The Golden Ratio, often denoted by the Greek letter phi (ϕ or φ), is a special mathematical ratio approximately equal to 1.61803398875. It is also known by several other names, including the golden mean, golden section, and divine proportion. The Golden Ratio is defined as the ratio between two quantities such that the ratio of the sum of the quantities to the larger quantity is the same as the ratio of the larger quantity to the smaller one. Mathematically, if $$a$$ and $$b$$ are two quantities with $$a > b > 0$$, the Golden Ratio is expressed as:
$$
\frac{a + b}{a} = \frac{a}{b} = \phi
$$
This can be derived from the quadratic equation:
$$
\phi^2 = \phi + 1
$$
Solving this equation gives the positive value of $$\phi$$ as approximately 1.61803398875[1][2][3].
Relation to the Fibonacci Sequence
The Golden Ratio is closely related to the Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The Fibonacci sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. As the Fibonacci numbers increase, the ratio of successive Fibonacci numbers approaches the Golden Ratio. For example, the ratio of 21 to 13 is approximately 1.615, and the ratio of 34 to 21 is approximately 1.619. As the numbers get larger, these ratios get closer to 1.61803398875, the value of the Golden Ratio[8][9].
Applications of the Golden Ratio
The Golden Ratio appears in various fields, including:
Art and Architecture: It is used to create aesthetically pleasing compositions. Famous examples include the Parthenon in Greece and the works of Leonardo da Vinci[5][7][12].
Nature: The Golden Ratio can be observed in the arrangement of leaves, the branching of trees, the spiral patterns of shells, and the structure of flowers and fruits[16][17].
