Generate Smith Volterra Cantor Set

What is the Smith-Volterra-Cantor Set?

The Smith-Volterra-Cantor Set (also known as the Fat Cantor Set) is one of the most remarkable and counter-intuitive constructions in real analysis. Unlike standard Cantor sets where we remove a constant fraction of each remaining segment (which always results in a set of zero total length), the Smith-Volterra-Cantor set removes a fraction that decreases exponentially with each iteration.

This results in a fractal dust that has a positive length (measure), yet contains no intervals, has no interior points, and is completely disconnected! It was first constructed by the Irish mathematician Henry Smith in 1875 and independently by the Italian mathematician Vito Volterra in 1881.

Mathematical Construction

Starting with the unit interval $I_0 = [0, 1]$, the set is built recursively. At step $n \ge 1$, we remove a middle interval of length $w_n$ from each of the $2^{n-1}$ remaining intervals.

In the standard formulation with a scaling parameter $c \in (0, 2)$: $$w_n = \frac{c}{4^n}$$

• Step 1 ($n=1$): We remove the middle interval of length $c/4$ from $[0, 1]$, leaving two intervals of length $(1 - c/4)/2$.
• Step 2 ($n=2$): We remove the middle interval of length $c/16$ from each of the $2$ remaining segments.
• Step $n$: We remove a middle interval of length $c/4^n$ from each of the $2^{n-1}$ remaining segments.

Lebesgue Measure (Total Length)

We can compute the exact total length of all intervals removed from the unit interval. At step $n$, we remove $2^{n-1}$ gaps, each of length $c/4^n$. The sum of the lengths of all removed gaps at iteration infinity is: $$\sum_{n=1}^{\infty} 2^{n-1} \frac{c}{4^n} = \frac{c}{2} \sum_{n=1}^{\infty} \left(\frac{2}{4}\right)^{n-1} = \frac{c}{2} \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^{n-1}$$

Using the formula for the sum of an infinite geometric series: $$\text{Total Removed Length} = \frac{c}{2} \cdot \frac{1}{1 - 1/2} = c$$

Therefore, the total length (Lebesgue measure) of the remaining Smith-Volterra-Cantor set is: $$\text{Remaining Length} = 1 - c$$ For the standard case of $c = 0.5$ (where the removed length is $0.5$), the remaining fat Cantor set has a measure of exactly $0.5$. It is a fractal that occupies a substantial amount of space!