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Golden Ratio Calculator

φ (phi) ≈ 1.6180339887

Powers of φ

φ^(-3)

0.236068

φ^(-2)

0.381966

φ^(-1)

0.618034

φ^0

1

φ^1

1.618034

φ^2

2.618034

φ^3

4.236068

φ^4

6.854102

φ^5

11.09017

φ^6

17.944272

About the Golden Ratio

The Golden Ratio (φ): An irrational number approximately equal to 1.618033988749895. It appears when a line is divided so that the whole line to the longer segment equals the longer segment to the shorter segment.

Formula: φ = (1 + √5) / 2

Key Properties:

  • φ² = φ + 1 ≈ 2.618
  • 1/φ = φ - 1 ≈ 0.618
  • φ is related to Fibonacci sequence: F(n)/F(n-1) → φ

Found in: Art, architecture (Parthenon), nature (shells, flowers), human body proportions, financial markets.

What is the Golden Ratio?

The golden ratio, denoted by the Greek letter phi (φ), is a mathematical constant approximately equal to 1.618033988749895. It has fascinated mathematicians, artists, and scientists for millennia.

The Mathematical Definition

Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Mathematically:

a/b = (a+b)/a = φ ≈ 1.618033988749895

The exact value is:

φ = (1 + √5) / 2

Key Properties of Phi

  • Self-similar: φ² = φ + 1 ≈ 2.618
  • Reciprocal: 1/φ = φ - 1 ≈ 0.618
  • Continued fraction: φ = 1 + 1/(1 + 1/(1 + 1/...))
  • Irrational: Cannot be expressed as a simple fraction

Connection to Fibonacci Sequence

The ratio of consecutive Fibonacci numbers approaches phi as the numbers get larger:

  • 1/1 = 1.000
  • 2/1 = 2.000
  • 3/2 = 1.500
  • 5/3 = 1.667
  • 8/5 = 1.600
  • 13/8 = 1.625
  • 21/13 = 1.615
  • 34/21 = 1.619...

The Golden Ratio in Nature

  • Nautilus shells: Spiral growth follows golden ratio
  • Flower petals: Often in Fibonacci numbers (3, 5, 8, 13)
  • Sunflower seeds: Spiral patterns exhibit phi
  • Pinecones: Spirals in both directions
  • Hurricanes: Spiral structure approximates golden spiral

The Golden Ratio in Art & Architecture

  • Parthenon: Proportions approximate golden ratio
  • Leonardo da Vinci: Used in Vitruvian Man and other works
  • Salvador Dalí: "The Sacrament of the Last Supper"
  • Le Corbusier: Modulor system based on phi

Practical Applications

  • Design: Creating visually pleasing proportions
  • Photography: Rule of thirds approximates golden ratio
  • Typography: Font sizing relationships
  • Web design: Layout proportions and spacing
  • Finance: Fibonacci retracement levels (38.2%, 61.8%)