Golden Ratio Calculator

Results:

Understanding the Golden Ratio

Basic Concepts

The golden ratio (φ or phi) is an irrational mathematical constant approximately equal to 1.618033988749895. It occurs when a line segment is divided into two parts such that the ratio of the longer part to the shorter part equals the ratio of the whole segment to the longer part.

φ = (1 + √5)/2 ≈ 1.618033988749895

a/b = (a+b)/a = φ

φ² = φ + 1

1/φ = φ - 1 ≈ 0.618033988749895

Properties and Characteristics

Mathematical Properties

  • Algebraic properties
  • Continued fraction representation
  • Fibonacci sequence relationship
  • Geometric constructions
  • Logarithmic spiral connection
  • Pentagonal symmetry
  • Recursive nature
  • Irrational number properties

Applications

  • Art composition
  • Architectural design
  • Natural phenomena
  • Financial markets
  • Music theory
  • Photography
  • Web design
  • Product design

Advanced Topics

Geometric Applications

  • Golden rectangle construction
  • Golden spiral generation
  • Pentagon and pentagram relationships
  • Platonic solids
  • Golden angle
  • Golden triangles
  • Logarithmic spirals
  • Fractal patterns

Mathematical Analysis

Golden Rectangle Area: A = a² × φ

Golden Spiral: r = e^(θ/φ)

Golden Angle: 2π/φ² ≈ 137.508°

Golden Triangle: sin(36°) = 1/(2φ)

Natural Phenomena

  • Plant growth patterns
  • Shell spirals
  • DNA structure
  • Galaxy formation
  • Human body proportions
  • Crystal structures
  • Wave patterns
  • Population growth

Artistic Applications

Rule of Thirds: 1.6666... ≈ φ

Dynamic Rectangles: w/h = φ

Harmonic Ratios: φⁿ series