Golden Ratio

1 Introduction

This Rmd file talks about the Golden Ratio and how it is defined in mathematical equations. The source of the article can be found here.

The golden ratio, which is often referred to as the golden mean, divine proportion, or golden section, is a special attribute, denoted by the symbol ϕ, and is approximately equal to 1.618. The study of many special formations can be done using special sequences like the Fibonacci sequence and attributes like the golden ratio.

This ratio is found in various arts, architecture, and designs. Many admirable pieces of architecture like The Great Pyramid of Egypt, Parthenon, have either been partially or completely designed to reflect the golden ratio in their structure. Great artists like Leonardo Da Vinci used the golden ratio in a few of his masterpieces and it was known as the “Divine Proportion” in the 1500s. Let us learn more about the golden ratio in this lesson.

2 Defining the Golden Ratio

The golden ratio, which is also referred to as the golden mean, divine proportion, or golden section, exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity between the two. With reference to this definition, if we divide a line into two parts, the parts will be in the golden ratio if:

The ratio of the length of the longer part, say “a” to the length of the shorter part, say “b” is equal to the ratio of their sum “(a + b)” to the longer length.

Refer to the following diagram for a better understanding of the above concept:

It is denoted using the Greek letter ϕ, pronounced as “phi”. The approximate value of ϕ is equal to 1.61803398875… It finds application in geometry, art, architecture, and other areas. Thus, the following equation establishes the relationship for the calculation of golden ratio: ϕ = a/b = (a + b)/a = 1.61803398875… where a and b are the dimensions of two quantities and a is the larger among the two.

2.1 Golden Ratio Definition

When a line is divided into two parts, the long part that is divided by the short part is equal to the whole length divided by the long part is defined as the golden ratio.Mentioned below are some Golden Ratio examples.

The golden ratio is widely used in architecture and art. Many amazing buildings, like the Great Mosque of Kairouan, are designed using this ratio. Artists such as Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat also incorporated the golden ratio into their artworks.

Great Mosque of Kairouan

3 Formulating the Golden Ratio

The golden ratio formula helps us figure out the value of the golden ratio. It’s a special equation used to find this unique number that appears in nature and art.

3.1 Golden Ratio Equation

From the definition of the golden ratio,

\(a/b = (a + b)/a = ϕ\)

From this equation, we get two equations:

\[a/b = ϕ → (1)\]

\[(a + b)/a = ϕ → (2)\]

From equation (1),

\[a/b = ϕ\]

\[⇒ a = b\]

Substitute this in equation (2),

\[(bϕ + b)/bϕ = ϕ\]

\[b( ϕ + 1)/bϕ = ϕ\]

\[(ϕ + 1)/ϕ = ϕ\]

\[1 + 1/ϕ = ϕ\]

\[1 + 1/ϕ = ϕ\]

\[∴ ϕ = 1 + 1/ϕ\]

Thus, from the above derivation we can see how the Golden Ratio equation is derived.

4 Conclusion

In conclusion, the golden ratio, approximately equal to 1.618, is a special attribute denoted by the symbol ϕ. It is found in various arts, architecture, and designs, and is reflected in structures like the Great Pyramid of Egypt and the Parthenon. Artists such as Leonardo Da Vinci used it in their masterpieces. The ratio is defined by the relationship between two quantities, and its applications can be seen in geometry, art, architecture, and other areas.