The Golden Ratio

(Difference between revisions)
 Revision as of 10:52, 17 May 2012 (edit)← Previous diff Current revision (03:00, 12 December 2012) (edit) (undo) (67 intermediate revisions not shown.) Line 1: Line 1: - {{Image Description + {{Image Description Ready |ImageName=The Golden Ratio |ImageName=The Golden Ratio - |Image=180px-Pentagram-phi.svg.png + |Image=Goldenratio.jpg - |ImageIntro=This pentagram is designed using two isosceles triangles that exhibit the golden ratio. One triangle has a base of length 1 and legs of length \varphi, while the other triangle has a base of length \varphi and legs of length 1. . The term ''golden ratio'' refers to the ratio 1: golden number. The golden number, often referred to as ''phi'' is numerically equal to $\frac{1 + \sqrt{5}}{2} \approx 1.61803399 \dots =\varphi$. + |ImageIntro=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 (letter)|phi]] ($\varphi$) represents the golden ratio. Its value is: + ${\varphi}=\frac{1 + \sqrt{5}}{2} = 1.61803399...$.
+ 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
- ==Music== - *Bartok/other composers use the Fibonacci sequence in their work. - *The golden ratio is even found in the makings of the scroll of the violin. - [[Image:Image19.gif|200px|]] [http://jwilson.coe.uga.edu/emt669/student.folders/frietag.mark/homepage/goldenratio/goldenratio.html "Golden Ratio"], Retrieved on 12 May 2012.[[Image:Violin-scrolls.jpg|200px|]] [http://hu.wikipedia.org/wiki/Fájl:Violin-scrolls.jpg "Violin-Scrolls.jpg"], Retrieved on 12 May 2012. + ==The Golden Ratio in Nature== + ===Spirals & Phyllotaxis=== + [[Image:Sunflower head.jpeg|px200]] - ==Nature== + 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. - *Fibonacci (or Golden) spirals are seen in many elements of nature including plants, waves, and outer space. + [[Image:Pinecone3.gif]] - [[Image:SpiralsInNature 2.jpg|150px]] + - [http://fredfields.blogspot.com/2010/08/using-golden-rectangle.html "Spirals in Nature"], Retrieved on 16 May 2012. + - ==Art and Architecture== + 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. - *Salvador Dalí uses the golden ratio in his painting, 'The Sacrament of the Last Supper' + - [[Image:SacramentoftheLastSupper.jpg|300px]] + - *Leonardo di Vinci's 'Mona Lisa' is also said to exhibit the golden ratio + - [[Image:Golden-ratio.jpg|200px]] + - [http://math-fail.com/2010/02/fibonacci-spiral.html "Mona Lisa"], Retrieved on 16 May 2012. + - ===Misconceptions about the Golden Ratio=== + |AuthorName=Joyce Han - 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?''==== + - [[Image:Golden ratio parthenon.jpg|300px]] + - [http://lotsasplainin.blogspot.com/2008/01/wednesday-math-vol-8-phi-golden-ratio.html "Parthenon"], Retrieved on 16 May 2012. + - + - + - ==Math== + - *Triangles + - [[Image:Goldenrhombs1.GIF|400px|center]] + - + - The Golden Ratio $\varphi$ is used to construct the golden triangle, an isoceles triangle that has legs of length $\varphi$ and base length of 1. It is above and to the left. Similarly, the golden gnomon has base ${\varphi}$ and legs of length 1. It is shown above and to the right. + - + - [[Image:180px-Pentagram-phi.svg.png]] + - + - These triangles can be used to from a pentagram, which has several golden ratio proportions. + - + - :$\frac{\mathrm{red} }{\mathrm{green} } = \frac{\mathrm{green} }{\mathrm{blue} } = \frac{\mathrm{blue} }{\mathrm{magenta} } = \varphi .$ + - + - [[Image:Penrose-4.jpg|250px]] [[Image:Penrose-21.jpg|250px]] + - + - These triangles can be used to form fractals and are one of the only ways to tile a plane using pentagonal symmetry. + - |ImageDesc==Mathematical Representations of the Golden Ratio= + - + - ==A Geometric Representation== + - [[Image:2302.JPG|300px]] + - + - From a geometric sense, the golden ratio can be defined using a line segment divided into two sections, of lengths a and b, respectively. If a and b are appropriately chosen, the ratio of b to a is the same as the ratio of a + b to b. + - + - The value of this ratio, denoted $\varphi$, turns out not to depend on the particular values of a and b, as long as they satisfy the proportion above. + - + - + - + - + - ==An Algebraic Representation== + - 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 + - :$r=\frac{a+b}{a}=1+\frac{b}{a} =1+\cfrac{1}{a/b}=1+\frac{1}{r}$. + - + - :$r=1+\frac{1}{r}$ + - + - 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 $r = \frac{1 \pm \sqrt{5}} {2}$. + - + - Because the ratio has to be a positive value, + - + - :$r=\frac{1 + \sqrt{5}}{2} \approx 1.61803399 \dots =\varphi$. + - + - + - + - ==Continued Fraction Representation and [[Fibonacci sequence|Fibonacci Sequences]]== + - The golden ratio can also be written as what is called a '''continued fraction''' by using recursion. + - + - {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText= |FullText=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 + - + - $\varphi -1= \cfrac{1}{\varphi }$. + - + - Solving for $\varphi$ gives + - + - $\varphi =1+ \cfrac{1}{\varphi }$. + - + - Now use recursion and 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, + - + - $\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\varphi}}}$ + - + - + - + - $\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\cdots}}}$ + - + - 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 $\varphi$ by 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. + - + - In fact, we can prove this relationship using mathematical [[Induction]]. + - + - {{Switch|link1=Click to show proof|link2=Click to hide proof|1= |2= + - + - 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 $f_n=f_{n-1}+f_{n-2}$ where $f_1=1$,$f_2=1$, and so $f_3=1+1=2$, $f_4=1+2=3$ and so on. + -
+ - + - We want to show that $x_n=\frac{f_{n+1}}{f_n}$ for all n. + - + - First, we establish our [[Induction|base case]]. We see that $x_1=1=\frac{1}{1}=\frac{f_2}{f_1}$, and so the relationship holds for the base case. + - + - Now we assume that $x_k=\frac{f_{k+1}}{f_{k}}$ for some $1 \leq k < n$ (This step is the [[Induction|inductive hypothesis]]). We will show that this implies that $x_{k+1}=\frac{f_{(k+1)+1}}{f_{k+1}}=\frac{f_{k+2}}{f_{k+1}}$. + - + -

+ - + - By our definition of $x_n$, we have + - + - $x_{k+1}=1+\frac{1}{x_k}$. + - + - By our inductive hypothesis, this is equivalent to + - + - $x_{k+1}=1+\frac{1}{\frac{f_{k+1}}{f_{k}}}$. + - + - Now we only need to complete some simple algebra to see + - + - $x_{k+1}=1+\frac{f_k}{f_{k+1}}$ + - + - $x_{k+1}=\frac{f_{k+1}+f_k}{f_{k+1}}$ + - + - Noting the definition of $f_n=f_{n-1}+f_{n-2}$, we see that we have + - + - $x_{k+1}=\frac{f_{k+2}}{f_{k+1}}$ + - + - Since that was what we wanted to show, we see that the terms in our continued fraction are represented by ratios of Fibonacci numbers. + - + - The exact continued fraction is $x_{\infty} = \lim_{n\rightarrow \infty}\frac{f_{n+1}}{f_n} =\varphi$. + - + - }}|NumChars=75}} + - + - + - + - ==Proof of the Golden Ratio's Irrationality== + - + - {{SwitchPreview|ShowMessage=Click to expand|HideMessage=Click to hide|PreviewText= |FullText= + - 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. Writing this all the way out 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 $(b-a)$ and $a$ gives us that + - + - $\frac{b}{a}=\frac{a}{b-a}$. + - + - Since $\varphi=\frac{b}{a}$, we can see that $\varphi=\frac{a}{b-a}$. + - + - Since we have assumed that a and b are integers, we know that b-a must also be an integer. Furthermore, since [itex] a - |ToDo=-animation |InProgress=Yes |InProgress=Yes }} }}

Current revision

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.

Basic Description

==The Golden Ratio as an Irrational Number==

The Golden Ratio in the Arts

Parthenon

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

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.




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.

Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

Have questions about the image or the explanations on this page?