Editing Pi (section)

=== In the Mandelbrot set ===
[[File:Mandel zoom 00 mandelbrot set.jpg|alt=An complex black shape on a blue background.|thumb|The [[Mandelbrot set]] can be used to approximate {{pi}}.]]

An occurrence of {{pi}} in the [[fractal]] called the [[Mandelbrot set]] was discovered by David Boll in 1991. He examined the behaviour of the Mandelbrot set near the "neck" at {{math|(−0.75, 0)}}. When the number of iterations until divergence for the point {{math|(−0.75, ''ε'')}} is multiplied by {{mvar|ε}}, the result approaches {{pi}} as {{mvar|ε}} approaches zero. The point {{math|(0.25 + ''ε'', 0)}} at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of {{mvar|ε}} tends to {{pi}}.<ref>{{cite journal |last1=Klebanoff |first1=Aaron |year=2001 |title=Pi in the Mandelbrot set |journal=Fractals |volume=9 |issue=4 |pages=393–402 |url=http://home.comcast.net/~davejanelle/mandel.pdf |archive-url=https://web.archive.org/web/20111027155739/http://home.comcast.net/~davejanelle/mandel.pdf |archive-date=27 October 2011 |access-date=14 April 2012 |doi=10.1142/S0218348X01000828 |url-status=dead}} {{pb}} {{cite book |last=Peitgen |first=Heinz-Otto |author-link=Heinz-Otto Peitgen |title=Chaos and fractals: new frontiers of science |publisher=Springer |year=2004 |pages=801–803 |isbn=978-0-387-20229-7}}</ref>