The Golden Ratio

• In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.
• The Golden ratio is a special ratio. Phi (lowercase) is used to refer to this ratio, and the value is , which is approximately 1.618.
  The uppercase phi () is used for the reciprocal of the golden ratio, which is 1/.
  

• There are many different ways in which the Fibonacci sequence is related to the Golden Ratio. Firstly, the further you go to the right of the sequence, the more the ratio of one term to the one before it estimates the Golden Ratio. The table below shows the first few numbers and their ratios:

First number	Second number	Ratio
0	1	-
1	1	1.0000
1	2	2.0000
2	3	1.5000
3	5	1.6667
5	8	1.6000
8	13	1.6250
13	21	1.6154
21	34	1.6190
34	55	1.6176
55	89	1.6182
89	144	1.6180

 

• The golden ratio also determines how attractive a person is.
  For example, the American pop singer and actress Jessica Simpson is attractive because the proportion of her face fits geometrically on the human face mask which conforms to some aspect of the Golden Ratio.
  

• Do you think the Golden Ratio "naturally" occurs in nature and art/architecture or do people "force" it to be there?

Obviously, nature doesn't know about the golden ratio, but for art and architecture, which are created by people, in order to look "good" and well proportioned, yes they "force" it to be there.

Facts about Fibonacci

1,1,2,3,5,8, 13, 21, 34, ...,

un represents the nth term of the Fibonacci sequence, show that

u

n =

1

+ 5 n ? 1 ? 5 n

2

., n 5n = 1, 2, 3, ...x, y, x + y, ...

Petals on flowers

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number: 

• 3 petals: lily, iris
• 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
• 8 petals: delphiniums
• 13 petals: ragwort, corn marigold, cineraria,
• 21 petals: aster, black-eyed susan, chicory
• 34 petals: plantain, pyrethrum
• 55, 89 petals: michaelmas daisies, the asteraceae family

http://www.world-mysteries.com/sci_17.htm

If

"I'm not smart, but I like to observe. Millions saw the apple fall, but Newton was the one who asked why." - Baruch

"I'm not smart, but I like to observe. Millions saw the apple fall, but Newton was the one who asked why."     - Baruch

Observation takes time, concentration and curiosity. After observing the siuation, everything becomes clear, at least more than it was. Taking the time to see, taking the time to appreciate, taking the time to think, taking the time to live, that makes evrything having more sense.
You don't have to be smart to be logical; logical mind wait to observe all the pieces of the situation, before acting.
A lot of us live their lives, as they can, without asking and understanding why life is as it is, that's why people like Newton became famous, they were able to ask the good existentials questions of life.