Golden Ratio

Dhristi JEE 2022-24

The Golden Ratio

The Golden Ratio, in mathematics, also known as Golden section, divine proportion, or Golden mean, is an irrational number, which is denoted by the Greek letter “phi” or “φ”. The Golden Ratio or Golden number is defined as the ratio of a line segment, which is cut into two pieces of unequal lengths, where the ratio of the whole segment to the longest segment is equal to the ratio of the longer segment to the shorter segment. The Golden ratio value or golden number is the irrational number \[\frac{(1+\sqrt{5})}{2}\] which is approximately 1.618. 


History of Golden Ratio

The history of the Golden ratio can be traced back to ancient times, where Greek mathematicians like Euclid and Pythagoras spent endless hours researching the equation and its properties. The Greek mathematician Euclid mentions the Golden ratio in the elements, where he implemented some propositions of the ratio. He used to call it the extreme and mean ratio. 

The Golden section frequently appeared in geometrical calculations, including the Pentagrams and Pentagons. The Ancient mathematician Hippasus of 5th century B.C. discovered that the Golden number or divine proportion was neither a whole number nor a fraction, which surprised the Pythagoreans. 

Over the past decades, many mathematicians have studied the Golden number’s importance, uses, and properties and have applied it to many mathematical formulas and calculations. In the 18th century, mathematicians, including Abraham de Moivre, Daniel Bernoulli, and Leonhard Euler, used the Golden ratio formula to discover the value of Fibonacci numbers. In the 1960s, Steve Baer discovered the Zome construction system based on the Golden ratio formula. 


Applications And Usage of Golden Section

The Golden section can be applied in various fields of studies starting from art and architecture to nature. Below are some of the most important uses of the divine proportion.

Art: Most painters and artists used the Golden section in their artistic masterpieces in the ancient era. They used the ratio to add beauty and make their art in the perfect proportion. Mathematicians like Luca Pacioli used the Golden section to provide pleasing and harmonious proportions for paintings. He also found Catholic religious significance in the ratio, for which he also titled the paintings after the ratio. 

Another great artist, Leonardo da Vinci also adopted Pacioli’s Golden section in his paintings to bring out a perfect proportion in them. Leonardo da Vinci’s famous painting, the Mona Lisa, is based on the Golden section and is considered the most beautiful painting having a perfect facial proportion. 

Designs and Books: In the books of early ages, one can find the divine proportion, which is in the ratio 5:3, and is scarce. We can find the divine proportion in many ancient manuscripts and incunables, which were printed in European countries. Even today, you can also find the golden section in many designs, including playing cards, posters, postcards, light switches, and televisions. 

Music: The golden section also plays a crucial role in the music industry, and many famous music composers and singers use it in their musical masterpieces. Famous French composer Erik Satie used the golden section in a few of his songs, including the Sonneries de la Rose. 

Nature: We can also observe the golden section in various aspects of nature. According to Johannes Kepler, the Golden ratio in nature can be seen in the propagation of plants and progenitive acts of animals.

Many other scientists and researchers have found evidence of the golden section in natural activities claiming it to be a universal law of nature.  

Apart from the fields mentioned above, the golden section is also used to study the perfect facial proportion. According to scientists, persons with a golden ratio face are considered more beautiful and appealing than others. They consider that there should be a proportionate gap between all the facial aspects to make a person look appealing. 


Golden Ratio Calculator

The golden ratio calculator is a valuable yet straightforward calculation method that helps you identify the shorter segment, the longer segment, and the combined value of the line segment with the help of a simple formula. If we consider a line segment with the longer segment a and shorter segment b, the golden section can be calculated by the formula: (a+b)/a = a/b. 

You can easily calculate the golden ratio of any two quantities by hand; here are the steps:

  • First, take a greater side or value and mark it as “a”.

  • Again take a smaller side or value and mark it as “b”.

  • Now, input all the values as per the formula; (a+b)/a = a/b.

  • Calculate a+b and divide the result by the value of a.

  • Calculate a/b.

  • If the answer is approximately equal to 1.618, then your quantities are in golden proportion. 


What is the Golden Ratio Rectangle?

While studying the concepts of the golden section, we frequently come across the term golden ratio rectangle, but what is it? Let’s find out. A golden ratio rectangle or golden rectangle is a rectangle whose length is denoted by a+b and width is denoted by a. Here a is the longer side, and b is the shorter side. It is used in art and architectural designs to bring out perfect proportions in constructions and paintings.  


Facts And Examples of Golden Ratio

Above, we discussed the golden ratio, its application, and calculation; now, let’s discuss some of the golden ratio examples and go through some amazing facts about the golden ratio. 

Below are some golden ratio examples that will help you understand the concept of the golden number.

  • You can find the pattern of the golden section in architectural wonders, such as The Great Pyramid of Giza.

  • You can also find the golden section in the famous Mona Lisa painting by Leonardo da Vinci. 

  • You can also find the golden section in the petals of flowers. The petals of a flower always follow the Fibonacci series, which is closely related to the golden section.

  • The galaxy’s spiral shape is an excellent example of the golden section, where each spiral arm is approximately 12 degrees.


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Now, here are some fantastic and astonishing facts about the golden ratio.

  • The golden ratio has many names, including the golden section, golden proportion, divine proportion, medial section, extreme, and mean ratio, etc.

  • The golden section occurs only when the formula of an equation is equal to the number phi, which is equal to 1.618.

  • We can find the golden section in things around us, and many forms of nature also prove that the golden section is a universal law.

  • The value of the golden section is a continued fraction, and therefore it is denoted by the “phi” symbol. 

FAQs on Golden Ratio

Q1. What is a Golden Rectangle?

Answer: The golden rectangle is actually a rectangle that has a length of a+b and a width of a. This rectangle is most commonly observed in art, as it has also been believed that it's the most captivating to the eye of all such rectangles. There is also a golden rectangle calculator which is a convenient tool to find the golden rectangle instead of working it manually. It also has its presence as the golden ratio in nature as well as the golden ratio in architecture. In other words, the golden ratio is observed in various forms of architecture and some concepts of nature, such as in the arrangement of leaves in some trees or decorative plants. The golden proportion is even spotted in regular pentagons.

Q2. How Do We Use the Golden Ratio to Improve Designs?

Answer: You can spot the Golden Ratio when you divide a line into two parts and the longer part (x) divided by the smaller part (y) is equivalent to the sum of (x) + (y) divided by (x), which both equal 1.618. This formula will enable us to create shapes, signboards, logos, layouts, and more.


Considering this idea, we can also create a golden rectangle. Take a square and multiply its one side by 1.618 to obtain a new shape: i.e. a rectangle with well-balanced proportions.

Q3. What is a Beauty Ratio?

Answer: A beauty ratio is actually the golden ratio in beauty. Also known as the golden face ratio, it means the ideal outcome—as described by the golden ratio—is approximately 1.6, which implies that a beautiful person's face is about 1 ½ times longer than it is wide. This also suggests that if the numbers are matching or equal, a person is considered more beautiful.

Q4. Define the golden ratio.

In mathematics, the golden ratio or golden number is an irrational number denoted by the Greek symbol “phi” or “φ.” It is also known as the golden section, golden proportion, medial section, and divine proportion. The value of the golden section is equal to 1.618. It is a continued fraction and therefore is denoted by the symbol “phi”. The golden section has many applications and can be found in many aspects of nature. Many renowned artists and mathematicians have used the ratio in their Works.

Q5. What is a golden rectangle? How to calculate the golden section?

A golden rectangle, also known as the golden ratio rectangle, is a rectangle where the longer part of a side is denoted by the letter “a” and the shorter part of the same side is denoted by the letter “b”. Hence, in the golden rectangle, the length of the side is denoted by a+b, and the width is denoted by a. If we find the ratio of the length to its width, we obtain the golden section. It is used in many architectural marvels by engineers and designers. To calculate the golden section, we take two line segments, where the longer segment of a side is denoted by a, and the shorter segment of the same is denoted by b. Now, the formula for the golden section is (a+b)/a = a/b. Now, we have to place all the values in their respective positions. Solving the equation, you will find the result equal to the golden number or “phi”.

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