3D fractals represent some of the most visually stunning and mathematically fascinating objects in computer graphics. This series explores how to render these infinite structures in real-time using GLSL shaders, combining ray marching with distance estimation to create breathtaking visualizations.

Series Overview

Each article in this series:

• Explains the mathematical foundation of a 3D fractal
• Provides interactive shader examples you can modify in real-time
• Demonstrates ray marching techniques for efficient rendering
• Shows optimization strategies for real-time performance

Articles in This Series

Classic 3D Fractals

1. Mandelbulb - The 3D extension of the Mandelbrot set. Explore infinite detail in a bulbous, organic structure using spherical coordinates and power iterations.

2. Quaternion Julia Sets - 4D quaternion fractals projected into 3D space. Discover swirling, organic forms with complex rotational symmetries.

3. Menger Sponge - The 3D generalization of the Sierpinski carpet. A recursive geometric fractal with infinite surface area and zero volume.

4. Mandelbox - A box-folded fractal combining sphere inversion with scaling. Creates intricate geometric patterns with sharp edges and smooth curves.

Geometric Fractals

5. Sierpinski Tetrahedron - The 3D version of the Sierpinski triangle. Recursive tetrahedral subdivision creating a fractal with fascinating self-similarity.

6. 3D IFS Fractals - Iterated Function Systems in 3D space. Create complex organic and geometric forms through affine transformations.

Advanced Techniques

7. Distance Estimation Methods - Techniques for efficiently computing distances to fractal surfaces. Essential for fast ray marching.

8. Hybrid Fractals - Combining multiple fractal formulas to create new structures. Explore the creative possibilities of fractal composition.

Why 3D Fractals?

3D fractals are:

• Visually stunning: Infinite detail and organic beauty
• Mathematically rich: Deep connections to complex analysis, topology, and chaos theory
• Computationally fascinating: Require clever algorithms for real-time rendering
• Artistically inspiring: Used in films, games, and digital art

Ray Marching for Fractals

All fractals in this series use ray marching with distance estimation:

1. Distance Estimation: Compute an approximate distance from any point to the fractal surface
2. Ray Marching: Step along rays using the distance estimate for efficient traversal
3. Surface Detection: When close enough, calculate normals and lighting
4. Optimization: Use bounding spheres, early termination, and adaptive step sizes

Interactive Examples

All articles include interactive 3D shader examples using the <GLSLIDE> component. You can:

• Modify parameters in real-time and see results instantly
• Adjust camera angles with mouse movement
• Explore the fractal by zooming and rotating
• Experiment with different formulas and parameters

Interactive Previews

Click the buttons below to switch between different fractal demos. Only one demo runs at a time for optimal performance:

Mathematical Foundations

Complex Numbers → Quaternions

2D fractals use complex numbers: z = a + bi

3D fractals extend to quaternions: q = w + xi + yj + zk

Distance Estimation

For fractals defined by iteration z_{n+1} = f(z_n), the distance estimate is:

d ≈ |z_n| * log(|z_n|) / |z'_n|

Where z'_n is the derivative of the iteration.

Ray Marching Algorithm

float raymarch(vec3 ro, vec3 rd) {
    float t = 0.0;
    for (int i = 0; i < maxSteps; i++) {
        vec3 p = ro + rd * t;
        float d = distanceEstimate(p);
        if (d < epsilon) return t; // Hit
        t += d;
        if (t > maxDist) break; // Miss
    }
    return -1.0;
}

Performance Optimization

Rendering 3D fractals in real-time requires optimization:

• Bounding spheres: Skip empty space quickly
• Adaptive step sizes: Larger steps in empty regions
• Early termination: Stop when close enough or too far
• Level of detail: Reduce iterations for distant objects

References

Academic Papers

• "The Mandelbulb: First 'True' 3D Image of the Mandelbrot Set?" by Daniel White and Paul Nylander
• "Distance Estimation Method for Rendering 3D Fractals" by Inigo Quilez
• "Quaternion Julia Sets" by John C. Hart et al.

Online Resources

• Inigo Quilez - Distance Estimation
• Fractal Forums - Community discussions
• Mandelbulb.com - Gallery and resources

Related Series

• Graphics Gems GLSL Series Index - Classic graphics algorithms
• Introduction to GLSL and Shaders - GLSL fundamentals