G LDEN

NUMBER

HALUK YAVUZCmpe220 Presentation

What is golden ratio?Is it a stupid irrational number like pi?Why human being likes it soo much?How can it creates a sense of beauty?How far into areas outside of mathematics it reaches?Is the design of the life based on this irrational number?

Of course I cannot answer this questions.The only thing I can do is to give some information about it and make you to think about answers of this quetions.

•GФlden number is denoted by Ф.•Ф=1/ Ф+1 so Ф^2= Ф+1 and lastly we have Ф^2- Ф-1=0•When we solve this equation roots will be

Let’s talk about its mathematical meaning and it curious properties.

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•First equals to 1.618033988... (this is regarded as Golden ratio)•Second equals to -0.618033988...(this is equal to negative of the reciprocal of the first one that is -1/ Ф.)•Since it is a root of polynomial it is not a transcendental number like π•It can be also derived from trigonometric functions likeФ =2sin(3π/10) Ф =2cos(π/5)Sqrt(3-Ф)= 2sin(π/5)•1/ Ф= Ф’ which is equal to the fractional part of the Ф, thisis called as silver ratio.

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All these pretty things are equal to Ф.

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"How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?" Does it seem familiar?

Of course, this is the famous rabbit problem from Liber Abaci, or Book of the Abacus of Leonardo of Pisa.

This situation creates a sequence: 1, 1, 2, 3, 5, 8, 13, 21...

which is called Fibonacci series.

More mathematically it is represented as 21 nnn NumNumNumAll sequences like this are called Lucas Sequences.

Fibonacci is a special Lucas sequence.

Whenever you take two subsequent numbers from any Lucas sequence and divide them, you get that magical value of Ф, or the Golden Mean.

The higher the numbers you use are, the closer to you get. Since the numbers go on forever, Ф does too. It's the most irrational number.

Lets see the result when we divide two consecutive numbers of Fibonacci sequence.

3/2 = 1.55/3 ≈ 1.6668/5 ≈ 1.6377/233 ≈ 1.618025751073

987/610 ≈ 1.6180327868852

The further up the Fibonacci series you go the closer approximation to the golden ratio you will get. This can be shown as a limit formula which is displayed below:

)()1(lim

nFibonFibo

nФ

The graphics of the ratios:

Another interesting relation

between fibonacci and phi

is Ф^n=fibo(n)*Ф+fibo(n-1)

n f(n) Ф^n

1 1 Ф= Ф+0

2 1 Ф^2=Ф+1

3 2 Ф^3=2*Ф+14 3 Ф^4=3*Ф+25 5 Ф^5=5*Ф+36 8 Ф^6=8*Ф+57 13 Ф^7=13*Ф+88 21 Ф^8=21*Ф+139 34 Ф^9=34*Ф+21

Proof of this propertycan be done by using thisФ^n= Ф^(n-1)+Ф^(n-2)

Our pretty golden rectangle

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Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells

The golden ratio is linked to this spiral as the distance from the centre of the spiral to where the spiral touches a corner of one of the squares increases in multiples of 1.618.

On the polar axis the equation of curve [r,Θ] represent a point.Also [rФ, Θ+π/2] represents a point.

When we disregard the statirng point of the curve we get the equations r= Ф^n and r=Θ^(n* π/2).

=Ф =Ф

You can do this infinitely.And allways you get the same ratio.Ф

There are such inifinitenumber of shapes that have golden ratio.

Besides its mathematical and geometric properties, actually human beings very like it in daily life.German physicist and psychologist Gustav Theodor Fechner in the 1860s conducted some experimets about the effects of the golden ratio on our senses. Fechner's experiment was simple: ten rectangles varying in their length-to-width ratios were placed in front of a subject, who was asked to select the most pleasing one. The results showed that 76% of all choices centered on the three rectangles having ratios of 1.75, 1.62, and 1.50, with a peak at the "Golden Rectangle" (with ratio 1.62). Fechner went further and measured the dimensions of thousands of rectangular-shaped objects (windows, picture frames in the museums, books in the library), and claimed (in his book Vorschule der Aesthetik) to have found the average ratio to be close to the Golden Ratio.

Which one is the best rectangle ?

Psychologist Judith Langlois of the University of Texas at Austin and her collaborators tested the idea that a facial configuration that is close to the population average is fundamental to attractiveness. Langlois digitized the faces of male and female students and mathematically averaged them, creating two-, four-, eight-, sixteen-, and thirty-two-face composites. College students were then asked to rate the individual and composite faces for attractiveness.

Langlois found that the 16- and 32-averaged faces were rated significantly higher than individual faces. Langlois explained her findings as being broadly based on natural selection (physical characteristics close to the mean having been selected during the course of evolution), and on "prototype theory" (prototypes being preferred over non-prototypes).

Science writer Eric Haseltine claimed (in an article in Discover magazine in September 2002) to have found that the distance from the chin to the eyebrows in Langlois's 32-composite faces divides the face in a Golden Ratio. A similar claim was made in 1994 by orthodontist Mark Lowey, then at University College Hospital in London. Lowey made detailed measurements of fashion models' faces. He asserted that the reason we classify certain people as beautiful is because they come closer to Golden Ratio proportions in the face than the rest of the population.

Many disagree with both Langlois's and Lowey's conclusions. Psychologist David Perret of the University of St. Andrews, for example, published in 1994 the results of a study that showed that individual attractive faces were preferred to the composites. Furthermore, when computers were used to exaggerate the shape differences away from the average, those too were preferred. Perret claimed to have found that his beautiful faces did have something in common: higher cheek bones, a thinner jaw, and larger eyes relative to the size of the face.

An even larger departure from the "averageness" hypothesis was found in a study by Alfred Linney from the Maxillo Facial Unit at University College Hospital. Using lasers to make precise measurements of the faces of top models, Linney and his colleagues found that the facial features of the models were just as varied as those in the rest of the population.

“I will certainly not attempt to make the ultimate sense of sex appeal in an article on the Golden Ratio. I would like to point out, however, that the human face provides us with hundreds of lengths to choose from. If you have the patience to juggle and manipulate the numbers in various ways, you are bound to come up with some ratios that are equal to the Golden Ratio. “

Mario Livio University of Cambridge.

The golden ratio meter. You can do the best by using this meter.

Can you see the white and ligth blue lines.These also have the same ratio: Ф

Phi appears in the Solar System and The Universe

Preapared by HALUK YAVUZ

Bibliography

http://library.thinkquest.org/C005449/home.html

http://www.friesian.com/golden.htm

http://goldennumber.net/