Golden Angle Phyllotaxis

Explore the mathematical beauty of plant seed arrangement - Polar formulas: θ = n × 137.5°, r = c√n

Parameter Controls

Mathematical Principles

Angle Formula: θ = n × α
Radius Formula: r = c√n
Cartesian Coordinates: x = r·cos(θ), y = r·sin(θ)

What is Phyllotaxis?

Phyllotaxis is the study of the arrangement of leaves, seeds, petals, and other plant parts on plant stems. The most famous is the golden angle arrangement pattern, widely found in sunflowers, pinecones, pineapples, and other plants.

Golden Angle 137.5°

Golden Angle = 360°/φ² ≈ 137.5°, where φ = (1+√5)/2 ≈ 1.618 is the golden ratio. This angle produces the most compact and uniform seed arrangement.

Fibonacci Spirals

Sunflower seeds typically form two sets of spirals (clockwise and counterclockwise), with spiral numbers being adjacent Fibonacci numbers, such as 34 and 55, or 55 and 89.

Sensitivity Test

Try adjusting the angle by just 0.1° or 0.2° from 137.5°, and you'll see the spiral pattern change dramatically, forming straight-arm spirals. This shows that the golden angle is the optimal arrangement angle.

Nature's Efficiency

This arrangement allows plants to maximize seed count, optimize space utilization and light reception, representing an optimal solution in the evolutionary process.

Examples in Nature

  • Sunflower: Seeds arranged by golden angle, typically with 34+55 or 55+89 spirals
  • Pinecone: Scale arrangements follow the same mathematical pattern
  • Pineapple: Fruit eye arrangements show spirals in three directions
  • Romanesco Broccoli: Self-similar spiral structures demonstrating fractal geometry