Editing Mathematical coincidence (section)

==== Concerning ''π'' ====
* The second [[convergent (continued fraction)|convergent]] of ''π'', [3; 7] = 22/7 = 3.1428..., was known to [[Archimedes]],<ref name=beckmann/> and is correct to about 0.04%.  The fourth convergent of ''π'', [3; 7, 15, 1] = [[355/113]] = 3.1415929..., found by [[Zu Chongzhi]],<ref>
{{cite book
 | author = Yoshio Mikami
 | title = Development of Mathematics in China and Japan
 | publisher = B. G. Teubner
 | page = 135
 | year = 1913
 | url = https://books.google.com/books?id=4e9LAAAAMAAJ&q=intitle:Development+intitle:%22China+and+Japan%22+355 }}</ref> is correct to six decimal places;<ref name=beckmann>{{Cite book
 | title = A History of Pi
 | author = Petr Beckmann
 | publisher = Macmillan
 | year = 1971
 | isbn = 978-0-312-38185-1
 | pages = 101, 170
 | url = https://books.google.com/books?id=TB6jzz3ZDTEC&q=pi+113+355++digits&pg=PA101
 }}</ref> this high accuracy comes about because π has an unusually large next term in its continued fraction representation: {{pi}} = [3; 7, 15, 1, 292, ...].<ref>{{Cite book
 | title = CRC concise encyclopedia of mathematics
 | author = Eric W. Weisstein
 | publisher = CRC Press
 | year = 2003
 | isbn = 978-1-58488-347-0
 | page = 2232
 | url = https://books.google.com/books?id=_8TyhSqHUiEC&q=pi+113+355++292+convergent&pg=PA2232
 }}</ref>
* A coincidence involving ''π'' and the [[golden ratio]] ''φ'' is given by <math>\pi \approx 4 / \sqrt{\varphi} = 3.1446\dots</math>. Consequently, the square on the middle-sized edge of a [[Kepler triangle]] is similar in perimeter to its circumcircle. Some believe one or the other of these coincidences is to be found in the [[Golden Ratio#Egyptian pyramids|Great Pyramid of Giza]], but it is highly improbable that this was intentional.<ref>
{{cite book
 | title = The Shape of the Great Pyramid
 | author = Roger Herz-Fischler
 | publisher = Wilfrid Laurier University Press
 | year = 2000
 | isbn = 978-0-889-20324-2
 | page = 67
 | url = https://books.google.com/books?id=_8TyhSqHUiEC&q=pi+113+355++292+convergent&pg=PA2232
}}</ref>
* There is a sequence of [[six nines in pi]] beginning at the 762nd decimal place of its decimal representation. For a randomly chosen [[normal number]], the probability of a particular sequence of six consecutive digits—of any type, not just a repeating one—to appear this early is 0.08%.<ref name="ArndtHaenel">{{Citation |last1=Arndt |first1=J. |title=Pi – Unleashed |page=3 |year=2001 |location=Berlin |publisher=Springer |isbn=3-540-66572-2 |name-list-style=amp |last2=Haenel |first2=C.}}.</ref> Pi is conjectured, but not known, to be a normal number.
* The first [[Feigenbaum constant]] is approximately equal to <math>\tfrac{10}{\pi-1}</math>, with an error of 0.0015%.