Golden Ratio (φ) is an irrational mathematical constant approximately equal to 1.6180339887. It appears in mathematics, nature, architecture, and art, often associated with aesthetically pleasing proportions.
Definition
The golden ratio is defined as:
- φ = (1 + √5) / 2 ≈ 1.618
It satisfies the equation:
- φ² = φ + 1
Mathematical Properties
- Self-Similarity: φ is the only positive number that satisfies φ² = φ + 1.
- Continued Fraction Representation: φ can be expressed as:
- φ = 1 + 1/(1 + 1/(1 + 1/(1 + ...))).
- Limit of Fibonacci Ratio: The ratio of consecutive Fibonacci numbers converges to φ:
- lim (F(n+1) / F(n)) = φ as n → ∞.
Golden Ratio in Geometry
- Golden Rectangle: A rectangle where the ratio of the longer side to the shorter side is φ.
- Golden Spiral: A logarithmic spiral that grows outward by a factor of φ for every quarter turn.
- Pentagon and Star: The golden ratio appears in the proportions of a regular pentagon and a five-pointed star (pentagram).
Applications
The golden ratio appears in various fields:
- Mathematics: Fibonacci numbers, continued fractions, prime number distribution.
- Art and Architecture: Used in the Parthenon, Da Vinci’s "Vitruvian Man", and Renaissance art.
- Nature: Found in flower petal arrangements, pinecones, and shells (e.g., Nautilus shell).
- Financial Markets: Fibonacci retracement levels in technical analysis.
Comparison with Other Ratios
| Ratio | Approximate Value | Appearance |
|---|---|---|
| Golden Ratio (φ) | 1.618 | Fibonacci sequence, art, nature |
| Silver Ratio (δ) | 2.414 | Some geometric tilings |
| Pi (π) | 3.1416 | Circle circumference-to-diameter ratio |
Golden Ratio in Fibonacci Sequence
The Fibonacci sequence is closely related to the golden ratio:
- The ratio of successive Fibonacci numbers approaches φ.
- The nth Fibonacci number can be computed using Binet’s Formula:
- F(n) = (φⁿ - (1 - φ)ⁿ) / √5.