quaternion_julia_fractal

Quaternion Julia Set Fractal

This example generates and visualizes a quaternion Julia set fractal, demonstrating the combination of quaternion mathematics with visualization.

What You'll Learn

• Understanding quaternion Julia sets
• Iterating quaternion functions
• Visualizing 4D fractals in 3D space
• Color mapping based on iteration counts

Prerequisites

• V compiler installed (download here)
• VSL library installed (installation guide)
• Basic understanding of fractals (helpful but not required)

Running the Example

# Navigate to this directory
cd examples/quaternion_julia_fractal

# Run the example
v run main.v

Expected Output

The example generates an interactive 3D plot showing:

• Fractal pattern: Quaternion Julia set visualized as colored points
• Color mapping: Points colored by iteration count (darker = more iterations)
• 2D slice: Shows a 2D slice through 4D quaternion space
• Console output: Progress and fractal parameters

Code Walkthrough

1. Julia Set Iteration

import vsl.quaternion

fn julia_iterate(c quaternion.Quaternion, z quaternion.Quaternion, max_iter int) int {
    mut current := z.copy()
    for i in 0..max_iter {
        current = current.multiply(current).add(c)  // z = z² + c
        if current.abs() > 2.0 {
            return i  // Escaped
        }
    }
    return max_iter  // Didn't escape
}

This implements the Julia set iteration: z = z² + c. Points that escape (magnitude > 2) are colored by iteration count.

2. Grid Generation

import vsl.quaternion

fn julia_iterate(c quaternion.Quaternion, z quaternion.Quaternion, max_iter int) int {
    mut current := z.copy()
    for i in 0..max_iter {
        current = current.multiply(current).add(c)
        if current.abs() > 2.0 {
            return i
        }
    }
    return max_iter
}

resolution := 30
range_val := 2.0
w_fixed := 0.0
z_fixed := 0.0
c := quaternion.quaternion(-0.123, 0.745, 0.0, 0.0)
max_iter := 50

// Sample 2D slice: fix w=0, z=0, vary x and y
for i in 0..resolution {
    for j in 0..resolution {
        x_val := -range_val + (2.0 * range_val * f64(i) / f64(resolution - 1))
        y_val := -range_val + (2.0 * range_val * f64(j) / f64(resolution - 1))
        z_init := quaternion.quaternion(w_fixed, x_val, y_val, z_fixed)
        iterations := julia_iterate(c, z_init, max_iter)
        // ... store results
    }
}

We sample a 2D slice through 4D quaternion space by fixing two components and varying the others.

3. Color Mapping

Points are colored based on how many iterations it took to escape, creating the fractal pattern.

Mathematical Background

Julia Sets

A Julia set is the set of points that don't escape to infinity under iteration of a complex function. For quaternions, we use:

zₙ₊₁ = zₙ² + c

Where:

• z is a quaternion
• c is a constant quaternion
• Points that escape (|z| > 2) are colored by iteration count

4D Visualization

Quaternions are 4D, but we visualize them by:

• Taking 2D slices (fixing 2 components)
• Using color to represent additional information
• Creating multiple slices for full visualization

Experiment Ideas

Try modifying the example to:

• Change constant c: Different values create different fractal patterns
• Increase resolution: Higher resolution = more detail (slower)
• Change slice plane: Fix different components (w, x, y, z)
• Modify iteration function: Try z³ + c or other functions
• Create animation: Generate multiple slices for animation

Performance Notes

• Higher resolution increases computation time quadratically
• Max iterations affects both detail and computation time
• For interactive exploration, start with resolution=20-30

Related Examples

• quaternion_rotation_3d - Basic quaternion operations
• noise_fractal_2d - Other fractal examples
• plot_scatter3d_1 - 3D plotting basics

Related Tutorials

• Quaternion Introduction
• 3D Visualization

Troubleshooting

Slow generation: Reduce resolution or max_iter

Plot doesn't open: Ensure web browser is installed

No fractal pattern: Try different constant c values

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Explore more fractal and quaternion examples in the examples directory.