the golden ratio. In each case, |
<math>\frac{{\color{Red}\mathrm{red}}+\color{Blue}\mathrm{blue}}{{\color{Blue}\mathrm{blue}} }= \frac{{\color{Blue}\mathrm{blue}} }{{\color{Red}\mathrm{red}} }= \varphi . </math> | <math>\frac{{\color{Red}\mathrm{red}}+\color{Blue}\mathrm{blue}}{{\color{Blue}\mathrm{blue}} }= \frac{{\color{Blue}\mathrm{blue}} }{{\color{Red}\mathrm{red}} }= \varphi . </math> | ||
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[[Image:1byrrectangle1.jpg|500px]][[Image:Pentagon_final.jpg|300px]] | [[Image:1byrrectangle1.jpg|500px]][[Image:Pentagon_final.jpg|300px]] | ||
- | The golden number, | + | The golden number, φ, is used to construct the '''golden triangle,''' an isoceles triangle that has legs of length <math>\varphi \times r</math> and base length of <math>1 \times r</math> where <math>r</math> can be any constant. It is above and to the left. Similarly, the '''golden gnomon''' has base <math>\varphi \times r</math> and legs of length <math>1 \times r</math>. It is shown above and to the right. These triangles can be used to form regular pentagons (pictured above) and <balloon title="A pentagram is a five pointed star made with 5 straight strokes">pentagrams.</balloon> |
The pentgram below, generated by the golden triangle and the golden gnomon, has many side lengths proportioned in the golden ratio. | The pentgram below, generated by the golden triangle and the golden gnomon, has many side lengths proportioned in the golden ratio. | ||
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- | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText=How can we derive the value of | + | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText=How can we derive the value of φ from its characteristics as a ratio? We may algebraically solve for the ratio (φ) by observing that ratio satisfies the following property by definition: |
:<math>\frac{b}{a} = \frac{a+b}{b} = \varphi</math>|FullText= | :<math>\frac{b}{a} = \frac{a+b}{b} = \varphi</math>|FullText= | ||
Let <math> r </math> denote the ratio : | Let <math> r </math> denote the ratio : | ||
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The golden ratio can also be written as what is called a '''continued fraction,'''a fraction of infinite length whose denominator is a quantity plus a fraction, which latter fraction has a similar denominator, and so on. This is done by using <balloon title="Recursion is the method of substituting an equation into itself">recursion</balloon>. | The golden ratio can also be written as what is called a '''continued fraction,'''a fraction of infinite length whose denominator is a quantity plus a fraction, which latter fraction has a similar denominator, and so on. This is done by using <balloon title="Recursion is the method of substituting an equation into itself">recursion</balloon>. | ||
- | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText= |FullText=We have already solved for | + | {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText= |FullText=We have already solved for φ using the following equation: |
<math>{\varphi}^2-{\varphi}-1=0</math>. | <math>{\varphi}^2-{\varphi}-1=0</math>. | ||
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<math>\varphi = 1.61803399...\,</math> | <math>\varphi = 1.61803399...\,</math> | ||
- | As you go farther along in the Fibonacci sequence, the ratio between the consecutive terms approaches the golden ratio. Many real world applications of the golden ratio are related to the Fibonacci sequence. For more real-world applications of the golden ratio [[Fibonacci Numbers|click here!]] | + | As you go farther along in the Fibonacci sequence, the ratio between the consecutive terms approaches the golden ratio. Many real-world applications of the golden ratio are related to the Fibonacci sequence. For more real-world applications of the golden ratio [[Fibonacci Numbers|click here!]] |
- | In fact, we can prove | + | In fact, we can prove that the ratio between terms in the Fibonacci sequence approaches the golden ratio by using mathematical [[Induction]]. |
{{Switch|link1=Click to show proof|link2=Click to hide proof|1= |2= | {{Switch|link1=Click to show proof|link2=Click to hide proof|1= |2= | ||
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we only need to show that each of the terms in the continued fraction is the ratio of Fibonacci numbers as shown above. | we only need to show that each of the terms in the continued fraction is the ratio of Fibonacci numbers as shown above. | ||
- | First, let <math> x_1=1</math>, <math> x_2=1+\frac{1}{1}=1+\frac{1}{x_1} </math>, <math> x_3= 1+\frac{1}{1+\frac{1}{1}}=1+\frac{1}{x_2} </math> and so on so that <math> x_n=1+\frac{1}{x_{n-1}} </math>. | + | First, let |
+ | :<math> x_1=1</math>, | ||
+ | :<math> x_2=1+\frac{1}{1}=1+\frac{1}{x_1} </math>, | ||
+ | :<math> x_3= 1+\frac{1}{1+\frac{1}{1}}=1+\frac{1}{x_2} </math> and so on so that | ||
+ | :<math> x_n=1+\frac{1}{x_{n-1}} </math>. | ||
- | These are just the same truncated terms as listed above. Let's also denote the terms of the Fibonacci sequence as <math> | + | These are just the same truncated terms as listed above. Let's also denote the terms of the Fibonacci sequence as |
+ | :<math> s_n=s_{n-1}+s_{n-2} </math> where <math>s_1=1</math>,<math>s_2=1</math> | ||
<br> | <br> | ||
- | We want to show that <math> x_n=\frac{ | + | We want to show that |
+ | :<math> x_n=\frac{s_{n+1}}{s_n} </math> for all n. | ||
- | First, we establish our [[Induction|base case]]. We see that <math> x_1=1=\frac{1}{1}=\frac{ | + | First, we establish our [[Induction|base case]]. We see that |
+ | :<math> x_1=1=\frac{1}{1}=\frac{s_2}{s_1} </math>, and so the relationship holds for the base case. | ||
- | Now we assume that <math> x_k=\frac{ | + | Now we assume that |
+ | :<math> x_k=\frac{s_{k+1}}{s_{k}} </math> for some <math> 1 \leq k < n </math> (This step is the [[Induction|inductive hypothesis]]). We will show that this implies that | ||
+ | :<math> x_{k+1}=\frac{s_{(k+1)+1}}{s_{k+1}}=\frac{s_{k+2}}{s_{k+1}} </math>. | ||
<br><br> | <br><br> | ||
- | By our definition of < | + | By our definition of ''x<sub>n</sub>'', we have |
- | <math> x_{k+1}=1+\frac{1}{x_k} </math>. | + | :<math> x_{k+1}=1+\frac{1}{x_k} </math>. |
By our inductive hypothesis, this is equivalent to | By our inductive hypothesis, this is equivalent to | ||
- | <math>x_{k+1}=1+\frac{1}{\frac{ | + | :<math>x_{k+1}=1+\frac{1}{\frac{s_{k+1}}{s_{k}}}</math>. |
Now we only need to complete some simple algebra to see | Now we only need to complete some simple algebra to see | ||
- | <math> x_{k+1}=1+\frac{ | + | :<math> x_{k+1}=1+\frac{s_k}{s_{k+1}} </math> |
- | <math> x_{k+1}=\frac{ | + | :<math> x_{k+1}=\frac{s_{k+1}+s_k}{s_{k+1}} </math> |
- | Noting the definition of <math> | + | Noting the definition of <math>s_n=s_{n-1}+s_{n-2}</math>, we see that we have |
<math> x_{k+1}=\frac{f_{k+2}}{f_{k+1}} </math> | <math> x_{k+1}=\frac{f_{k+2}}{f_{k+1}} </math> | ||
- | + | So by the principle of mathematical induction, we have shown that the terms in our continued fraction are represented by ratios of consecutive Fibonacci numbers. | |
- | The exact continued fraction is <math> x_{\infty} = \lim_{n\rightarrow \infty}\frac{f_{n+1}}{f_n} =\varphi </math>. | + | The exact continued fraction is |
+ | :<math> x_{\infty} = \lim_{n\rightarrow \infty}\frac{f_{n+1}}{f_n} =\varphi </math>. | ||
}}|NumChars=75}} | }}|NumChars=75}} | ||
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Now, since we know | Now, since we know | ||
- | <math> \frac{b}{a}=\frac{a+b}{b} </math> | + | :<math> \frac{b}{a}=\frac{a+b}{b} </math> |
- | we see that <math> b^2=a(a+b) </math> by cross multiplication. | + | we see that <math> b^2=a(a+b) </math> by cross multiplication. Foiling this expression gives us <math> b^2=a^2+ab </math>. |
- | Rearranging this gives us <math> b^2-ab=a^2 </math>, which is the same as <math> b(b-a)=a^2 </math>. | + | Rearranging this gives us <math> b^2-ab=a^2 </math>, which is the same as :<math> b(b-a)=a^2 </math>. |
- | Dividing both sides of the equation by | + | Dividing both sides of the equation by ''a(b-a)'' gives us |
- | <math> \frac{b}{a}=\frac{a}{b-a} </math>. | + | :<math> \frac{b}{a}=\frac{a}{b-a} </math>. |
- | Since <math> \varphi=\frac{b}{a} </math>, | + | Since <math> \varphi=\frac{b}{a} </math>, this means <math> \varphi=\frac{a}{b-a} </math>. |
Since we have assumed that a and b are integers, we know that b-a must also be an integer. Furthermore, since <math> a<b </math>, we know that <math> \frac{a}{b-a} </math> must be in lower terms than <math> \frac{b}{a} </math>. | Since we have assumed that a and b are integers, we know that b-a must also be an integer. Furthermore, since <math> a<b </math>, we know that <math> \frac{a}{b-a} </math> must be in lower terms than <math> \frac{b}{a} </math>. | ||
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+ | } | ||
+ | |Field=Algebra | ||
+ | |InProgress=No | ||
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Revision as of 10:16, 7 June 2012
The Golden Ratio |
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The Golden Ratio
- The golden number, often denoted by lowercase Greek letter "phi", is . The term golden ratio refers to the ratio : 1. The image to the right is a warped representation of dividing and subdividing a rectangle into the golden ratio. The result is 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.
Contents |
Basic Description
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. 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 uses 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. 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.Misconceptions about the Golden Ratio
In his paper, Misconceptions about the Golden Ratio, George Markowsky investigates many claims about the golden ratio appearing in man-made objects and in nature. Specifically, he claims that the golden ratio does not appear in the Parthenon or the Great Pyramids, two of the more common beliefs. He also disputes the belief that the human body exhibits the golden ratio. To read more, click here!
What do you think?
A Geometric Representation
The Golden Ratio in a Line Segment
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 φ. The line segment above (left) exhibits the golden proportion. The line segments above (right) are also examples of the golden ratio. In each case,
- The golden rectangle is made up of line segments exhibiting the golden proportion. Remarkably, when a square is cut off of the golden rectangle, the remaining rectangle also exhibits the golden proportions. This continuing pattern is visible in the golden rectangle above.
Triangles
The golden number, φ, is used to construct the golden triangle, an isoceles triangle that has legs of length and base length of where can be any constant. It is above and to the left. Similarly, the golden gnomon has base and legs of length . 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.
These triangles can be used to form fractals and are one of the only ways to tile a plane using pentagonal symmetry. Two fractal examples are shown below.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Algebra, Geometry
Mathematical Representations of the Golden Ratio
An Algebraic Derivation of Phi
How can we derive the value of φ from its characteristics as a ratio? We may algebraically solve for the ratio (φ) by observing that ratio satisfies the following property by definition:
Let denote the ratio :
- .
So
- which can be rewritten as
- thus,
Multiplying both sides by , we get
which can be written as:
- .
Applying the quadratic formula , we get .
The ratio must be positive because we can not have negative line segments or side lengths. Because the ratio has to be a positive value,
- .
Continued Fraction Representation and Fibonacci Sequences
The golden ratio can also be written as what is called a continued fraction,a fraction of infinite length whose denominator is a quantity plus a fraction, which latter fraction has a similar denominator, and so on. This is done by using recursion.
We have already solved for φ using the following equation:
.
We can add one to both sides of the equation to get
.
Factoring this gives
.
Dividing by gives us
.
Adding 1 to both sides gives
.
Substitute in the entire right side of the equation for in the bottom of the fraction.
Substituting in again,
This last infinite form is a continued fraction
If we evaluate truncations of the continued fraction by evaluating only part of the continued fraction (the finite displays above it), replacing by 1, we produce the ratios between consecutive terms in the Fibonacci sequence.
Thus we discover that the golden ratio is approximated in the Fibonacci sequence.
As you go farther along in the Fibonacci sequence, the ratio between the consecutive terms approaches the golden ratio. Many real-world applications of the golden ratio are related to the Fibonacci sequence. For more real-world applications of the golden ratio click here!
In fact, we can prove that the ratio between terms in the Fibonacci sequence approaches the golden ratio by using mathematical Induction.
,
we only need to show that each of the terms in the continued fraction is the ratio of Fibonacci numbers as shown above.
First, let
- ,
- ,
- and so on so that
- .
These are just the same truncated terms as listed above. Let's also denote the terms of the Fibonacci sequence as
- where ,
We want to show that
- for all n.
First, we establish our base case. We see that
- , and so the relationship holds for the base case.
Now we assume that
- for some (This step is the inductive hypothesis). We will show that this implies that
- .
By our definition of x_{n}, we have
- .
By our inductive hypothesis, this is equivalent to
- .
Now we only need to complete some simple algebra to see
Noting the definition of , we see that we have
So by the principle of mathematical induction, we have shown that the terms in our continued fraction are represented by ratios of consecutive Fibonacci numbers.
The exact continued fraction is
- .
Proof of the Golden Ratio's Irrationality
Remarkably, the Golden Ratio is irrational, despite the fact that we just proved that is approximated by a ratio of Fibonacci numbers. We will use the method of contradiction to prove that the golden ratio is irrational.
Suppose is rational. Then it can be written as fraction in lowest terms , where a and b are integers.
Our goal is to find a different fraction that is equal to and is in lower terms. This will be our contradiction that will show that is irrational.
First note that the definition of implies that since clearly and the two fractions must be equal.
Now, since we know
we see that by cross multiplication. Foiling this expression gives us .
Rearranging this gives us , which is the same as :.
Dividing both sides of the equation by a(b-a) gives us
- .
Since , this means .
Since we have assumed that a and b are integers, we know that b-a must also be an integer. Furthermore, since , we know that must be in lower terms than .
Since we have found a fraction of integers that is equal to , but is in lower terms than , we have a contradiction: cannot be a fraction of integers in lowest terms. Therefore cannot be expressed as a fraction of integers and is irrational.
For More Information
- Markowsky. “Misconceptions about the Golden Ratio.” College Mathematics Journal. Vol 23, No 1 (1992). pp 2-19.
Teaching Materials
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References
- ↑ "Parthenon", Retrieved on 16 May 2012.
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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