golden rectangle

The golden rectangle is a rectangle where the ratio of the longer side to the shorter side is the same as the ratio of the sum of the sides to the longer side. This proportion is known as the golden ratio, approximately 1.618.

Mathematically, if the shorter side is 1 and the longer side is x, the relationship can be expressed as:

[ \frac{x+1}{x} = \frac{x}{1} ]

This equation leads to the quadratic equation (x^2 - x - 1 = 0), with the positive solution being the golden ratio, often represented by the Greek letter phi ((\phi)). The concept of the golden ratio and its application in geometry, such as the golden rectangle, dates back to the ancient Greeks, with Euclid describing it in his Elements. The golden ratio is also closely linked to the Fibonacci sequence, where the ratio of successive numbers approaches the golden ratio.12