The mask of perfection

[Vishesh]

The Golden Ratio has widely known as 'nature's building blocks.The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).

 

The ratio of successive pairs tends to the so-called golden section

(GS) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.

 

The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,

is well known in many different areas of mathematics and science.

However, it is quite amazing that the Fibonacci number patterns occur so frequently in nature ( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature".

 

 

Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.

 

Later, Leonardo da Vinci painted Mona Lisa's face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.

 

 

 

Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio, though there is much debate about whether he was conscious of this. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.

 

Even today, the golden ratio is in human-made objects all around us. Look at almost any Christian cross; the ratio of the vertical part to the horizontal is the golden ratio. To find a golden rectangle, you need to look no further than the credit cards in your wallet.

 

Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the golden shapes, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 article in the journal Perception, professor Christopher Green,

of York University in Toronto, discusses several experiments over the years that have shown no measurable preference for the golden rectangle, but notes that several others have provided evidence suggesting such a preference exists.

 

Regardless of the science, the golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the golden section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the golden ratio. Even a cross section of the most common form of human DNA fits nicely into a golden decagon. The golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to spark interest in the mathematical community.

 

Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have.

 

What are your veiws on this topic?

[Garry Denke]

Some CS books define the Fibonacci sequence by:

 

F[0] = F[1] = 1

F[n+2] = F[n] + F[n+1] ; (n >= 0)

 

Then the sequence begins:

 

n 0 1 2 3 4 5 6 7 8 9 10

F[n] 1 1 2 3 5 8 13 21 34 55 89

 

Mathematical books and papers shift the sequence by one element:

 

F[0] = 0 ; F[1] = 1

F[n+2] = F[n] + F[n+1] ; (n >= 0)

 

n 0 1 2 3 4 5 6 7 8 9 10

F[n] 0 1 1 2 3 5 8 13 21 34 55

 

One consequence of the mathematical definition is:

 

GCD(F[m], F[n]) = F[GCD(m, n)]

 

For example:

 

GCD(F[9], F[6]) = GCD(34, 8) = 2 = F[3] = F[GCD(9, 6)]

 

This property does not hold with the CS definition.

 

With the mathematical definition:

 

F[n] = alpha^n - beta^n / alpha - beta

where alpha = (1 + sqrt(5))/2 and beta = (1 - sqrt(5))/2

 

Using this to define F[n] for negative n gives:

 

n 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

F[n] 0 1 -1 2 -3 5 -8 13 -21 34 -55

 

Note that F[n] = 0 only for n = 0

 

The recurrence F[n+2] = F[n] + F[n+1] is valid for negative n.

 

Garry Denke

[C1ay]

We discussed phi a little here.....

[Turtle]

___We have also discussed phi in its geometric derivation from the vesica piscis:

http://hypography.com/forums/showthread.php?t=1902&highlight=Vesica+piscis

 

___I also dangle(d) the offer of a Katabatak Red Ribbon Of Reaction for the first to post the Katabatak pattern of the Fibonacci sequence in base ten; post #58 here:

http://hypography.com/forums/showthread.php?t=1343

___As I have now broached the changing of base in regard to the Katabataks, I note that the Katabatk pattern is not the same length for the Fibonacci numbers when you change the base.

[Tormod]

Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.

 

...

 

Later, Leonardo da Vinci painted Mona Lisa's face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.

 

...

 

Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio

 

...

 

Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.

 

Not to throw an ice bucket on this thread but I would like to ask for proof of these claims - especially the part about the musical pieces (how do you measure the golden ratio in music?).

 

While it is "common knowledge" that it is true, science has shown that it is not correct. According to Mario Livio, author of "The Golden Ratio: The Story of Phi, the World's Most Astonishing Number", there is not a single piece of art that perfectly matches the golden ratio.

 

The golden ratio may surely be observed in many things - if you look for it, and especially if some leeway is given to whether a perfect match is found or not.

 

The golden ratio is something that can be approximated, but not achieved, as (like Pi) phi is an irrational number. That is why it is found in nature but not in manmade objects (unless it is aimed for with high precision).

 

BTW, a credit card has the dimensions 8.5x5.4 centimeters (roughly measured with my metal ruler). This totals 13.9 centimeters. The ratio between 13.9/8.5 (ie, the ratio between the whole and it's longest section) is 1.63529.

 

The ratio between the longest section and the smallest is 8.5/5.4 = 1.5740.

 

Phi is 1.61803. That's not a very good match in either case. A credit card is not a golden rectangle.

 

For more on the golden ratio:

http://en.wikipedia.org/wiki/Golden_ratio