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Decimal expansion of volume of the Weeks manifold.
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%I #29 Dec 16 2024 07:58:05

%S 9,4,2,7,0,7,3,6,2,7,7,6,9,2,7,7,2,0,9,2,1,2,9,9,6,0,3,0,9,2,2,1,1,6,

%T 4,7,5,9,0,3,2,7,1,0,5,7,6,6,8,8,3,1,5,9,0,1,4,5,0,6,7,7,5,7,5,2,9,3,

%U 4,1,8,2,7,7,4,1,5,7,2,1,0,3,1,2,3,1,5,6,7,2,6,4,3,3,3,3,0,3,5,8,0,4,1,8,0

%N Decimal expansion of volume of the Weeks manifold.

%C Gabai, Meyerhoff and Milley show that the Weeks manifold is the unique closed orientable hyperbolic 3-manifold of smallest volume. This constant gives the volume. - _Jeremy Tan_, Nov 19 2016

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/lr/lr.html">Volumes of hyperbolic 3-manifolds</a>, PDF linked to from "Mathematical Constants", 9/5/2004. [Broken link]

%H D. Gabai, R. Meyerhoff and P. Milley, <a href="https://arxiv.org/abs/0705.4325">Minimum volume cusped hyperbolic three-manifolds</a>, arXiv:0705.4325 [math.GT], 2007; J. Amer. Math. Soc. 22 (2009), 1157-1215. MR2525782

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Weeks_manifold">Weeks manifold</a>

%F Equals Im(dilog(z0)+log(|z0|)*log(1-z0)) where z0 = 0.8774.. + 0.7448..i is the root of z^3-z^2+1 with Im(z)>0. - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 15 2007

%e 0.9427073627769277209212996030922116475903...

%t z0 = Roots[z^3 - z^2 + 1 == 0, z][[3, 2]]; RealDigits[ Im[ PolyLog[2, z0] + Log[ Abs[ z0]] Log[1 - z0]], 10, 111][[1]] (* _Robert G. Wilson v_, Nov 19 2016 *)

%o (PARI) z0=polroots(z^3-z^2+1)[3]; imag(dilog(z0)+log(abs(z0))*log(1-z0)) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 15 2007

%K cons,nonn

%O 0,1

%A _Jonathan Vos Post_, Mar 13 2007

%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 15 2007

%E Title updated by _Jeremy Tan_, Nov 19 2016