Editing Sierpiński carpet (section)

==Wallis sieve==
[[File:Wallis sieve iteration 3.png|thumb|Third iteration of the Wallis sieve]]

A variation of the Sierpiński carpet, called the '''Wallis sieve''', starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the {{mvar|i}}th step by subdividing each square into {{math|(2''i'' + 1)<sup>2</sup>}} (the [[square number|odd squares]]<ref>{{Cite OEIS |A016754 |Odd squares: a(n) {{=}} (2n+1)^2. Also centered octagonal numbers.}}</ref>) smaller squares and removing the middle one. By the [[Wallis product]], the area of the resulting set is {{sfrac|{{pi}}|4}}, unlike the standard Sierpiński carpet which has zero limiting area. Although the Wallis sieve has positive [[Lebesgue measure]], no subset that is a [[Cartesian product]] of two sets of real numbers has this property, so its [[Jordan measure]] is zero.<ref>{{cite journal | last = Rummler | first = Hansklaus | doi = 10.2307/2324662 | issue = 9 | journal = [[The American Mathematical Monthly]] | mr = 1247533 | pages = 858–860 | title = Squaring the circle with holes | volume = 100 | year = 1993| jstor = 2324662 }}</ref>