Golden Ratio

Have you ever wondered why specific patterns in nature or designs in art look interesting or balanced? One idea that helps describe these patterns is the Golden Ratio. The Golden Ratio is a mathematical constant, approximately equal to 1.618. It appears in various mathematical constructions and can also be observed in some artworks, architectural designs, and natural patterns, though not always perfectly. In this article, we will explore what the Golden Ratio is, how it is defined in mathematics, and where it is commonly referenced in real-world contexts.




History of Golden Ratio

 

• 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, represented by ϕ (phi) and approximately equal to 1.618, has several interesting mathematical properties that make it unique. These properties appear in algebra, geometry, ratios, and in number patterns such as the Fibonacci sequence.
 

• \(ϕ=1+\frac{ϕ}{1} \), which means ϕ is the only positive number that is equal to 1 plus its own reciprocal.
   
• The Golden Ratio satisfies the quadratic equation: \(ϕ^2=ϕ+1\). his is actually how the value of ϕ is derived. Solving the equation gives:  
  \(ϕ=1+\frac{\sqrt {5}}{2}\).
   
• If a line is divided into two segments in the Golden Ratio, the ratio of: 
  The whole line to the longer part is equal to the longer part to the shorter part. This ratio always equals ϕ.
   
• The Golden Ratio shows up naturally in several geometric figures, like the regular pentagon, the golden rectangle, and the golden triangle. 
   
• As you divide consecutive Fibonacci numbers: \(\frac{F_{n+1}}{F_n}\),   the value gets closer and closer to ϕ as the numbers grow larger. This is called convergence to the Golden Ratio.

The Golden Ratio formula describes a special way of dividing a line into two parts. When a line splits, then:

• The ratio of the whole line to the longer segment is the same as:
   
• The ratio of the longer segment to the shorter segment. 

This common ratio is called the Golden Ratio, represented by the symbol ϕ (phi). Mathematically, if the longer part is a and the shorter part is b, then the total length is a + b. The Golden Ratio condition can be written as:
\(\frac{a + b}{a} = \frac{a}{b} = ϕ\).

Equation of The Golden Ratio
 

We know that, \(\frac{a + b}{a} = \frac{a}{b}\).

This common ratio is the Golden Ratio, represented by ϕ (phi).

So we can rewrite the equation as: \(\frac{a + b}{a} = ϕ\) and \(\frac{a}{b} = ϕ\).
Since both expressions equal ϕ, we can use either one to derive the formula.
We will use the first: 
\(\frac{a + b}{a} = ϕ\).

Step 1: Split the numerator.

\(\frac{a + b}{a} = \frac{a}{a} + \frac{b}{a}\)

\(1+\frac{ b}{a} = ϕ\)

Step 2: Use the relationship between segments. 
According to the golden ratio condition, \(\frac{a}{b} = ϕ\).
By taking the reciprocal: \(\frac{b}{a} = \frac{1}{ϕ}\).
We can substitute this into the earlier equation, 
\(1 + \frac{1}{ϕ} = ϕ\).

Step 3: Form a quadratic equation. 
Multiply both sides by φ to remove the fraction: 
\(ϕ+1=ϕ^2\)
By rearranging, we get, \(ϕ^2−ϕ−1=0\). 
This is the Golden Ratio equation.
 

Step 4: Solve the quadratic equation. 
By using the quadratic formula: 
\(\phi = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
For the equation \(ϕ^2−ϕ−1=0\),
a = 1, b = -1 and c = -1. 
By substituting the values, 

\(ϕ= \frac{1± 1^2−4(1)(−1)}{2}\)

\(ϕ= \frac{1± 1+4}{2}\)

\(ϕ= \frac{1± 5}{2}\)
 

Step 5: Choose the positive solution. 
Since the Golden Ratio represents a positive length ratio, we take the positive solution:
\(\phi = \frac{1 + \sqrt{5}}{2} \).

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

 

• Hit and trial method
  
   
• Golden ratio equation

Hit and Trial Method: 
 

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. 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.
 

• 2nd 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.

 

• 3rd 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.

Golden Ratio Equation: 
 

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, \(a = 1, b = -1, c = -1,\)

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

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

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

\(= {1 \pm \sqrt{5} \over 2 }\)

So, \(\phi= {1 \pm \sqrt{5} \over 2 }\)

​

The Fibonacci sequence is a series of numbers where each term (from the third onward) equals the sum of the two preceding terms. Starting from 0 and 1, the sequence goes: 
0,1,1,2,3,5,8,13,21,…

If you take two consecutive Fibonacci numbers \(F_1 \) and \(F_{n+1}\), and form the ratio: \(\frac{F_{n+1}}{F_n}\). 
As n increases, this ratio approaches the Golden Ratio, ϕ (about 1.618). This means that the larger the Fibonacci numbers, the more their ratio approximates ϕ.

Let us look at some examples in pairs:

 

Golden Rectangle
 

A Golden Rectangle is a special type of rectangle whose side lengths are in the Golden Ratio, represented by the symbol ϕ (phi). This means that if the longer side of the rectangle is a and the shorter side is b, then: \(\frac{a}{b} = ϕ≈1.618\). This proportion makes the Golden Rectangle unique among rectangles.

How a golden rectangle is formed? 
 

To construct a golden rectangle: 

Step 1: Start with a square of side length b.

Step 2: Attach a rectangle to one side of the square such that the new, larger rectangle has side lengths a and b.

Step 3: If the ratio \(\frac{a}{b} \) equals the Golden Ratio, the resulting figure is a Golden Rectangle.
This idea comes directly from the Golden Ratio equation:
\(\frac{a}{b} = a = \frac{b}{a} = ϕ\).

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.
  
   
• Parents and teachers can show everyday objects like rectangles, notebooks, cards, screen ratios, etc., and compare their sides. Even if they are not perfect Golden Rectangles, they help explain the idea of proportionality.
  
   
• Walk students through the equation \(\frac{a + b}{a} = \frac{a}{b}\), and help them see how this leads to \(\phi= {1 \pm \sqrt{5} \over 2 }\).
  
   
• Let students generate a simple Fibonacci list and compute ratios of consecutive terms. This hands-on activity helps them see how the ratios approach ϕ naturally.
  
   
• Please encourage students to draw squares, extend them into rectangles, and observe how removing squares forms similar shapes. This reinforces the idea of self-similarity in Golden Rectangles.
  
   
• Use graph paper, geometric apps, or drawing tools to help students construct Golden Rectangles or visualize spirals. Visual demonstrations make abstract concepts more concrete.

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