Golden Spiral & Phyllotaxis Calculator
Choose a seed count and watch the golden-angle spiral pattern seen in sunflowers, pinecones, and daisy heads build itself, point by point.
Divergence angle & spiral preview
Each seed is rotated 137.507764° from the last — the same divergence angle that packs sunflower and pinecone seed heads with no gaps or overlaps.
A pattern built one small rotation at a time
There’s no master plan behind a sunflower’s seed head — each new seed is simply placed a little farther out and rotated by the same fixed angle from the one before it. What makes the golden angle special is what happens over hundreds of repetitions of that one simple rule: because 137.5° can’t be well approximated by any small fraction of a circle, the seeds never fall into a small number of straight radial lines with gaps between them. Try a “nicer” angle like 90° or 120° in your head and you can see why it would fail — those divide evenly into a circle, so every fourth or third seed would land in exactly the same few directions.
The visible spiral arms in the preview above are an optical side effect of this packing rule, not something calculated directly — your eye naturally connects the nearest neighbors into curves.
Frequently Asked Questions
What is the golden angle, and where does 137.5° come from?
The golden angle divides a full circle in golden proportion: it's 360° × (2 − φ), which works out to about 137.5077640500°. It's the angle you get by taking the smaller of the two arcs when a circle's circumference is split so the two arcs are in golden ratio to each other.
Why do sunflowers (and pinecones, and many other plants) use this exact angle?
The golden angle is, in a precise mathematical sense, the 'most irrational' angle — it's the hardest angle to approximate closely with a simple fraction of a full turn. When new seeds or leaves are added one at a time, each rotated by this angle from the last, that property is what prevents any two spiral arms from ever lining up and leaving a gap or a wasted overlap. This is Vogel's model (H. Vogel, 1979), the standard mathematical description of the pattern.
How are the point coordinates calculated?
Each seed i (starting from 0) is placed at radius c × √i and rotated i × 137.5077640500° from the starting direction, then converted to x/y coordinates with ordinary trigonometry: x = r·cos(θ), y = r·sin(θ). The square-root radius is what keeps the seeds evenly spaced by area as the pattern grows outward, rather than bunching up near the center or spreading too thin at the edge.
Is this the same as the Fibonacci spiral drawn from nested squares?
They're related but not identical. The classic 'Fibonacci spiral' picture — quarter-circle arcs inscribed in a sequence of squares sized 1, 1, 2, 3, 5, 8… — is a geometric approximation of a true logarithmic (golden) spiral. This tool instead uses Vogel's model, which places discrete points (seeds) rather than drawing a continuous curve, but both are built from the same golden-ratio mathematics.
Educational visualization of Vogel’s phyllotaxis model. Real plants approximate this pattern closely but not perfectly — biological growth introduces variation this idealized model doesn’t capture.