Cantor Set

Cantor Set
Fields: Topology and Fractals
Image Created By: Keith Peters

Cantor Set

A Cantor set is a simple fractal that laid the foundation for modern topology. The picture at right is an artistic representation of the Cantor set.

Basic Description

Creation of the Cantor Set

We highlight the first steps of construction here.

We begin with a straight line, of length 1.

We remove the middle third of the line.

With the two newly separate line segments, we again remove the middle third of these lines.

Here we have all three stages and more ...

The process repeats onward infinitely, and what remains is the Cantor set. This construction is an example of an iterated process.

Properties

Self Similar and Infinite

Two important insights can be made from the infinitely iterating construction process. The first is that the set's construction goes on indefinitely. The second being its self similarity; when we look closer at any individual part, we find a copy of the original image.

By being infinitely self-similar, the Cantor set is a fractal

by definition.

The main image displayed is an artistic representation of the creation Cantor set using yolks. Each line of yolks has a diameter 1/3 of the yolk above.

Length

At every step in the construction process, we are removing 1/3 of the previous total length. This apparent since we are removing a third of the total length at every step. We may then relate the total length of the set at any given step by the function, $L_n=L_0 \centerdot (2/3)^n$, where $L_0$ is the initial length, $L_n$ is the nth iteration length, and n is the number of iterations of the process.

For example, let's assume we have a starting length of 1. Then,

$L_0=1, L_1=\frac{2}{3}, L_2 = \frac{4}{9}, L_3= \frac{8}{27}, L_4= \frac{16}{81}$

Since a true Cantor set is created by repeating this process an infinite number of times, we can calculate its final length as limit; $\lim_{n \to \infty}L_0 \centerdot (2/3)^{n} = 0$.

Because the length of a Cantor set is 0, the Cantor set cannot contain sections of non-zero length. However, while a Cantor set has no length, it still contains an infinite number of points. This is because we do not remove endpoints when we remove the middle thirds.

For example, our first iteration removed the length between the points 1/3 and 2/3, but we did not remove the points 1/3 and 2/3 themselves. In mathematical notation, the part of the line we removed is $1/3<x<2/3$.

After each iteration, there are $2^n$ end points created and never removed later. By repeating this process an infinite number of times, we will create an infinite number of endpoints.

Not all points contained within the set are endpoints, however. Points such as 1/4 and 3/10 remain in the Cantor set despite being non-endpoints.

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Topological Properties

This picture is actually a tertiary Cantor set, a Cantor set is any li [...]

[Click to hide A More Mathematical Explanation]

Topological Properties

This picture is actually a tertiary Cantor set, a Cantor set is any line that displays the properties of being a perfect set and nowhere dense.

Teaching Materials

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Future Directions for this Page

• A zooming animation, similar to the one on Sierpinski's triangle, that illustrates how the set is infinitely self similar.
• A more fleshed out explanation about the topology parts from someone who has taken the course?
  

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  [[Description::A Cantor set is a simple fractal that laid the foundation for modern topology.  The picture at right is an artistic representation of the Cantor set.|]]