Numbers › Golden ratio

The golden ratio (φ)

A rectangle so balanced that cutting off a square leaves the same shape behind. Shows up everywhere — flowers, shells, even faces.

The golden rectangle

Cut off a square — what's left is another golden rectangle. Forever.

φ ≈ 1.6180

Depth5

What's special about φ?

φ ≈ 1.618. It satisfies φ² = φ + 1. That self-similar property is why golden rectangles nest inside themselves forever — and why the golden spiral is the limiting shape of Fibonacci squares.

Find it in nature

Sunflower seeds, pinecones, nautilus shells, the curl of a fern — all settle around the golden ratio because that's the most efficient packing.