Sunflower Seed Spiral

By the end of this lab you'll be able to:

• Use the golden angle (≈ 137.5°) to place dots in a sunflower spiral pattern
• Understand why the golden angle produces optimal packing
• Calculate dot positions using r = sqrt(n) and angle n * 137.5°

The seeds in a real sunflower head are arranged by the golden angle — 137.508°. This lab reproduces that spiral exactly. The same packing appears in pinecones, pineapples, and artichokes.

Welcome to the Sunflower!

Sunflowers discovered the golden angle millions of years before mathematicians did. It packs the most seeds into the smallest space — nature's optimal packing algorithm! Let's code it together!

How It Works

Seed n is placed at polar coordinates r = scale * sqrt(n), angle n * golden_angle. Converting to x, y: x = r * cos(angle), y = r * sin(angle). The colors cycle through a sunny palette for each "arm" of the spiral.

Sample Code

import turtle
import math
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()

golden_angle = 137.508
scale = 8
colors = ['gold', 'darkorange', 'goldenrod', 'orange']

for n in range(200):
    angle = math.radians(n * golden_angle)
    r = scale * math.sqrt(n)
    x = r * math.cos(angle)
    y = r * math.sin(angle)
    monty.penup()
    monty.goto(x, y)
    monty.pendown()
    size = max(2, 8 - r / 20)
    monty.dot(int(size), colors[n % len(colors)])

What Do You Think Will Happen?

200 dots placed one at a time, each rotated exactly 137.5° from the last. Will they form a spiral, a grid, or something that looks like a real sunflower? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

A real sunflower pattern — you can see the clockwise and counter-clockwise spirals that sunflowers are famous for. Were you right?

How It Works

math.sqrt(n) for the radius spreads seeds evenly outward — without it all seeds would cluster near the edge. The golden angle 137.508° is the most "irrational" angle possible, meaning no seed ever lands directly in front of a previous one. This ensures maximum spread and minimum wasted space.

Explanation Table

Line	What it does
golden_angle = 137.508	The golden angle — creates optimal packing
r = scale * math.sqrt(n)	Spread seeds outward at a uniform rate
x = r * math.cos(angle)	Convert polar → x coordinate
y = r * math.sin(angle)	Convert polar → y coordinate
size = max(2, 8 - r/20)	Larger dots at center, smaller at edge

Learning Check

Your Turn — Try a Different Angle

Change golden_angle = 137.508 to golden_angle = 138. Predict what will happen to the packing — better or worse than the golden angle? Run it to compare!

▶ Run ↻ Reset

At 138°, the seeds cluster into radial spokes — the packing is worse than 137.508°! The golden angle truly is special: even a tiny change produces visible clustering.

Experiments

1. Try 90°. Change to golden_angle = 90. You'll know it worked when only 4 spokes appear (every 4th seed lands in the same direction).

2. Use 400 seeds. Change range(200) to range(400). You'll know it worked when the spiral extends much further and looks denser.

3. Use a single color. Replace the colors list with just 'forestgreen'. You'll know it worked when all dots are the same color.

4. Try the exact golden ratio angle. The golden ratio φ ≈ 1.618; the golden angle = 360° × (1 − 1/φ) ≈ 222.5°. Try this value and see if it differs from 137.508°.

Nature's Algorithm!

You coded the same packing algorithm nature uses in sunflowers, pinecones, and galaxies! The golden angle appears everywhere in biology because it maximizes resource efficiency. Up next: Spider Web — concentric rings connected by radial threads.