Golden Ratio

The golden ratio is the ratio between two numbers, which is equal to 1.618. The golden ratio is denoted using the symbol ɸ. The golden ratio is also known as the golden number, divine proportion, and golden proportion. 

Consider a line which 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 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 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. He referred to the paintings of Leonardo da Vinci, “The Last Supper” and ”Vitruvian Man,” to explain the golden ratio. 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) = ɸn - (1 -ɸ )n /√5

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 1/1.6 = 0.625. Term 1 is 0.625

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 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 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 

Using the value a = 1, b = -1, c = -1, 
 

So, ɸ = (1 + √5) / 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 + √5 / 2 or ɸ = 1 + 1/ɸ 
   
• By understanding the concept of the golden ratio that is
   
• 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.