Base 3 representation of Cantor Set

Base-3 Representation of the Cantor Set

• Definition: The Cantor set consists of all numbers in the interval [0, 1] whose base-3 (ternary) representation contains only the digits 0 and 2. Numbers containing the digit 1 are excluded.
• Step-by-Step Construction:
  • Start with the interval [0, 1]. Every number x in this interval can be written in base-3 as x = 0.a₁a₂a₃..., where aₖ ∈ {0, 1, 2}.
  • Remove the open middle third (1/3, 2/3). Numbers in this interval have their first digit a₁ = 1 in base-3. The remaining intervals are [0, 1/3] and [2/3, 1].
  • Remove the middle third of each remaining interval. Numbers in these intervals have their second digit a₂ = 1. The remaining intervals are [0, 1/9], [2/9, 1/3], [2/3, 7/9], and [8/9, 1].
  • Repeat this process infinitely. At each step, remove all numbers whose base-3 representation contains 1 in the corresponding digit aₖ.
• Final Cantor Set in Base-3:
  • The Cantor set consists of all numbers in [0, 1] whose base-3 digits are restricted to 0 and 2.
  • Examples of numbers in the Cantor set include 0.000...₃ = 0, 0.222...₃ = 1, 0.202020...₃ = 2/3, and 0.200200...₃ = 8/27.
  • Any number x in the Cantor set can be written as x = Σ (aₖ / 3ᵏ) for k = 1 to ∞, where aₖ ∈ {0, 2}.
• Key Properties:
  • Uncountable Infinity: The Cantor set is uncountably infinite because there are infinitely many sequences of 0s and 2s in base-3.
  • Measure Zero: Despite being uncountable, the Cantor set has a total "length" of 0, as the total length removed (sum of the middle thirds) adds to 1.
  • Self-Similarity: The Cantor set is fractal-like, with each segment resembling the whole.
  • Binary Representation: By replacing 0 with 0 and 2 with 1, the Cantor set corresponds directly to binary sequences, making it easier to study its structure.

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