The Menger Sponge is the 3D generalization of the Sierpinski carpet, discovered by Karl Menger in 1926. It's a fascinating geometric fractal with paradoxical properties: infinite surface area yet zero volume, making it a perfect example of a fractal dimension between 2 and 3.

Interactive Demo

Move your mouse to rotate the camera and explore the infinite detail. Scroll to adjust the iteration count and see how the structure becomes more complex:

Mathematical Foundation

Construction Process

The Menger Sponge is constructed recursively:

1. Start: A cube
2. Divide: Split into 3×3×3 = 27 smaller cubes
3. Remove: Remove the center cube and the 6 face-center cubes (7 total)
4. Repeat: Apply the same process to the remaining 20 cubes
5. Infinite: Continue ad infinitum

Distance Estimation

We use a fold-based approach for efficient computation:

float mengerSpongeDE(vec3 pos, int iterations) {
    vec3 p = pos;
    float scale = 1.0;
    
    for (int i = 0; i < iterations; i++) {
        // Fold into first octant
        p = abs(p);
        
        // Scale by 3
        p *= 3.0;
        scale *= 3.0;
        
        // Remove center boxes
        if (p.x > 1.0 && p.y > 1.0) p.xy = vec2(2.0) - p.xy;
        if (p.x > 1.0 && p.z > 1.0) p.xz = vec2(2.0) - p.xz;
        if (p.y > 1.0 && p.z > 1.0) p.yz = vec2(2.0) - p.yz;
        
        p -= 1.0;
    }
    
    return sdBox(p, vec3(0.5)) / scale;
}

Fractal Dimension

The Menger Sponge has a fractal dimension of:

$D = \frac{\log(20)}{\log(3)} \approx 2.727$

This is between 2 (surface) and 3 (volume), reflecting its paradoxical nature. The dimension is calculated using the scaling factor (3) and the number of copies (20) that remain after each iteration.

Properties

• Infinite surface area: Each iteration multiplies the surface area
• Zero volume: The removed cubes eventually remove all volume
• Self-similarity: Every part looks like the whole
• Hausdorff dimension: Approximately 2.727

Visual Characteristics

• Geometric precision: Sharp edges and corners
• Infinite detail: Zooming in reveals the same pattern
• Symmetry: Cubic symmetry in all three axes
• Complexity: Each iteration adds 20× more detail

Optimization

The fold-based approach is highly efficient:

• No recursion: Iterative algorithm
• Fast convergence: Distance estimate is accurate
• GPU-friendly: Parallelizable per pixel

Applications

• Mathematical visualization: Teaching fractal geometry
• Architectural inspiration: Self-similar structures
• Game environments: Procedural geometric worlds
• Art: Geometric abstract art

References

• Menger Sponge on Wikipedia
• "Fractal Geometry of Nature" by Benoit Mandelbrot
• Inigo Quilez - Distance Functions

Related Articles

• GLSL 3D Fractals - Sierpinski Tetrahedron - Related geometric fractal
• GLSL 3D Fractals Series Index - Return to series index