Spirals in nature
Nautilus shells, sunflower heads, hurricanes, galaxies — nature reaches for spirals again and again. The logarithmic spiral grows by a constant ratio; at the golden ratio, it becomes optimal. Explore why 137.5° is the most irrational angle, and watch it pack seeds with no wasted space.
The logarithmic spiral
A logarithmic spiral has the equation r = aebθ, where b controls the growth rate. At every point, the spiral crosses radial lines at the same angle — it is equiangular. When b = ln(φ)/(pi/2), the spiral grows by the golden ratio φ = 1.618... every quarter turn. This is the spiral approximated by nautilus shells and ram’s horns.
The golden angle
Divide a circle by the golden ratio: 360° / φ ≈ 222.5°. The smaller arc is 137.508° — the golden angle. If each new seed in a flower head rotates by this angle from the previous one, the result is optimal packing with no wasted space. Any rational approximation would leave gaps in radial lines; the golden angle is the “most irrational” number, so it avoids all of them.
Fibonacci spiral
Draw squares with Fibonacci side lengths (1, 1, 2, 3, 5, 8, 13...) and inscribe a quarter-circle arc in each. The result approximates a logarithmic spiral with golden-ratio growth. It is not exactly a logarithmic spiral — it is piecewise circular — but the approximation improves with each term as the Fibonacci ratio converges to φ.