Editing Cantor set (section)

== Variants ==
[[File:cantor_set_radial.svg|thumb|Radial plot of the first ten steps<ref>{{cite web | url=http://gist.github.com/curran/74cb4d255acf072633a2df0d9b9be7c3 | title=Radial Cantor Set }}</ref>]]

===Smith–Volterra–Cantor set===
{{main|Smith–Volterra–Cantor set}}

Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle.  In the case where the middle {{sfrac|8|10}} of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s.  If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder <math>(1-f)^n\to 0</math> as <math>n\to\infty</math> for any <math>f</math> such that <math>0<f\leq 1</math>.

On the other hand, "fat Cantor sets" of positive measure can be generated by removal of smaller fractions of the middle of the segment in each iteration.  Thus, one can construct sets [[Homeomorphism|homeomorphic]] to the Cantor set that have positive Lebesgue measure while still being nowhere dense.  If an interval of length <math>r^n</math> (<math>r\leq 1/3</math>) is removed from the middle of each segment at the ''n''th iteration, then the total length removed is <math display="inline">\sum_{n=1}^\infty 2^{n-1}r^n=r/(1-2r)</math>, and the limiting set will have a [[Lebesgue measure]] of <math>\lambda=(1-3r)/(1-2r)</math>.  Thus, in a sense, the middle-thirds Cantor set is a limiting case with <math>r=1/3</math>.  If <math>0<r<1/3</math>, then the remainder will have positive measure with <math>0<\lambda<1</math>.  The case <math>r=1/4</math> is known as the [[Smith–Volterra–Cantor set]], which has a Lebesgue measure of <math>1/2</math>.

=== Cantor dust ===
<!-- This is linked to by the redirect "Cantor dust"-->
'''Cantor dust''' is a multi-dimensional version of the Cantor set. It can be formed by taking a finite [[Cartesian product]] of the Cantor set with itself, making it a [[Cantor space]]. Like the Cantor set, Cantor dust has [[Measure zero|zero measure]].<ref>{{cite book|author=Helmberg, Gilbert|title=Getting Acquainted With Fractals|publisher=Walter de Gruyter|year=2007|isbn=978-3-11-019092-2|page=46|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA46}}</ref>
[[File:Cantorcubes.gif|thumb|right|250px|[[Cantor cube]]s recursion progression towards Cantor dust]]
{|
|[[Image:Cantor dust.svg|thumb|'''Cantor dust''' (2D)]]
|[[Image:Cantors cube.jpg|thumb|'''Cantor dust''' (3D)]]
|}

A different 2D analogue of the Cantor set is the [[Sierpinski carpet]], where a square is divided up into nine smaller squares, and the middle one removed. The remaining squares are then further divided into nine each and the middle removed, and so on ad infinitum.<ref>{{cite book|author=Helmberg, Gilbert|title=Getting Acquainted With Fractals|publisher=Walter de Gruyter|year=2007|isbn=978-3-11-019092-2|page=48|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA48}}</ref> One 3D analogue of this is the [[Menger sponge]].