The Cantor set
The Cantor set begins with a line segment and repeatedly removes the open middle third from every remaining segment. What survives looks sparse, yet its mathematics is remarkably rich.
A construction by removal
After one step, two closed thirds remain; after two, four ninths; after n steps, 2ⁿ segments of length 3⁻ⁿ remain. The Cantor set is the intersection of all stages, containing points that are never removed.
After the first step two intervals of length one third remain; after the second there are four of length one ninth, and after step n there are 2ⁿ intervals of length 3⁻ⁿ. The rule is complete even though no display can draw the limiting set. A point belongs precisely when it never falls into a removed middle third. This view turns the construction into an exact selection procedure rather than merely a sequence of thinner lines.
Zero length, uncountably many points
The removed lengths sum to one, so the remaining set has total length zero. Nevertheless it is uncountable. This contrast helped dismantle the intuition that a set with many points must occupy a correspondingly large length.
The total length remaining after n stages is (2/3)ⁿ and tends to zero. Nevertheless, the limiting set contains uncountably many points: every infinite sequence of left and right choices selects one point. Zero length therefore does not mean “almost no points”; it means no one-dimensional extent in Lebesgue measure. The separation between cardinality and measure is one reason the set became historically instructive.
Digits reveal the self-similarity
In base three, Cantor-set points can be represented using only digits 0 and 2, with the usual care for dual expansions. Two half-sized logical choices occur at every step, giving similarity dimension log 2 / log 3.
In ternary notation, the selected numbers are exactly those that can be written without the digit 1: 0 chooses the left branch and 2 the right. Geometry becomes a symbolic sequence. The first n digits determine an interval at stage n, while the remaining digits refine the location inside it. Self-similarity appears not only in the picture, but as a shift acting on infinite sequences.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Cantor SetWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
Fractal dimension
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Sierpiński triangle and carpet
How repeated removal creates exact self-similarity, vanishing area and a bridge from the Cantor set to higher dimensions.
When curves became ‘pathological’
Why continuous nowhere-smooth and space-filling curves disturbed classical intuition—and later became central examples.