Phyllotaxis Spiral

What is Phyllotaxis?

Phyllotaxis (Greek: "leaf arrangement") describes how botanical organs such as seeds, leaves, or petals are placed around a stem. Nature favours arrangements that minimise overlap and maximise sunlight exposure — most often the golden-angle spiral.

The Golden Angle

The golden ratio φ ≈ 1.618 034 leads to the golden angle:

θ_g = 360° × (φ − 1) ≈ 137.5078°

Placing new elements at this irrational fraction of a full turn ensures no two points line up, producing quasi-uniform packing.

Parametric Formula

for n = 0, 1, 2, …
    r = c · √n        // radial distance grows with √n
    θ = n · θ_g       // rotation by n golden angles
    x = r · cos θ
    y = r · sin θ

The constant c controls spacing. In the demo we randomise c (4–8) and sometimes even tweak the angle slightly to show diverse yet related patterns.

Botanical Examples

• Sunflower florets
• Pinecone scales
• Succulent rosettes

Using the Visualiser

• Pause / resume to watch points being dropped.
• New Spiral randomises c and the angle for fresh formations.
• Colour hue cycles with n, giving a Bauhaus-inspired gradient.