Fractals and Hausdorff Dimension

Fractals are geometric objects that exhibit self-similarity at every scale. Unlike smooth curves (dimension 1) or filled surfaces (dimension 2), fractals can have non-integer dimensions — the Hausdorff dimension.

Box-Counting Dimension

The box-counting (Hausdorff) dimension is defined as:

$D = -\lim_{\varepsilon \to 0} \frac{\log N(\varepsilon)}{\log \varepsilon}$

where $N(\varepsilon)$ is the number of boxes of side length $\varepsilon$ needed to cover the fractal.

Classic Fractal Dimensions

Cantor Set

Start with [0,1]. At each iteration, remove the middle third from every interval. After $n$ iterations: $2^n$ intervals of length $(1/3)^n$ remain. Self-similarity ratio $r = 1/3$, copies $N = 2$, so $D = \log(2)/\log(3)$.

Koch Curve

Start with a line segment. Replace each segment with four segments of length $1/3$. Self-similarity: $N=4$ copies at ratio $r=1/3$, so $D = \log(4)/\log(3)$.

Sierpiński Triangle

Start with a filled triangle. Remove the central triangle at each step, leaving 3 copies at half the scale. $D = \log(3)/\log(2)$.

Box-Counting in Practice

Given a set of points, estimate $D$ by counting how many grid boxes of size $\varepsilon$ contain at least one point. Plot $\log N$ vs $\log(1/\varepsilon)$; the slope is $D$.

Implementation

Implement the following functions:

import math

def hausdorff_dim_cantor():
    # Return log(2)/log(3)
    ...

def hausdorff_dim_koch():
    # Return log(4)/log(3)
    ...

def hausdorff_dim_sierpinski():
    # Return log(3)/log(2)
    ...

def cantor_set_count(n_iterations):
    # Return number of intervals after n iterations = 2^n
    ...

def box_count_estimate(points_x, epsilon):
    # Count distinct grid boxes of size epsilon containing points
    ...