Geometric

Cantor set

Georg Cantor described this set in 1883, decades before the word “fractal” existed. Take the unit interval [0, 1]. Remove the open middle third (1/3, 2/3). From each remaining third, remove its open middle third. Repeat forever. The intersection is the Cantor set, a nowhere-dense, perfect, uncountable set of measure zero.

At a glance

Equivalent definitions

• Removal construction: iteratively remove open middle thirds from [0, 1].
• Base-3 characterization: the points in [0, 1] whose base-3 (ternary) expansion contains no “1” digit.
• Iterated function system: attractor of f0(x) = x/3 and f2(x) = (x + 2)/3.
• Symbolic dynamics: continuous image of {0, 2}^ℕ under base-3 expansion.

Why measure zero but uncountable

At step n, the total length of remaining intervals is (2/3)n, which goes to 0. So the Cantor set has Lebesgue measure zero. But every point with a ternary expansion in {0, 2}survives, and there are uncountably many such sequences (bijection with the set of binary sequences). So Cantor set has the cardinality of the continuum despite occupying no length, our first hint that “dimension” needs to mean more than just cardinality or measure.

Generalizations

• Fat Cantor (Smith-Volterra-Cantor): remove smaller portions each step; positive measure but still nowhere dense.
• Cantor dust: 2D / 3D analogues by taking products of Cantor sets.
• Random Cantor sets: random middle-portion removal; appear in branching processes and percolation.

References

• Cantor, G., “Über unendliche, lineare Punktmannigfaltigkeiten,” Math. Ann., 1883.
• Falconer, K., Fractal Geometry, Wiley, 2014.
• Edgar, G., Measure, Topology, and Fractal Geometry, Springer, 2008.
• Sierpinski triangle · Koch snowflake

Try it

Run an interactive playground at /tools/cantor.