The golden ratio, often referred to as the golden mean, divine proportion, or golden section, is a remarkable mathematical constant symbolized by the Greek letter ϕ and approximately equal to 1.618. The exploration of various special patterns and formations frequently involves the use of unique sequences such as the Fibonacci sequence and properties associated with the golden ratio.
This ratio holds a special place in the realms of art, architecture, and design. Many iconic architectural wonders, including the Great Pyramid of Egypt and the Parthenon, have been either partially or wholly designed to incorporate the golden ratio into their structures. Renowned artists such as Leonardo da Vinci also employed the golden ratio in some of their masterpieces, earning it the epithet “Divine Proportion” during the 1500s. In this lesson, we delve deeper into the fascinating world of the golden ratio.
What is the Golden Ratio?
The golden ratio, also known as the golden mean, divine proportion, or golden section, is a mathematical concept that emerges when two quantities exhibit a specific relationship. This relationship occurs when the ratio of the two quantities equals the ratio of their sum to the larger of the two quantities. To put it simply, if we divide a line into two segments, those segments will be in the golden ratio if:
The length of the longer segment, denoted as “a,” divided by the length of the shorter segment, denoted as “b,” is equal to the ratio of the sum of the two lengths, “(a + b),” to the length of the longer segment.
For a clearer grasp of this concept, please refer to the accompanying diagram below:
[Diagram: A line divided into two segments, with the longer segment labeled as “a,” the shorter segment labeled as “b,” and the relationship described as (a/b) = ((a + b)/a).]
The golden ratio is symbolized by the Greek letter ϕ, pronounced as “phi.” Its approximate value is approximately 1.61803398875… This mathematical concept finds applications in various fields, including geometry, art, and architecture. The relationship defining the golden ratio can be expressed through the equation: ϕ = a/b = (a + b)/a = 1.61803398875…, where “a” and “b” represent the dimensions of two quantities, with “a” being the larger of the two.
Golden Ratio Definition
The golden ratio in architecture and art is a concept where, when a line is divided into two segments, the longer part divided by the shorter part equals the whole length divided by the longer part. This ratio has been prominently applied in various architectural and artistic examples:
In the realm of architecture, numerous marvels, including the Great Mosque of Kairouan, have been constructed with careful attention to incorporating the golden ratio into their structures.
In the world of art, esteemed artists such as Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat utilized the golden ratio as a significant attribute in their creations.
Golden Ratio Formula
The value of the golden ratio can be calculated using a specific formula. This equation is derived to establish a general formula for determining the golden ratio.
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
How to Calculate the Golden Ratio?
Calculating the value of the golden ratio can be achieved through various methods. Let’s begin with a fundamental approach.
Hit and trial method
To calculate the value of the golden ratio, you can follow a series of steps. Begin with an arbitrary initial value for the constant, and then proceed as follows in each iteration:
- Calculate the multiplicative inverse of your guessed value, i.e., 1/value. This result becomes our first term.
- Calculate another term by adding 1 to the multiplicative inverse obtained in step 1.
- Ensure that both terms derived from the above steps are approximately equal. If they are not, repeat the process until you obtain values that are nearly equal.
- For subsequent iterations, utilize the value obtained in step 2 as your new assumed value, and continue the process.
This iterative approach will progressively yield a more accurate approximation of the golden ratio.
For example,
To calculate the golden ratio using an iterative method, we start with an initial guess of 1.5 since ϕ = 1 + 1/ϕ, which implies that ϕ must be greater than 1. Here’s the step-by-step calculation:
Iteration 1:
- Term 1 = Multiplicative inverse of 1.5 = 1/1.5 = 0.6666…
- Term 2 = Multiplicative inverse of 1.5 + 1 = 0.6666… + 1 = 1.6666…
- Since the terms are not equal, we proceed to the next iteration using the assumed value equal to Term 2.
Iteration 2:
- Term 1 = Multiplicative inverse of 1.6666… = 1/1.6666… = 0.6…
- Term 2 = Multiplicative inverse of 1.6666… + 1 = 0.6… + 1 = 1.6…
- The terms are still not equal, so we continue.
Iteration 3:
- Term 1 = Multiplicative inverse of 1.6… = 1/1.6… = 0.625…
- Term 2 = Multiplicative inverse of 1.6… + 1 = 0.625… + 1 = 1.625…
- The terms are not equal yet, so we proceed.
Iteration 4:
- Term 1 = Multiplicative inverse of 1.625… = 1/1.625… = 0.6153…
- Term 2 = Multiplicative inverse of 1.625… + 1 = 0.6153… + 1 = 1.6153…
- The terms are still not equal.
We continue this process iteratively until we reach the desired equality of terms, which approximates the golden ratio.
The more iterations you perform, the closer the approximate value will get to the accurate one. However, it’s worth noting that there are more efficient methods available for calculating the precise value of the golden ratio.
Golden Ratio Equation
Another method to determine the value of the golden ratio is by solving the golden ratio equation step by step:
Starting with:
ϕ = 1 + 1/ϕ
Multiply both sides by ϕ:
ϕ^2 = ϕ + 1
Rearrange the equation:
ϕ^2 – ϕ – 1 = 0
This equation is a quadratic equation and can be solved using the quadratic formula:
ϕ = (-b ± √(b^2 – 4ac)) / (2a)
Substituting the values a = 1, b = -1, and c = -1, we get:
ϕ = (1 ± √(1 + 4)) / (2)
Simplifying further, we take the positive value because ϕ represents a ratio of lengths and cannot be negative:
ϕ = (1 + √5) / 2
So, the value of ϕ, the golden ratio, is equal to (1 + √5) / 2.
