Julia Set Pixel Art

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

• Modify the Mandelbrot iteration to compute a Julia set by swapping z and c
• Understand how changing one complex constant c produces dramatically different shapes
• Compare connected vs. disconnected Julia sets (Fatou dust)

The Julia set for c = -0.7 + 0.27i — a connected, swirling shape sometimes called the "Douady rabbit." Change c and the whole shape transforms; some values give spirals, others give dust.

Welcome to the Julia Set!

In the Mandelbrot set, c changes for each pixel and z starts at 0. In a Julia set, c is fixed and z starts at each pixel's position. Swapping these two roles gives an entirely different fractal! Let's code it together!

How It Works

For each pixel (px, py), set z = (pixel's complex position) and c = -0.7 + 0.27j (fixed). Repeatedly apply z = z² + c. Color by escape time. The set of points that never escape is the Julia set.

Sample Code

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

cols = 60
rows = 50
pixel = 5
max_iter = 25
c = complex(-0.7, 0.27)

colors = ['black', 'darkblue', 'navy', 'blue', 'mediumblue',
          'royalblue', 'deepskyblue', 'cyan', 'limegreen',
          'yellow', 'orange', 'red', 'white']

for py in range(rows):
    for px in range(cols):
        zr = -1.5 + px * 3.0 / cols
        zi = -1.25 + py * 2.5 / rows
        z = complex(zr, zi)
        count = 0
        while abs(z) <= 2 and count < max_iter:
            z = z * z + c
            count += 1
        color_idx = min(count, len(colors) - 1)
        monty.goto(-150 + px * pixel, -125 + py * pixel)
        monty.dot(pixel + 1, colors[color_idx])

What Do You Think Will Happen?

The Julia set for c = -0.7 + 0.27i is a connected blob with a swirling boundary. Will it look more like the Mandelbrot, or completely different? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

The Julia set looks very different from the Mandelbrot — it's a symmetric, swirling connected shape. Were you right?

How It Works

The key difference from Mandelbrot: - Mandelbrot: z = 0, c = pixel_position — tests each point as a parameter - Julia: z = pixel_position, c = fixed — tests each point as an initial value

The Julia set is connected if and only if c is inside the Mandelbrot set.

Explanation Table

Line	What it does
c = complex(-0.7, 0.27)	Fixed parameter — determines the Julia shape
z = complex(zr, zi)	Starting value = current pixel position
z = z * z + c	Same iteration as Mandelbrot — but c never changes
zr = -1.5 + px * 3.0 / cols	Map pixel to complex z-plane x-coordinate

Learning Check

Your Turn — Try a Different c Value

Change c = complex(-0.7, 0.27) to c = complex(0.285, 0.01). Each c value creates a completely different Julia set. Predict whether it'll look similar or very different — then run it to explore!

▶ Run ↻ Reset

c = 0.285 + 0.01i is very close to the edge of the Mandelbrot set — the resulting Julia set has a feathery, spiraling boundary.

Experiments

1. Try c = complex(-0.4, 0.6). This creates a "dragon" shape. You'll know it worked when you see a jagged, asymmetric form.

2. Try c = complex(0.0, 1.0). This is outside the Mandelbrot set — the Julia set becomes "Fatou dust" (disconnected scattered points). You'll know it worked when most pixels are colored, not black.

3. Higher max_iter. Change max_iter = 50. More iterations reveal finer boundary detail. You'll know it worked when the transition between colors is smoother.

4. Widen the view. Change zr = -2.0 + px * 4.0 / cols and zi = -1.67 + py * 3.33 / rows. You'll know it worked when the full Julia set fits in the frame.

Two Fractals, One Rule!

The Mandelbrot and Julia sets use exactly the same iteration — just with z and c swapped! That tiny change gives you an infinite family of different fractals from one formula. Up next: Sierpiński Carpet — 8-way recursive subdivision.