The Golden Ratio
The golden ratio φ = (1 + √5) / 2 ≈ 1.618 is the most irrational number, appearing in geometry, art, architecture, and nature. Explore golden rectangles, Fibonacci spirals, and the phyllotaxis patterns found in sunflowers and pine cones.
Fibonacci Convergence
Properties of φ
How it works
The golden ratio φ is the positive root of x² = x + 1, giving φ = (1 + √5)/2 ≈ 1.6180339887. It is sometimes called the “most irrational number” because its continued fraction representation [1; 1, 1, 1, ...] converges more slowly than that of any other irrational — its rational approximations are the worst possible.
A golden rectangle has sides in the ratio φ:1. When you cut a square from one end, the remaining rectangle is again a golden rectangle, rotated 90°. This self-similar subdivision can repeat forever. Drawing quarter-circle arcs through each successive square produces the golden spiral, which approximates a logarithmic spiral with growth factor φ per quarter turn.
The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ... has the property that consecutive ratios Fn+1/Fn converge to φ. This connection arises because φ is the dominant eigenvalue of the matrix [[1,1],[1,0]] that generates the Fibonacci recurrence.
In nature, the golden angle (360°/φ² ≈ 137.5°) appears in phyllotaxis — the arrangement of leaves, seeds, and petals. When successive botanical elements are placed at golden angle increments, they pack optimally without overlap. This produces the familiar spiral patterns in sunflower seed heads, pine cones, and artichokes, where you can count Fibonacci numbers of spirals in each direction (e.g., 34 clockwise and 55 counter-clockwise).