Gieseking's constant

The Gieseking constant is a mathematical constant that the maximum volume of hyperbolic indicating tetrahedron. It is named after Hugo Gieseking (1887–1915), who in 1912 constructed the Gieseking manifold from such a tetrahedron by fusing side surfaces . Colin Adams was able to show in 1987 that the Gieseking manifold is the unambiguous non-compact hyperbolic 3- manifold with minimal volume. The Gieseking constant is also called the Lobachevsky constant after Nikolai Lobatschewski .

definition

The Gieseking constant is defined as

${\displaystyle G=\int _{0}^{2\pi /3}\ln \left(2\cos {\tfrac {t}{2}}\right){\mathrm {d} }t=1,01494\;16064\;09653\;62502\;12025\;54274\;52028\;59416\;89307\;53029\;\ldots }$

(see the sequence A143298 in OEIS ).

Further representations

Alternative spellings of the Gieseking constant are

${\displaystyle G={\rm {Cl_{2}}}\left({\frac {1}{3}}\pi \right)}$,

where is the Clausen function , ${\displaystyle {\rm {Cl_{2}}}}$

${\displaystyle G={\frac {1}{36}}i\left(\pi ^{2}-36{\rm {Li_{2}}}(-(-1)^{2/3})\right)={\frac {1}{2}}i\left({\rm {Li_{2}}}((-1)^{2/3})-{\rm {Li_{2}}}((-1)^{1/3})\right)}$,

where is the (classical) dilogarithm , ${\displaystyle {\rm {Li_{2}}}}$

${\displaystyle G=D\left({\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}\right)}$,

where is the Bloch-Wigner dilogarithm , ${\displaystyle D}$

${\displaystyle G=3\Lambda \left({\frac {\pi }{3}}\right)}$,

where is the Lobachevsky function , and ${\displaystyle \Lambda }$

${\displaystyle G={\frac {9-\psi _{1}\left({\tfrac {2}{3}}\right)+\psi _{1}\left({\tfrac {4}{3}}\right)}{4{\sqrt {3}}}}}$,

where is the trigamma function . ${\displaystyle \psi _{1}}$

Series development

The Gieseking constant has the series expansion

${\displaystyle G={\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{k=0}^{\infty }{\frac {1}{(3k+2)^{2}}}+\sum _{k=1}^{\infty }{\frac {1}{(3k+1)^{2}}}\right)={\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}\pm \ldots \right)}$.

literature

• Colin C. Adams: The newest inductee in the number hall of fame , Mathematics Magazine 71, December 1998, pp. 341–349 (English; Zentralblatt review )
• Steven R. Finch: Mathematical constants . Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , p. 233 (English; Finch's website for the book with errata and addenda: Mathematical Constants . )

Web links

• Eric W. Weisstein : Gieseking's Constant . In: MathWorld (English).

Individual evidence

1. ^ Adams: The newest inductee in the number hall of fame . 1998 (english)
2. ^ John W. Milnor : Hyperbolic geometry: The first 150 years . In: Bulletin of the AMS , January 6, 1982, pp. 9–24 (English; Zentralblatt review ; "This works out as 1.0149416 ...." on p. 20)
3. ↑ Hugo Gieseking : Analytical investigations on topological groups . L. Wiegand, Hilchenbach 1912 (inaugural dissertation at the Westfälische Wilhelms-Universität Münster; with curriculum vitae until 1911; yearbook review )
4. ^ Colin C. Adams: The noncompact hyperbolic 3-manifold of minimal volume . In: Proceedings of the AMS , 100, August 1987, pp. 601-606 (English; Zentralblatt review ; " v = 1.01494 ...." on p. 602)
5. ↑ Steven R. Finch: Volumes of Hyperbolic 3-Manifolds . ( Memento of the original from September 19, 2015 in the Internet Archive ; PDF; 366 kB) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. September 5, 2004, p. 4