✨ Golden Ratio Calculator

Calculate the divine proportion φ (phi). Explore Fibonacci sequences, golden rectangles, and apply the golden ratio to design and mathematics.

φ
1.618033988749
The Golden Ratio (Phi)

🔢 Golden Ratio Calculations

🌀 Fibonacci Sequence

Each number approaches the golden ratio when divided by the previous number

📐 Golden Rectangle

1.618 : 1

Width to Height ratio = φ (1.618...)

🎯 Preset Values

🔧 Actions

📊 Quick Ratios

φ
1.618
1/φ
0.618
φ²
2.618
φ - 1
0.618

🎨 Golden Ratio Applications

Discover how the golden ratio appears in nature, art, architecture, and modern design. Learn practical applications for your creative projects.

🎨
Art & Design
Canvas proportions, composition layouts, spacing between elements, and creating visually pleasing artwork using golden ratio principles.
🏗️
Architecture
Building facades, room proportions, window placements, and structural elements designed with golden ratio for aesthetic harmony.
💻
Web Design
Layout grids, sidebar widths, content areas, typography scales, and user interface elements based on golden ratio proportions.
📐
Typography
Font size relationships, line heights, paragraph widths, and typographic hierarchies using golden ratio for readable text.
📸
Photography
Composition rules, subject placement, cropping ratios, and frame proportions for aesthetically pleasing photographs.
🌿
Nature
Flower petals, shell spirals, tree branches, human body proportions, and natural patterns following golden ratio.
🎵
Music
Musical intervals, composition structure, rhythm patterns, and harmonic relationships based on golden ratio mathematics.
💰
Trading
Fibonacci retracements, market analysis, support/resistance levels, and financial chart analysis using golden ratio.

📚 Golden Ratio Guide

Master the mathematical principles and practical applications of the golden ratio. From ancient Greek mathematics to modern design systems.

📐 Mathematical Foundation

Understanding the mathematical basis of the golden ratio and its unique properties in geometry and algebra.

  • φ = (1 + √5) / 2: Mathematical definition
  • ≈ 1.618033988749: Decimal approximation
  • φ² = φ + 1: Unique algebraic property
  • 1/φ = φ - 1: Reciprocal relationship
  • Continued Fraction: 1 + 1/(1 + 1/(1 + ...))

🌀 Fibonacci Connection

The deep relationship between Fibonacci numbers and the golden ratio in mathematical sequences.

  • Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34...
  • Ratio Convergence: F(n+1)/F(n) → φ
  • Binet's Formula: F(n) = (φⁿ - ψⁿ)/√5
  • Lucas Numbers: Related sequence with φ properties
  • Golden Spiral: Quarter circles in Fibonacci rectangles

🎨 Design Applications

Practical implementation of golden ratio principles in visual design and user interface creation.

  • Layout Grids: Column and row proportions
  • Element Sizing: Buttons, cards, containers
  • Spacing Systems: Margins and padding ratios
  • Typography: Font size and line height scales
  • Color Balance: Proportion of different hues

🏛️ Historical Context

The golden ratio's appearance throughout history in art, architecture, and cultural artifacts.

  • Ancient Greece: Parthenon and geometric studies
  • Renaissance Art: Leonardo da Vinci's compositions
  • Modern Architecture: Le Corbusier's Modulor
  • Islamic Art: Geometric patterns and designs
  • Contemporary: Apple, Twitter, and logo designs

💻 Implementation Tools

Technical methods for applying golden ratio calculations in development and design workflows.

  • CSS Calculations: calc() functions with φ values
  • Design Systems: Token-based proportional scaling
  • Grid Frameworks: Golden ratio based layouts
  • Prototyping: Design tool plugins and extensions
  • APIs: Programmatic golden ratio calculations

🔬 Scientific Occurrences

Natural phenomena and scientific discoveries where the golden ratio appears spontaneously.

  • Plant Growth: Leaf arrangements and phyllotaxis
  • Animal Forms: Shell spirals and body proportions
  • Crystals: Molecular structures and lattices
  • Astronomy: Planetary orbits and galaxy spirals
  • Human Body: Facial features and limb ratios