The Golden Ratio

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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 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. [1]


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.
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.[2]


Misconceptions about the Golden Ratio

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 to the right.

Markowsky has determined the above dimensions to be incorrect.
Markowsky has determined the above dimensions to be incorrect.
Does the Mona Lisa exhibit the golden ratio?
Does the Mona Lisa exhibit the golden ratio?

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 golden rectangles appears in the Mona Lisa. 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, click here!

What do you think?

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?

[3]

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

An Algebraic Derivation of Phi

How can we derive the value of \varphi from its characteristics as a ratio? We may algebraically solve for the ratio (\varphi) by observing that ratio satisfies the following property by definition:

\frac{b}{a} = \frac{a+b}{b} = \varphi

Let  r denote the ratio :

r=\frac{a}{b}=\frac{a+b}{a}.

So

r=\frac{a+b}{a}=1+\frac{b}{a} which can be rewritten as
1+\cfrac{1}{a/b}=1+\frac{1}{r} thus,
r=1+\frac{1}{r}

Multiplying both sides by r, we get

 {r}^2=r+1

which can be written as:

r^2 - r - 1 = 0 .

Applying the quadratic formula , we get r = \frac{1 \pm \sqrt{5}} {2}.

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,

r=\frac{1 + \sqrt{5}}{2} = 1.61803399... =\varphi.




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 \varphi using the following equation:

{\varphi}^2-{\varphi}-1=0.

We can add one to both sides of the equation to get

{\varphi}^2-{\varphi}=1.

Factoring this gives

 \varphi(\varphi-1)=1 .

Dividing by \varphi gives us

\varphi -1= \cfrac{1}{\varphi }.

Adding 1 to both sides gives

\varphi =1+ \cfrac{1}{\varphi }.

Substitute in the entire right side of the equation for  \varphi in the bottom of the fraction.

\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{\varphi } }

Substituting in again,

\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\varphi}}}


\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\cdots}}}

Continuing this substitution forever, the last infinite form is a continued fraction

If we evaluate truncations of the continued fraction by evaluating only part of the continued fraction, replacing \varphi with 1, we produce the ratios between consecutive terms in the Fibonacci sequence.

\varphi \approx 1 + \cfrac{1}{1} = 2

\varphi \approx 1 + \cfrac{1}{1+\cfrac{1}{1}} = 3/2

\varphi \approx 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1} } } = 5/3

\varphi \approx 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1+\cfrac{1}{1}}}} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{2}}} =1 + \cfrac{1}{1 + \cfrac{2}{3}} = 8/5

Thus we discover that the golden ratio is approximated in the Fibonacci sequence.

1,1,2,3,5,8,13,21,34,55,89,144...\,

1/1 = 1
2/1 = 2
3/2 = 1.5
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538462...
34/21 = 1.61904762...
55/34 = 1.61764706...
89/55 = 1.61818182...


\varphi = 1.61803399...\,

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.

Since we have already shown that

 \varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\cdots}}} ,

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

 x_1=1,
 x_2=1+\frac{1}{1}=1+\frac{1}{x_1} ,
 x_3= 1+\frac{1}{1+\frac{1}{1}}=1+\frac{1}{x_2} and so on so that
 x_n=1+\frac{1}{x_{n-1}} .

These are just the same truncated terms as listed above. Let's also denote the terms of the Fibonacci sequence as

 s_n=s_{n-1}+s_{n-2} where s_1=1,s_2=1,s_3=2,s_4=3 etc.


We want to show that

 x_n=\frac{s_{n+1}}{s_n} for all n.

First, we establish our base case. We see that

 x_1=1=\frac{1}{1}=\frac{s_2}{s_1} , and so the relationship holds for the base case.

Now we assume that

 x_k=\frac{s_{k+1}}{s_{k}} for some  1 \leq k < n (This step is the inductive hypothesis). We will show that this implies that
 x_{k+1}=\frac{s_{(k+1)+1}}{s_{k+1}}=\frac{s_{k+2}}{s_{k+1}} .



By our assumptions about x1,x2...xn, we have

 x_{k+1}=1+\frac{1}{x_k} .

By our inductive hypothesis, this is equivalent to

x_{k+1}=1+\frac{1}{\frac{s_{k+1}}{s_{k}}}.

Now we only need to complete some simple algebra to see

 x_{k+1}=1+\frac{s_k}{s_{k+1}}
 x_{k+1}=\frac{s_{k+1}+s_k}{s_{k+1}}

Noting the definition of s_n=s_{n-1}+s_{n-2}, we see that we have

 x_{k+1}=\frac{s_{k+2}}{s_{k+1}}

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

 x_{\infty} = \lim_{n\rightarrow \infty}\frac{s_{n+1}}{s_n} =\varphi .



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 \varphi is rational. Then it can be written as fraction in lowest terms  \varphi = b/a, where a and b are integers.

Our goal is to find a different fraction that is equal to  \varphi and is in lower terms. This will be our contradiction that will show that  \varphi is irrational.

First note that the definition of  \varphi = \frac{b}{a}=\frac{a+b}{b} implies that  b > a since clearly  b+a>b and the two fractions must be equal.


Now, since we know

 \frac{b}{a}=\frac{a+b}{b}

we see that  b^2=a(a+b) by cross multiplication. Foiling this expression gives us  b^2=a^2+ab .

Rearranging this gives us  b^2-ab=a^2 , which is the same as : b(b-a)=a^2 .

Dividing both sides of the equation by a(b-a) gives us

 \frac{b}{a}=\frac{a}{b-a} .

Since  \varphi=\frac{b}{a} , this means  \varphi=\frac{a}{b-a} .

Since we have assumed that a and b are integers, we know that b-a must also be an integer. Furthermore, since  a<b , we know that  \frac{a}{b-a} must be in lower terms than  \frac{b}{a} .

Since we have found a fraction of integers that is equal to  \varphi , but is in lower terms than  \frac{b}{a} , we have a contradiction:  \frac{b}{a} cannot be a fraction of integers in lowest terms. Therefore  \varphi 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.




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References

  1. "Golden ratio", Retrieved on 20 June 2012.
  2. "Misconceptions about the Golden Ratio", Retrieved on 24 June 2012.
  3. "Parthenon", Retrieved on 16 May 2012.

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




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