Golden Ratio

The Golden Ratio is a special way to compare two lengths. If a line is split into two parts so that the longer part divided by the shorter part gives the same number as the whole line divided by the longer part, that number is about 1.618. We call it the Golden Ratio, and use the Greek letter ɸ (phi) to represent it.

 

Consider a line that is divided into two parts. The total length of the line divided by the long part is equal to the length of the long part divided by the short part. That means, if the long part is “a” and the short part is “b”, then the total length is \(a + b\)

The concept of the golden ratio has been applied in many fields such as mathematics, arts, sculpting, and architecture. The Greek sculptor and mathematician Phidias studied Phi and used it in sculpting.


At the same time, Plato used the golden ratio (not known by name) to study mathematical relationships. In 300 BCE, Euclid in “Elements” discussed a line dividing in the extreme and mean ratio that means the line dividing at 0.6180399…  

 

During the renaissance, Luca Pacioli, in his work De Divina Proportione, called the golden ratio the Divine Proportion.


Around the same time, German mathematician Johannes Kepler identified the importance of the golden ratio, calling it a “great treasure” of geometry.


The first person to coin the term golden ratio is Martin Ohm in “Die reine Elementar-Mathematik” (The Pure Elementary Mathematics).


During the 1900s, an American mathematician named Mark Barr used the Greek letter Phi(ɸ) to symbolize the golden ratio. Phi (ɸ) was named after the Greek sculptor Phidias. It is believed that he used the golden ratio in his works.


For years, the golden ratio has fascinated mathematicians, artists, physicists, and many others. 

The golden ratio appears in the fields of algebra, geometry, and arts. In this section, we will learn about the golden rectangle, and how the golden ratio is related to the Fibonacci sequence.

 

Golden Rectangle: In the golden ratio, the ratio of the sides of the rectangle is equal to ɸ. These rectangles are formed by adding or removing a square. 

 

The golden ratio is connected with the Fibonacci sequence: We can find the Fibonacci series by using the golden ratio. That is, \(F(n) = \frac {(ɸⁿ – (1 – ɸ)ⁿ)} {\sqrt5}\)

There are different methods to find the Golden ratio. These are the methods we use

 

• Hit and trial method
  
   
• Golden ratio equation

In this method, we guess the value of the golden ratio. Follow these steps to find the value of the golden ratio:


Step 1: Guess the number and calculate the multiple inverse of the number. This is called term 1.


Step 2: To find term 2, we add 1 to term 1.


Step 3: If term 1 is equal to term 2, then it is the value of the golden ratio. If not, we will repeat the process till term 1 equals term 2.


Step 4:  The process is repeated till we get the value of the golden ratio. 


First iteration, 


Step 1: Let’s guess the value as 1.6. The multiple inverse of 1.6 is \(\frac {1}{1.6} = 0.625\). Term 1 is 0.625 (approx). 


Step 2: \(Term 2 = 1 + term 1\) 
\(= 1 + 0.625 = 1.625\)


Term 1 is not equal to term 2, so we will make the next guess.


Second iteration, 


Step 1: Let’s guess the value as 1.666. The multiple inverse of 1.666 is \(\frac {1}{1.666} = 0.6\)
Term 1 is 0.6


Step 2: \(Term 2 = 1 + term 1 \)
\(= 1 + 0.6 = 1.6\)

 

The guessed value is not equal to term 2


Third iteration, 


Step 1: Let’s guess the value as 1.625. The reciprocal of 1.625 is \(\frac {1}{1.625} = 0.61538\)
Term 1 is 0.6


Step 2: \(Term 2 = 1 + term 1\)
\(= 1 + 0.61538 = 1.61538\)


The estimated value is not equal to term 2


The value of the golden ratio is 1.618.

To find the value of the golden ratio another method we use is the golden ratio equation. As we know, ɸ = 1 + 1/ɸ

To find the value of ɸ

We multiply both sides by ɸ, ɸ2 = ɸ + 1

ɸ2 - ɸ - 1 = 0

Using a quadratic equation that is \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Using the value \(a = 1, b = -1, c = -1,\) 

ɸ = \( {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\( = {-(-1) \pm \sqrt{(-1)^2-4\times1 \times -1} \over 2 \times1}\)

\(= {1 \pm \sqrt{(1 + 4} \over 2 }\)

\(= {1 \pm \sqrt{5} \over 2 }\)
 

So, ɸ \(= {1 \pm \sqrt{5} \over 2 }\)

Working with the golden ratio and incorporating it in calculations can be tricky. In this section, let’s learn a few tips and tricks to better understand the golden ratio. 

 

• Memorizing the value of the golden ratio, which is 1.618. 
  
   
• Memorizing the formula of golden ratio that is ɸ  \(= {1 \pm \sqrt{5} \over 2 }\) or ɸ = 1 + 1/ɸ 
  
   
• By understanding the concept of the golden ratio, that is \({a + b \over a } = {a \over b} \) =  ɸ
  
   
• By using the Fibonacci series we can find the value of the golden ratio which is the ratio of consecutive Fibonacci series is equal to the golden ratio.
  
   
• Visualize the ratio. Draw rectangles or spirals to see the Golden Ratio in action. Make a rectangle with sides 1 and 1.618 units. Create a spiral by connecting squares. This is the Golden Spiral often found in shells and galaxies. Visual learning helps memory stick.

The golden ratio is widely used in art, architecture, nature, and design to create balance, harmony, and visually pleasing structures. 

 

1. Architecture: Many famous structures, like the Parthenon in Greece and the Great Pyramid of Giza, are believed to be designed using proportions close to the golden ratio for aesthetic appeal.
   
    
2. Art and design: Renaissance artists like Leonardo da Vinci used it in works such as the Vitruvian Man and The Supper to create balance and beauty.
   
    
3. Human body: The proportions of the face distance between eyes, nose, lips and body navel to height ratio often align closely with the golden ratio, which is associated with perceived beauty.
   
    
4. Stock market and trading: Traders use Fibonacci retracement levels, based on golden ratio percentages (61.8%, 38.2%), to predict potential reversals in stock prices.
   
    
5. Nature: The arrangement of leaves, flower petals, pine cones, and even the spiral shells of nails and nautilus often follow the Fibonacci sequence, which is linked to the golden ratio