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)
