Generate Generalized Cantor Set

What is a Generalized Cantor Set?

A Generalized Cantor Set (often called an $\alpha$-Cantor Set) is a family of fractals parameterized by the middle gap fraction $\alpha \in (0, 1)$. Rather than being forced to remove the exact middle third ($\alpha = 1/3 \approx 33.3\%$), this generator allows you to customize the width of the removed middle interval, leaving two symmetric intervals of length $(1-\alpha)/2$ on either side.

By varying $\alpha$ from very close to $0$ to very close to $1$, you can sweep the entire range of fractal dimensions between $0$ and $1$. This makes the generalized Cantor set an extremely powerful tool for studying the relationship between topological properties, measure theory, and fractal dimensionality.

Mathematical Formulation

Starting with a unit interval $I_0 = [0, 1]$, we construct the set by repeatedly performing the following operation on every remaining interval $[x, y]$ of length $L = y - x$:

• Remove the open middle interval of length $\alpha \cdot L$.
• This leaves two symmetric segments of length $L_{new} = \frac{1 - \alpha}{2} \cdot L$.

At iteration step $n$, there are $2^n$ segments, each of length: $$L_n = \left(\frac{1-\alpha}{2}\right)^n$$

Fractal Dimension

Because the set is symmetric and self-similar, its Hausdorff dimension $D$ can be calculated directly using the classic self-similarity formula: $$D = \frac{\ln(2)}{\ln\left(\frac{2}{1-\alpha}\right)}$$

By adjusting $\alpha$, we observe key limiting behaviors:

• As $\alpha \to 0$ (no gaps), the dimension $D \to 1.0$. The set approaches the continuous line segment $[0, 1]$.
• As $\alpha \to 1$ (entire middle removed), the dimension $D \to 0.0$. The set approaches a pair of isolated boundary points $\{0, 1\}$.
• When $\alpha = 1/3$, we get $D = \frac{\ln(2)}{\ln(3)} \approx 0.6309$, which is the dimension of the standard Cantor ternary set.