Cantor set

Fri, 16 Oct 2020 18:56:00 +0200

The Cantor ternary set is a remarkable subset of the real numbers named after German mathematician Georg Cantor who described the set in 1883. It has the same cardinality as \(\mathbb{R}\), yet it has zero Lebesgue measure.

The set can be constructed recursively by first considering the unit interval \(C_{0}=[0,1]\). In the next step, we divide the set into three equal parts and discard the open middle set. This leads to the new set \(C_{1}=[0,\tfrac{1}{3}]\cup[\tfrac{2}{3},1]\). This procedure is repeated on the two remaining subintervals \([0,\tfrac{1}{3}]\) and \([\tfrac{2}{3},1]\), and iteratively on all remaining subsets such that \(C_n = \tfrac{1}{3}C_{n-1} \cup (\tfrac{2}{3} + \tfrac{1}{3}C_{n-1})\). The Cantor set is defined by the limit as \(n\to\infty\), i.e., \(\mathcal{C} = \cap_{n=0}^{\infty} C_{n}\).

The recursion can be illustrated in python in the following way

  import numpy as np

  def transform_interval(x, scale=1.0, translation=0.0):
      return tuple(map(lambda z: z * scale + translation, x))

  def Cantor(n):
      if n==0: return {(0,1)}
      Cleft = set(map(lambda x: transform_interval(x, scale=1/3), Cantor(n-1)))
      Cright = set(map(lambda x: transform_interval(x, translation=2/3), Cleft))
      return Cleft.union(Cright)

Python on Klaus K. Holst

Cantor set