Feigenbaum Constant Approximations
A curious approximation to the Feigenbaum constant delta is given by
| pi+tan^(-1)(e^pi)=4.669201932..., |
(1)
|
where e^pi is Gelfond's constant, which is good to 6 digits to the right of the decimal point.
M. Trott (pers. comm., May 6, 2008) noted
| delta approx 2G+3, |
(2)
|
where G is Gauss's constant, which is good to 4 decimal digits, and
| [画像: delta approx 9/T, ] |
(3)
|
where T is the tetranacci constant, which is good to 3 decimal digits.
A strange approximation good to five digits is given by the solution to
| x^x=1333, |
(4)
|
which is
| x=e^(W(ln1333))=4.669202878..., |
(5)
|
where W(z) is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).
| [画像: delta approx (10)/(pi-1) ] |
(6)
|
gives delta to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).
M. Hudson (pers. comm., Nov. 20, 2004) gave
which are good to 17, 13, and 9 digits respectively.
Stoschek gave the strange approximation
| [画像: delta approx 4(1+(12^2)/(163)+(4·12^2+31)/(4·163^2)+...)/(1+(10^2)/(163)+(10^2+30)/(163^2)+...), ] |
(10)
|
which is good to 9 digits.
R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations
![画像:(e(e-1))/(1+exp{8[(1+e^(-8))^(3/2)-2]})](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FeigenbaumConstantApproximations/Inline33.svg){kind=link}
where e is the base of the natural logarithm and e^pi is Gelfond's constant, which are good to 3, 3, 5, 7, 9, and 10 decimal digits, respectively, and
![画像:tan[e+tan^(-1)(2/((e-1)^8e)-e^pi)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FeigenbaumConstantApproximations/Inline49.svg){kind=link}
which are good to 3, 3, 3, 4, 6, 8, and 8 decimal digits, respectively.
An approximation to mu_infty due to R. Phillips (pers. comm., Jan. 27, 2005) is obtained by numerically solving
for x, where phi is the golden ratio, which is good to 4 digits.
See also
Almost Integer, Feigenbaum ConstantExplore with Wolfram|Alpha
More things to try:
References
Friedman, E. "Problem of the Month (August 2004)." https://erich-friedman.github.io/mathmagic/0804.html.Referenced on Wolfram|Alpha
Feigenbaum Constant ApproximationsCite this as:
Weisstein, Eric W. "Feigenbaum Constant Approximations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FeigenbaumConstantApproximations.html