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. Try the standard Cantor Set for the classic ternary version, or the Asymmetric Cantor Set for unequal left and right ratios.

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.