A number that appears in sunflower seeds, galaxy spirals, the Parthenon, and the proportions of your own hand. The universe seems to have a favorite number.
Each number is the sum of the two before it. Hover to see the ratio approach φ.
Seeds spiral in two directions: typically 34 clockwise, 55 counterclockwise. Both Fibonacci numbers. This arrangement maximizes packing efficiency — no seed is wasted.
The arms of spiral galaxies follow logarithmic spirals closely related to φ. Gravity and angular momentum conspire to create the same pattern seen in a nautilus shell.
The ratio of your forearm to your hand approximates φ. Each finger bone to the next. The spiral of your ear. Biology converges on this ratio because it's efficient.
φ is the most irrational number — hardest to approximate with fractions. It's the limit of F(n+1)/F(n). It satisfies φ² = φ + 1. Its continued fraction is all 1s: [1; 1, 1, 1, ...]
The climax of many classical compositions falls at approximately the 61.8% mark — the golden ratio of the piece's duration. Bartok and Debussy used it deliberately.
The deep reason may be that φ represents optimal growth. When a plant adds leaves at the golden angle (137.5°), each leaf gets maximum sunlight. Evolution finds φ because φ finds efficiency.