The Cantor set

What just happened?

You started from the interval $[0,1]$. At each iteration, from every segment you remove the open middle third (at the start the interval $(1/3,\,2/3)$) — leaving two segments, each $1/3$ of the length. You repeat this forever. The full bar turns into an ever finer “dust” of points (the grey line is the full starting segment, from which the middles disappear).

The length shrinks geometrically to zero:

Since each segment leaves two with a combined length of $\tfrac{2}{3}$ of the previous one, the length is multiplied by $\tfrac{2}{3}$ each iteration. The ratio $\tfrac{2}{3} < 1$, so the sequence $L_n$ is convergent to zero — after infinitely many steps no “length” is left, even though the number of segments grows like $2^n$.

The paradox: length 0, yet uncountably many points

And yet the Cantor set is not empty — it contains uncountably many points, exactly as many as the whole interval $[0,1]$ (it has the cardinality of the continuum). These are the points that, in their ternary expansion, can be written without the digit 1. So we have a set of length (measure) zero, yet equinumerous with the whole interval — a paradox linking measure theory with Cantor’s set theory.

Self-similarity and fractal dimension:

Each of the 2 halves is a scaled-down copy of the whole (scale 1/3) — the set is self-similar. Its fractal dimension is $\log 2/\log 3 \approx 0.631$ — it lies between a point (0) and a line (1): the structure is “more than a set of points, less than an interval”.

Who discovered it?

The German mathematician Georg Cantor (1845–1918) described this set in 1883. He founded set theory and the notion of different “sizes” of infinity (cardinal numbers). The Cantor set is one of the first and most famous fractals — and at the same time a classic example of how unreliable intuition can be when it comes to infinity.

Why this matters:

This is a textbook example of a geometric sequence convergent to zero ($L_n = L_0 q^n$, $q = \tfrac{2}{3} < 1$) alongside a divergent sequence ($N_n = 2^n \to \infty$). It ties together limits of sequences, powers and fractal geometry — and teaches caution: “length 0” does not mean “empty”.

Remember: the length of the Cantor set shrinks by ×2/3 each iteration and tends to zero ($(2/3)^n \to 0$), and yet it still contains uncountably many points. Vanishing length, uncountable richness.