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A126774
Decimal expansion of volume of the Weeks manifold.
0
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, 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, 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
OFFSET
0,1
COMMENTS
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
LINKS
S. R. Finch, Volumes of hyperbolic 3-manifolds, PDF linked to from "Mathematical Constants", 9/5/2004. [Broken link, see cached copy below]
D. Gabai, R. Meyerhoff and P. Milley, Minimum volume cusped hyperbolic three-manifolds, arXiv:0705.4325 [math.GT], J. Amer. Math. Soc. 22 (2009), 1157-1215. MR2525782
Wikipedia, Weeks manifold
FORMULA
Formula: 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
EXAMPLE
0.9427073627769277209212996030922116475903...
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A155535 A099879 A351829 * A179587 A223709 A050016
KEYWORD
cons,nonn
AUTHOR
Jonathan Vos Post, Mar 13 2007
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 15 2007
Title updated by Jeremy Tan, Nov 19 2016
STATUS
approved