Fractal Dimension Calculator
Fractal Dimension Calculator computes box-counting dimension for 2D/3D fractals, visualizes self-similar structures.
Fractal Dimension Calculator
The Fractal Dimension Calculator computes the box-counting dimension (\\(D\\)) to quantify the complexity of fractal sets. The dimension is calculated as:
\\(D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log (1/\epsilon)}\\), where \\(N(\epsilon)\\) is the number of boxes of size \\(\epsilon\\) covering the fractal.
Fractal Dimension Calculator: Mandelbrot Set (2D)
Equation: \\(z_{n+1} = z_n^2 + c\\), where \\(c = x + iy\\) is a complex number.
Box-counting is applied to the boundary of the set where points escape after a number of iterations.
Fractal Dimension Calculator: Mandelbrot Example 1 (Full Set)
This example computes the fractal dimension of the Mandelbrot set’s boundary over a standard region.
- Inputs:
- Mode: 2D (Mandelbrot Set)
- Real Min (\\(x_{\text{min}}\\)): -2.0
- Real Max (\\(x_{\text{max}}\\)): 1.0
- Imag Min (\\(y_{\text{min}}\\)): -1.5
- Imag Max (\\(y_{\text{max}}\\)): 1.5
- Resolution: 400
- Max Iterations: 100
- Box Sizes: 4,8,16,32
- Evaluation Point (x,y): 0.0,0.0
- Expected Output:
- Iteration Count at (0.0,0.0): 100 (inside set)
- Fractal Dimension (\\(D\\)): ~2.0
- Visualization: The canvas shows the Mandelbrot set with a black interior and colored boundary, highlighting self-similar structures.
Fractal Dimension Calculator: Mandelbrot Example 2 (Zoomed Boundary)
This example zooms into a complex boundary region to compute a more precise fractal dimension.
- Inputs:
- Mode: 2D (Mandelbrot Set)
- Real Min (\\(x_{\text{min}}\\)): -0.75
- Real Max (\\(x_{\text{max}}\\)): -0.70
- Imag Min (\\(y_{\text{min}}\\)): 0.10
- Imag Max (\\(y_{\text{max}}\\)): 0.15
- Resolution: 400
- Max Iterations: 100
- Box Sizes: 4,8,16,32
- Evaluation Point (x,y): -0.725,0.125
- Expected Output:
- Iteration Count at (-0.725,0.125): ~50 (near boundary)
- Fractal Dimension (\\(D\\)): ~1.8
- Visualization: The canvas shows a zoomed-in view of the Mandelbrot boundary, revealing intricate self-similar patterns.
Fractal Dimension Calculator: Cantor Dust (3D)
Generates a 3D Cantor set by recursively dividing a cube, applying box-counting to estimate dimension.
Fractal Dimension Calculator: Cantor Dust Example 1
This example computes the fractal dimension of a 3D Cantor dust with moderate recursion depth.
- Inputs:
- Mode: 3D (Cantor Dust)
- Cube Size: 1.0
- Recursion Depth: 3
- Resolution: 100
- Max Iterations: 100
- Box Sizes: 4,8,16,32
- Evaluation Point (x,y,z): 0.5,0.5,0.5
- Expected Output:
- Presence at (0.5,0.5,0.5): Inside set
- Fractal Dimension (\\(D\\)): ~1.585 (theoretical: \\(\log(8)/\log(3) \approx 1.585\\))
- Visualization: The 3D plot shows a Cantor dust with cyan cubes, illustrating self-similarity in three dimensions.
Fractal Dimension Calculator: Cantor Dust Example 2 (Deeper Recursion)
This example increases recursion depth for a more detailed Cantor dust.
- Inputs:
- Mode: 3D (Cantor Dust)
- Cube Size: 1.0
- Recursion Depth: 4
- Resolution: 100
- Max Iterations: 100
- Box Sizes: 4,8,16,32
- Evaluation Point (x,y,z): 0.5,0.5,0.5
- Expected Output:
- Presence at (0.5,0.5,0.5): Inside set
- Fractal Dimension (\\(D\\)): ~1.585
- Visualization: The 3D plot shows a denser Cantor dust, emphasizing finer self-similar structures.