What is phyllotaxis?
Phyllotaxis means the arrangement of leaves, seeds, or petals on a plant. A plant wants each new leaf to receive light and each seed to use space efficiently. Repeating the same turn between new growth points is a wonderfully economical solution.
If the turn were a simple fraction of a circle—one half, one third, or one quarter—new points would repeatedly line up. The golden angle avoids those neat repetitions for longer than any nearby angle, so the points spread out instead of forming wasteful spokes.
Where does 137.507° come from?
Start with the golden ratio φ = (1 + √5) / 2. Dividing a full turn in the golden ratio leaves a smaller portion equal to about 0.381966 of the circle. Multiply that by 360° and you get the golden angle.
Using the smaller angle makes the direction of growth easy to see. Turning by the larger complement, 222.493°, produces the same set of positions in reverse rotational direction.
α = 360° × (1 − 1/φ) ≈ 137.507764°- φ ≈ 1.618034
- 1 − 1/φ ≈ 0.381966
- 360° × 0.381966 ≈ 137.507°
Build the pattern point by point
Number the points n = 0, 1, 2, 3… . Point n gets angle nα. Its distance from the center is proportional to √n, which keeps the average area per point roughly constant.
Convert the polar coordinates to x and y with x = r cos θ and y = r sin θ. That is the whole drawing algorithm. Color, dot size, and spacing can change the mood without changing the underlying structure.
θₙ = nα · rₙ = c√n · (x, y) = (r cos θ, r sin θ)
Why do Fibonacci spirals appear?
Your eye groups nearby points into curved arms called parastichies. The visible arm counts are often neighboring Fibonacci numbers such as 34 and 55 or 55 and 89.
Those numbers appear because ratios of consecutive Fibonacci numbers are increasingly accurate approximations of φ. As the pattern grows, a different neighboring pair becomes the easiest family of curves to see. The plant is not drawing Fibonacci spirals one by one; they emerge from the repeated angle.
- Small heads may show 13 and 21 arms.
- Larger heads often show 34 and 55, or 55 and 89.
- Counts vary in living plants because growth is physical, not perfectly mathematical.
A tiny experiment
Set the generator to 1,000 points and start near 137.507°. Move the angle by only 0.2°. Ordered arms begin to bunch into stronger spokes. Try 120° and the points collapse into three obvious directions; try 144° and a fivefold rhythm appears.
This sensitivity is the lesson: irrational turns distribute repeated events, while simple rational turns synchronize them. The same idea matters in sampling, animation, procedural art, and antenna layouts.
Turn the idea into an experiment.
The quickest way to understand a pattern is to change it and watch what happens.