Cantor Set

Definition

• The Cantor Set is a mathematical construct that lies on a single line segment.
• It was discovered in 1874 by Henry John Stephen Smith and later mentioned by German mathematician Georg Cantor in 1883.
• The most common construction is the Cantor ternary set.
• As a general rule, mathematics of the normal kind cannot be performed on the Cantor Set as the result will not be in the set.

Context

Cantor Ternary Set

• Start with the closed interval [0, 1].
• Remove the open middle third from this interval, leaving two line segments: \([0, \frac{1}{3}]\) and \([\frac{2}{3}, 1]\).
• Next, delete the open middle third from each of these remaining segments, resulting in four segments.
• Continue this process infinitely: remove the middle third from each remaining segment at every step.
• The Cantor ternary set contains all points in the interval [0, 1] that are not deleted during this process.
• Mathematician Benoit Mandelbrot whimsically described the Cantor set's construction as a bar curdling towards its extremities.

Recursive Formulas

  - We can describe the Cantor set recursively:
    - Let \(I_0 = [0, 1]\).
    - For each \(n \geq 0\), divide each remaining segment into thirds, and remove the middle third. Denote the resulting segments as \(I_n\).
    - The Cantor set is then defined as the infinite intersection: \(C = \bigcap_{n=0}^{\infty} I_n\).

Properties

• **Uncountability**: The Cantor set contains more points than the set of natural numbers—it's "more infinite."
• **Self-Similarity**: The Cantor set exhibits self-similarity, where each part resembles the whole.
• **Fractal Nature**: Its complement (the missing points) forms a fractal string.
• **Ternary Representation**: Each point in the Cantor set corresponds to a ternary (base 3) fraction without the digit 1.

Visualizing the Cantor Set

• Imagine an infinitely deep binary tree.
• Each point in the Cantor set has a unique path through this tree, turning left or right at each level based on its position relative to deleted segments.
• Representing left turns as 0 and right turns as 2 yields the ternary fraction for each point.

Source: Conversation with Copilot, 7/8/2024

(1) THE CANTOR SET - A BRIEF INTRODUCTION - Mathematics. https://www.math.uwaterloo.ca/~xzliu/cantor-set.pdf.
(2) Cantor set - Wikipedia. https://en.wikipedia.org/wiki/Cantor_set.
(3) Cantor Set: Generalization, Properties & Everything You Need To Know. https://calculus-help.com/2019/06/10/cantor-set/.
(4) A Generalized Cantor Theorem | SpringerLink. https://link.springer.com/chapter/10.1007/978-1-4899-3558-8_41.
(5) CANTOR SET Definition & Meaning | Dictionary.com. https://www.dictionary.com/browse/cantor-set.
(6) Cantor set - Simple English Wikipedia, the free encyclopedia. https://simple.wikipedia.org/wiki/Cantor_set.
(7) Cantor Set - Definition, Cardinality, Formula, and Applications. https://mathmonks.com/sets/cantor-set.