Difference between revisions of "Cantor set"
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Another equivalent representation for <math>\mathcal{C}</math> is: Start with the interval <math>[0,1]</math>, then scale it by <math>\frac{1}{3}</math>. Then join it with a copy shifted by <math>\frac{2}{3}</math>, and repeat ''ad infinitum''. | Another equivalent representation for <math>\mathcal{C}</math> is: Start with the interval <math>[0,1]</math>, then scale it by <math>\frac{1}{3}</math>. Then join it with a copy shifted by <math>\frac{2}{3}</math>, and repeat ''ad infinitum''. | ||
| − | Using this representation, <math>\mathcal{C}</math> can be rendered in [[LaTeX]]: <cmath>\newcommand{\cantor}{#1\phantom{#1}#1}\cantor{\cantor{\cantor{\cantor{.}}}}</cmath> | + | Using this representation, <math>\mathcal{C}</math> can be rendered in [[LaTeX]]: <cmath>\newcommand{\cantor}[1]{#1\phantom{#1}#1}\cantor{\cantor{\cantor{\cantor{.}}}}</cmath> |
{{stub}} | {{stub}} | ||
Revision as of 20:19, 4 June 2020
The Cantor set 
is a subset of the real numbers that exhibits a number of interesting and counter-intuitive properties. It is among the simplest examples of a fractal. Topologically, it is a closed set, and also a perfect set. Despite containing an uncountable number of elements, it has Lebesgue measure equal to 
.
The Cantor set can be described recursively as follows: begin with the closed interval ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, and then remove the open middle third segment 
, dividing the interval into two intervals of length 
. Then remove the middle third of the two remaining segments, and remove the middle third of the four remaining segments, and so on ad infinitum.
![[asy] int max = 7; real thick = 0.025; void cantor(int n, real y){ if(n == 0) fill((0,y+thick)--(0,y-thick)--(1,y-thick)--(1,y+thick)--cycle,linewidth(3)); if(n != 0) { cantor(n-1,y); for(int i = 0; i <= 3^(n-1); ++i) fill( ( (1.0+3*i)/(3^n) ,y+0.1)--( (1.0+3*i)/(3^n) ,y-0.1)--( (2.0+3*i)/(3^n) ,y-0.1)--( (2.0+3*i)/(3^n) ,y+0.1)--cycle,white); } } for(int i = 0; i < max; ++i) cantor(i,-0.2*i); [/asy]](http://latex.artofproblemsolving.com/8/e/7/8e7092d9d51c41b46f2c5eb4f1c7114e576d8d25.png)
Equivalently, we may define to be the set of real numbers between and 
with a base three expansion that contains only the digits and 
(including repeating decimals).
Another equivalent representation for is: Start with the interval , then scale it by . Then join it with a copy shifted by 
, and repeat ad infinitum.
Using this representation, can be rendered in LaTeX: ![\[\newcommand{\cantor}[1]{#1\phantom{#1}#1}\cantor{\cantor{\cantor{\cantor{.}}}}\]](http://latex.artofproblemsolving.com/3/4/6/346e4055102653c313f195cdce16e4983ec8d56b.png)
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