The golden ratio is derived from a mathematical relationship where a line is divided into two parts, a and b, such that the ratio of the whole line (a + b) to the longer part (a) is equal to the ratio of the longer part (a) to the shorter part (b). Mathematically, this can be expressed as:
a+ba=ab=Φ≈1.618\frac{a + b}{a} = \frac{a}{b} = \Phi \approx 1.618aa+b=ba=Φ≈1.618
This ratio can be found in various natural phenomena, such as the arrangement of leaves on a stem, the branching of trees, and the spirals of shells. Its prevalence in nature has led to its application in art and architecture, where it is often used to create aesthetically pleasing compositions.