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.

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

Example 1

Remove a square from a 1:φ rectangle. What remains?

Discovered by: Ancient Greeks (Pythagoreans) (c. 6th century BCE)

Euclid defined it as 'extreme and mean ratio'; called 'golden' since Renaissance.

Composition, proportions

Parthenon

Sunflowers, shells