The Cantor Set is a fractal set introduced by the German mathematician Georg Cantor in 1883. It is constructed from a segment of length 1 (usually the interval [0, 1]), and recursively, the middle third of each remaining segment is removed.
- Construction (algorithmic): Start with the interval [0, 1]. In the first step, remove the central third (1/3, 2/3), leaving two segments: [0, 1/3] and [2/3, 1]. In the next step, do the same with each remaining segment, and so on.
- Fractal Dimension: Although visually it results in a "very thin" set (infinite points), it is mathematically proven that its Hausdorff dimension is \(\displaystyle \frac{\ln 2}{\ln 3}\), approximately 0.6309.
- Lebesgue Measure: After removing infinitely many thirds, the total measure (length) of the resulting set is 0. That is, it occupies "zero space" on the line.
- Cardinality: Despite having a measure of 0, surprisingly, the Cantor Set contains infinitely many points; in fact, it has the same cardinality as the interval [0, 1].
This computational example allows you to see "step by step" how the Cantor Set is constructed, removing the middle third at each iteration. As you progress through many iterations, you will observe the fractal nature of the figure.