Golden Ratio Calculator
Calculate golden ratio segments, verify proportions, and explore the divine proportion
Calculate Golden Ratio
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Golden Ratio (φ)
Results
Fibonacci Sequence Approximation
The golden ratio appears as the limit of consecutive Fibonacci number ratios:
Golden Ratio Facts
Mathematical Value
φ = (1 + √5)/2 ≈ 1.618033988749...
Greek Letter
Denoted by φ (phi), named after sculptor Phidias
Unique Property
φ² = φ + 1 and 1/φ = φ - 1
Fibonacci Connection
Limit of Fₙ₊₁/Fₙ as n approaches infinity
Applications
Art & Architecture
Parthenon, Mona Lisa, modern designs
Nature
Flower petals, pinecones, shell spirals
Photography
Composition, rule of thirds alternative
Finance
Fibonacci retracements in trading
Tips & Usage
Golden rectangles can be subdivided infinitely
Tolerance for verification is ±0.001
Used in design for aesthetically pleasing proportions
Appears in pentagon geometry and regular star
Understanding the Golden Ratio
Mathematical Definition
The golden ratio 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.
Formula Derivation
If a > b > 0, then:
a/b = (a+b)/a = φ
Solving: φ² - φ - 1 = 0
φ = (1 + √5)/2
Unique Properties
- •φ² = φ + 1 (only positive number where this is true)
- •1/φ = φ - 1 ≈ 0.618 (golden ratio conjugate)
- •φ = [1; 1, 1, 1, 1, ...] (continued fraction)
Historical Significance
Ancient Greece
Euclid described the "extreme and mean ratio" around 300 BCE
Renaissance
Called "Divine Proportion" by Luca Pacioli (1509)
Modern Era
Term "golden ratio" first used by Martin Ohm (1835)
Geometric Constructions
Golden Rectangle
Rectangle with aspect ratio φ:1
- • Can be subdivided into smaller golden rectangles
- • Contains golden spiral when squares are removed
- • Found in art, architecture, and design
Golden Triangle
Isosceles triangle with golden ratio proportions
- • Base to side ratio is φ
- • Appears in regular pentagon
- • Self-similar when subdivided
Real-World Examples
Architecture
Parthenon facade, UN Building, modern skyscrapers
Art
Mona Lisa composition, Dalí's paintings, photography
Nature
Nautilus shells, sunflower seeds, tree branching