Difference between revisions of "Cantor set"
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| − | The '''Cantor set''' <math>\mathcal{C}</math> is a [[subset]] of the [[real number]]s that exhibits a number of interesting and counter-intuitive properties. | + | The '''Cantor set''' <math>\mathcal{C}</math> is a [[subset]] of the [[real number]]s that exhibits a number of interesting and counter-intuitive properties. It is among the simplest examples of a [[fractal]]. [[Topology|Topologically]], it is a [[closed set]], and also a [[perfect set]]. Despite containing an [[uncountable]] number of elements, it has [[Lebesgue measure]] equal to <math>0</math>. |
| − | The Cantor set can be described [[recursion|recursively]] as follows: begin with the | + | The Cantor set can be described [[recursion|recursively]] as follows: begin with the [[closed interval]] <math>[0,1]</math>, and then remove the [[open interval | open]] middle third segment <math>(1/3,2/3)</math>, dividing the [[interval]] into two intervals of length <math>\frac{1}{3}</math>. 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''. |
| − | Equivalently, we may define <math>\mathcal{C}</math> to be the set of real numbers between <math>0</math> and <math>1</math> with a [[base number | base]] three expansion that contains only the digits <math>0</math> and <math> | + | <center><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></center> | ||
| + | |||
| + | Equivalently, we may define <math>\mathcal{C}</math> to be the set of real numbers between <math>0</math> and <math>1</math> with a [[base number | base]] three expansion that contains only the digits <math>0</math> and <math>2</math> (including [[0.999...|repeating decimals]]). | ||
{{stub}} | {{stub}} | ||
Revision as of 19:39, 1 March 2010
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).
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