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What are Julia Sets?

Julia sets are closely related to the Mandelbrot set but offer a different perspective on fractal geometry. Named after French mathematician Gaston Julia, these sets represent a family of fractals, each defined by a specific complex parameter.
While the Mandelbrot set uses the formula z(n+1) = z(n)² + c where c varies and z starts at 0, Julia sets keep c constant and vary the starting value of z.

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The Key Difference from Mandelbrot

The distinction is implemented in the mandel_vs_julia function: 
static void mandel_vs_julia(t_complex *z, t_complex *c, t_fractol *fractol)
{
    if (!ft_strncmp(fractol->name, "Julia"))
    {
        // Julia: c is constant, z varies by pixel position
        c->x = fractol->julia_x;
        c->y = fractol->julia_y;
    }
    else
    {
        // Mandelbrot: c varies by pixel position, z starts at 0
        c->x = z->x;
        c->y = z->y;
    }
}

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Understanding the Algorithm

For Julia sets:
  1. The parameter c is fixed to values you specify (julia_x, julia_y)
  2. For each pixel, z starts at that pixel’s complex coordinate
  3. The iteration z = z² + c is applied repeatedly
  4. The coloring depends on whether z escapes to infinity
The Mandelbrot set can be thought of as a “map” of all possible Julia sets. Each point in the Mandelbrot set corresponds to a connected Julia set, while points outside correspond to disconnected “dust” Julia sets.

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The Two Parameters

Julia sets are defined by two real numbers that represent a complex parameter c = julia_x + julia_y*i:

julia_x

The Real ComponentTypically ranges from -2.0 to +2.0Controls the horizontal aspect of the fractal’s structure

julia_y

The Imaginary ComponentTypically ranges from -2.0 to +2.0Controls the vertical aspect of the fractal’s structure

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Running Julia Sets

To launch a Julia set, you must provide both parameters: 
./fractol Julia <julia_x> <julia_y>

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Example Commands

# Classic Julia set - creates spiral patterns
./fractol Julia -0.7 0.27015

# Dendrite Julia set - tree-like branching structure
./fractol Julia 0.0 1.0

# Douady's Rabbit - famous three-lobed shape
./fractol Julia -0.123 0.745

# Siegel Disk - smooth circular regions
./fractol Julia -0.391 -0.587

# San Marco Dragon
./fractol Julia -0.75 0.11

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Parameter Input

The parameters are parsed from command-line arguments: 
int main(int argc, char **argv)
{
    t_fractol fractol;
    
    if (argc == 4 && !ft_strncmp(argv[1], "Julia"))
    {
        fractol.name = argv[1];
        fractol.julia_x = atoi_plus(argv[2]);  // Parse first parameter
        fractol.julia_y = atoi_plus(argv[3]);  // Parse second parameter
        
        fractol_init(&fractol);
        fractol_render(&fractol);
        mlx_loop(fractol.mlx_connection);
    }
}
The atoi_plus function handles floating-point parsing, so you can use decimal values like -0.7 or 0.27015.

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Exploring Different Julia Sets

Connected Julia Sets (c inside Mandelbrot)
  • Form a single continuous shape
  • Examples: c = -0.7 + 0.27015i, c = 0.285 + 0.01i
  • Display intricate, organic patterns
Disconnected Julia Sets (c outside Mandelbrot)
  • Break into isolated fragments (“Fatou dust”)
  • Examples: c = 1.0 + 0.0i, c = -2.0 + 0.0i
  • Create scattered, chaotic patterns
All Julia sets have rotational symmetry of 180 degrees around the origin.Some special cases:
  • Real c values (julia_y = 0): bilateral symmetry across the x-axis
  • Imaginary c values (julia_x = 0): bilateral symmetry across the y-axis
  • c on the unit circle: may have additional rotational symmetries
Dendrite (c = i)
  • Tree-like branching structure
  • One of the most visually striking
Douady’s Rabbit (c ≈ -0.123 + 0.745i)
  • Named for its rabbit-like appearance
  • Famous in fractal literature
San Marco Dragon (c ≈ -0.75 + 0.11i)
  • Dragon-shaped spiral arms
  • Located near the Mandelbrot set’s “neck”
Like the Mandelbrot set, Julia sets are rendered using the escape-time algorithm with:
  • escape_value: 4 (default)
  • iterations_definition: 30 (default, adjustable with +/- keys)
  • Color mapping: PSYCHEDELIC_LIME to PSYCHEDELIC_MINT gradient
  • Points in the set: Rendered as WHITE

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Visual Exploration Tips

How to find interesting parameters:
  1. Pick points near the Mandelbrot set’s boundary for connected Julia sets
  2. Points inside the main cardioid create simple circular patterns
  3. Points near the “neck” (around -0.75, 0i) create dramatic spirals
  4. Points on the boundary create the most intricate structures

Beginner-Friendly

./fractol Julia -0.7 0.27
./fractol Julia -0.8 0.156
./fractol Julia 0.285 0.01
Clear, beautiful patterns that are easy to navigate

Advanced Exploration

./fractol Julia -0.4 0.6
./fractol Julia 0.285 0.0
./fractol Julia -0.835 -0.2321
Complex structures requiring patience to explore

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Mathematical Connection to Mandelbrot

The relationship between Mandelbrot and Julia sets is profound: 
// The same iteration formula, different variable roles:
// Mandelbrot: c = pixel_position, z starts at 0
// Julia: c = constant parameter, z = pixel_position

while (i < fractol->iterations_definition)
{
    z = sum_complex(sqare_complex(z), c);  // z = z² + c
    
    if ((z.x * z.x) + (z.y * z.y) > fractol->escape_value)
    {
        // Escaped - color based on iteration count
        color = map(i, PSYCHEDELIC_LIME, PSYCHEDELIC_MINT,
                fractol->iterations_definition);
        return;
    }
    i++;
}
Every point in the Mandelbrot set represents a Julia set parameter that produces a connected fractal. This makes the Mandelbrot set a “catalog” of all interesting Julia sets.

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Interactive Exploration

Once launched, use the same controls as the Mandelbrot set:
  • Zoom in/out: Mouse scroll wheel
  • Pan: Arrow keys or WASD
  • Detail adjustment: +/- keys (modify iteration count)
  • Exit: ESC key
Experiment with different parameters to discover the incredible variety of Julia set fractals. Each parameter pair creates an entirely unique universe to explore.

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