Golden Ratio

The golden ratio is a mathematical constant defined by a proportion in which the ratio of a whole segment to its larger part equals the ratio of the larger part to the smaller part.

The golden ratio is a mathematical constant defined by a proportion in which the ratio of a whole segment to its larger part equals the ratio of the larger part to the smaller part. It is an irrational number, usually denoted by the Greek letter phi (φ), and is approximately 1.618. It appears in geometry and is often discussed in relation to art, design, and nature.

Definition

The golden ratio is the proportion obtained when a line segment is divided into two unequal parts so that the ratio of the whole segment to the longer part equals the ratio of the longer part to the shorter part. This proportion is often written as φ and equals (1+√5)/2.

Mathematical properties

The golden ratio is an irrational number, so its decimal expansion continues without repeating. It is closely connected to algebraic equations, including x² - x - 1 = 0, and appears in several classical geometric contexts such as the pentagon, pentagram, decagon, and dodecahedron.

History and naming

The ratio has been studied since antiquity and is associated with Greek mathematics, especially Euclid's discussion of dividing a line segment in extreme and mean ratio. Modern writers use names such as golden ratio, golden section, golden mean, and divine proportion.

Applications and cultural references

The golden ratio is often cited in discussions of aesthetics, proportions, and design, and it is frequently mentioned in connection with nature, art, and architecture. These uses are common in popular and design literature, but they are distinct from the ratio's formal mathematical definition.