Visualization of Cantor set by Mathematica or Maple!

Question

We all know what it is the Cantor set. The Cantor set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third $(\frac{1}{3}, \frac{2}{3})$ from the interval $[0, 1]$, leaving two line segments: $[0, \frac{1}{3}] ∪ [\frac{2}{3}, 1]$. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: $[0, \frac{1}{9}] ∪ [\frac{2}{9}, \frac{1}{3}] ∪ [\frac{2}{3}, \frac{7}{9}] ∪ [\frac{8}{9}, 1]$. This process is continued ad infinitum, where the nth set. The Cantor ternary set contains all points in the interval $[0, 1]$ that are not deleted at any step in this infinite process. An explicit formula for the Cantor set is $$C=[0,1]\bigcup_{m=1}^{\infty}\bigcup_{k=0}^{3^{m-1}-1}(\frac{3k+1}{3^m},\frac{3k+2}{3^m})$$

I know Maple just for doing some calculus and some basic modeling. I am asking if someone can note me a program in which we visualize the Cantor set. As I don't know if this job can be done in Maple, I added Mathematica in the title. Maybe its environment is more powerful than Maple in this question. Thanks.

Answer 1

The Cantor $\frac13$ set is kind of hard to visualize, since it has infinitely many points but zero Lebesgue measure. The best visualization I could think of was to plot an approximation of its indicator function:

> f := a -> [seq(x/3, x in a), seq((2+x)/3, x in a)]:
> cantor := n -> (f@@n)([0,1]):
> delta2plot := points -> [seq(seq([x,y], y in [0,1,0]), x in points))]:
> plots:-listplot(delta2plot(cantor(5)), color=gray, view=[0..1,0..1]);

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Answer 2

For Mathematica see

• http://demonstrations.wolfram.com/CantorSet/
• http://library.wolfram.com/infocenter/Demos/139/