Golden Ratio Phi φ

Suppose we have a rectangle of width p and length p + q where p > 0, q > 0 and p > q aka p > q > 0.

Golden ratio Phi φ is the ratio of p / q when:

(p + q) / p = p / q
1 + q / p = p / q
1 + 1 / φ = φ
φ + 1 = φ²
φ² - φ - 1 = 0

Solve with quadratic formula
φ = (-(-1) ± √ ( (-1)² - 4(1)(-1) ) ) / 2(1)
φ = (1 ± √ 5 ) / 2

Since p > 0 and q > 0, so φ > 0,
φ = (1 + √ 5 ) / 2

φ ≈ 1.618

Easy place to see φ?

Jigsaw Puzzle

Notice those 1000-pieces jigsaw puzzle? What is its dimension in length and width? It is 40x25 pieces and 40 / 25 = 1.6 ≈ φ.

How does φ relate to angles?

Since φ = (1 + √ 5 ) / 2, cos 36° = (1 + √ 5) / 4, thus φ = 2 cos 36°.