Fermat Spiral Explorer
Fermat’s spiral (r = ±√θ) creates the pattern seen in sunflower seed heads. Adjust the divergence angle and watch how seeds pack into a disk. At the golden angle (~137.508°), optimal packing emerges and the visible spiral counts are always Fibonacci numbers.
Fermat’s spiral
Fermat’s spiral is defined by r = ±a√θ. Unlike an Archimedean spiral (where r grows linearly with θ), Fermat’s spiral fills a disk because consecutive points spread outward more slowly, packing tighter toward the center.
The golden angle
The golden angle is 360° / φ² ≈ 137.508°, where φ is the golden ratio. This angle is the “most irrational” number — its continued fraction converges as slowly as possible, which means no spoke pattern can form. The result is the most uniform packing achievable by a single-angle rule.
Parastichies and Fibonacci numbers
The visible spirals in a sunflower head are called parastichies. At the golden angle, the counts of clockwise and counterclockwise spirals are always consecutive Fibonacci numbers — 13 and 21, or 21 and 34, or 34 and 55, depending on how many seeds you count. This is not a coincidence: it is a direct consequence of the golden angle’s relationship to the Fibonacci sequence through the continued fraction of φ.
Rational angles and spokes
Try setting the angle to a simple fraction of 360° — like 120° (1/3 of a turn) or 144° (2/5 of a turn). Seeds land on a few radial lines, wasting space between them. The simpler the fraction, the fewer spokes. Irrational angles avoid this, but only the golden angle avoids it optimally.