π
Golden Ratio (φ)
★★☆☆☆Middle School
📖Definition
The golden ratio φ (phi) is when the ratio of whole to larger part equals the ratio of larger to smaller part.
📐Formulas
φ = \frac1 + √52 ≈ 1.6180339...
Value of golden ratio
φ² = φ + 1
Golden ratio property
(1)/(φ) = φ - 1
Reciprocal of golden ratio
lim_n → ∈fty \fracF_n+1Fₙ = φ
Relation to Fibonacci
✏️Examples
예제 1
Remove a square from a 1:φ rectangle. What remains?
📜History
Discovered by: Ancient Greeks (Pythagoreans) (c. 6th century BCE)
Euclid defined it as 'extreme and mean ratio'; called 'golden' since Renaissance.
⚡Applications
Art
Composition, proportions
Architecture
Parthenon
Nature
Sunflowers, shells
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