Golden Ratio

 Golden Ratio, Divine Ratio ((√5+1)/2)

There are a very few numbers which are as fascinating and intriguing as Golden Ratio. Mathematicians, Scientists, Engineers, Architects, Philosophers, Biologists, Artists, Musicians, Historians, Psychologists, numerologists and even Mystics have all been charmed by this number for more than 2000 years.

Before delving deep into the characteristics, properties, ubiquity and profundity of the number let us first examine what this number is all about.

What is Golden Ratio?

Let us take a line AB. Suppose if we choose a point C on the line AB such that   AC: CB = AB: AC.

Let AC =a, CB=b and AB=a+b. Here a>b.

A___________a_____________C________b_______B

   

    

Now a/b= (a+b)/a

Simplifying the above equation, we get

axa  = axb + bxb

a² = ab + b²

Dividing the equation by b²

We get

(a/b)²  = a/b + 1

Let us assume a/b = x .

Now ,   x²= x+1 

Solving the equation, we find the value of x = (√5+1)/2.Thus a/b= (√5+1)/2.

This ratio of a and b is known as Golden Ratio or Golden Section. This ratio is known to mankind for thousands of years. It is generally denoted by φ.

Φ = (√5+1)/2 

The approximate value of Φ is 1.6180339887…

This ratio is also known as Divine Ratio.

What is so golden/divine about this Golden/Divine Ratio? 

This is a ratio which pops up in different fields of Mathematics like Geometry, Algebra, Permutation and Combination, Number Theory, Trigonometry, Progressions etc so often that many mathematicians attached a lot of mystical properties to this number in addition to genuine mathematical properties..

Further occurrence of the number in the fields of biology, cosmology, religion, art also attracted other professionals to this widely found number in diverse fields.

The modern history of the Golden Ratio starts with Luca Pacioli (Italian Mathematician) in   1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

Over a period of time, people also started referring to it as Divine ratio considering its ubiquitous presence as a ratio in human and animal bodies, leaves and flowers, and also about galaxies among other fields as mentioned above.

It is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

Mathematical Properties of Golden Ratio:

The Golden Ratio, Φ, is an irrational number that has the following unique properties:

1. Taking the reciprocal of Φ and adding one yields Φ. phi=1/phi+1, or Φ=(1/Φ)+1.
                             Φ =    (1/Φ)+1 = 0.618…
2. Φ squared equals itself plus one. In other words, Φ² = Φ+1. These characteristics are indeed very interesting; it is the only number in the world which has such properties.
                            Φ² = Φ+1= 2.618….
3. If we convert the equation Φ² = Φ+1 into the equation Φ²-Φ-1=0 which is in the format ax² + bx + c = 0, we can solve using the quadratic equation formula, x= (-b ± √(b² - 4ac))/(2a). Solving the equation we get x = (1 ± √5)/2.
   Together, these two solutions are known as Phi (1.618033989) and phi (0.618033989). Phi and phi are reciprocals.
4. Another way of representing Golden  Ratio is                           

                                         

Solving the above expression we get the same value of Golden Ratio.

Golden Ratio and its connection with Fibonacci series:  

Leonardo Fibonacci discovered in 12^th Century, a simple Mathematical series that is the basis for a mind-boggling mathematical relationship with Golden Ratio.

Starting with 0 and 1, each new number in the series is simply the sum of the two terms before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

The ratio of each successive pair of numbers in the series approximates phi (1.618. . .), as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.

The ratios of the successive numbers in the Fibonacci series quickly converge on Phi.  After the 40th number in the series, the ratio is accurate to 15 decimal places.

1.618033988749895 . 

Mysterious Examples of Golden Ratio in Nature:

(1)    Shells: This shape, a rectangle in which the ratio of the sides a/b is equal to the Golden Ratio can result in a process that can be repeated into infinity — and which takes on the form of a spiral. It's called the logarithmic spiral that abounds in nature and is associated with Golden Ratio.

(2)    Spiral Galaxies: Spiral galaxies follow the same Fibonacci pattern which in turn is related to Golden Ratio. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees showing the ubiquity of Golden Ratio.

(3)    Hurricanes: Like Spiral Galaxies, Hurricanes also show the pattern of logarithmic spiral associated with the Golden Ratio.

(4)    Branches of trees: The Golden Ratio through Fibonacci sequence can be eerily seen in the way trees branch out or split. The main trunks of various trees grow vertically until they produce a branch, which creates two nodal points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems evincing Fibonacci pattern. A good example is the sneezewort. Root systems and algae also exhibit this pattern.

(5)    Pinecones: The seed pods of a pinecones are arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. The number of steps almost always matches a pair of consecutive Fibonacci numbers. 

(6)    Flower Petals: The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals; buttercups have five, the chicory's 21, the daisy's 34.All Fibonacci numbers.

(7)    Seed Heads of flowers: The heads of flowers also follow the Fibonacci pattern. Generally the seeds are produced at the center, and then move towards the outside to fill the vacant space in predetermined mathematical spiraling way in line with the Fibonacci series and Golden Ratio. Sunflowers provide a classic example of such logarithmic spirals.

(8)    Faces of humans and animals: Faces, both human and nonhuman, evince examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).It's worth noting that every person's body is different, but that averages across populations tend towards phi.

(9)     Fingers, Teeth, Arms, Legs: The length of our fingers, each section — from the tip of the base to the wrist — is larger than the preceding one by roughly the ratio of phi.Same relationship can be found in the teeth , arms and legs of human beings.

(10) DNA Molecules: In a recent study, it has been found that The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

(11) Reproduction of Honey Bees: Honey bees follow Fibonacci sequence in two interesting ways. The most beautiful example is by dividing the number of females in a colony by the number of males. The answer is generally very close to 1.618. Additionally, the family tree of honey bees also follows the Fibonacci pattern.

Counter Point about Golden Ratio: There are some unsubstantiated claims about occurrence of Golden ration in Egyptian Pyramids. Further there are Pseudo Numerologists who want to point out presence of Golden Ratio in even stock Markets for ulterior motives.

There is a trend to show presence of Golden Ratio in every conceivable situation. The mysticism about phi is actually a classic obsession for numerologists and Sacred Geometry. In many instances, attempts to find Golden ratio in different situations are no better than the manifestation of numerological mysticism. The readers are advised to be beware of such futile numerological mysticism.