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Mat. Zametki, 2009, Volume 86, Issue 2, Pages 190–201 (Mi mz8472)  

This article is cited in 8 scientific papers (total in 8 papers)

The Volume of the Lambert Cube in Spherical Space

D. A. Derevnina, A. D. Mednykhb

a Tumen State Academy of Architecture and Engineering
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: The Lambert cube $Q(\alpha,\beta,\gamma)$ is one of the simplest polyhedra. By definition, this is a combinatorial cube with dihedral angles $\alpha$, $\beta$, and $\gamma$ at three noncoplanar edges and with right angles at all other edges. The volume of the Lambert cube in hyperbolic space was obtained by R. Kellerhals (1989) in terms of the Lobachevskii function $\Lambda(x)$. In the present paper, we find the volume of the Lambert cube in spherical space. It is expressed in terms of the function
$$ \delta(\alpha,\theta)=\int_{\theta}^{\pi/2}\log(1-\cos2\alpha\cos2\tau)\frac{d\tau}{\cos2\tau}, $$
which can be regarded as the spherical analog of the function
$$ \Delta(\alpha,\theta)=\Lambda(\alpha+\theta)-\Lambda(\alpha-\theta). $$


Keywords: Lambert cube, spherical space, hyperbolic space, Lobachevskii function, Schläfli formula

DOI: https://doi.org/10.4213/mzm8472

Full text: PDF file (528 kB)
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English version:
Mathematical Notes, 2009, 86:2, 176–186

Bibliographic databases:

UDC: 514.135
Received: 30.07.2008
Revised: 31.12.2008

Citation: D. A. Derevnin, A. D. Mednykh, “The Volume of the Lambert Cube in Spherical Space”, Mat. Zametki, 86:2 (2009), 190–201; Math. Notes, 86:2 (2009), 176–186

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Kolpakov, A. D. Mednykh, M. G. Pashkevich, “A volume formula for $\mathbb Z_2$-symmetric spherical tetrahedra”, Siberian Math. J., 52:3 (2011), 456–470  mathnet  crossref  mathscinet  isi
    2. Baigonakova G.A., Godoi-Molina Maurisio, Mednykh A.D., “O geometricheskikh svoistvakh giperbolicheskogo oktaedra, obladayuschego $mmm$-simmetriei”, Vestn. Kemerovskogo gos. un-ta, 2011, no. 3-1, 13–18  elib
    3. Buser P., Mednykh A., Vesnin A., “Lambert cubes and the Löbell polyhedron revisited”, Adv. Geom., 12:3 (2012), 525–548  crossref  mathscinet  zmath  isi  elib  scopus
    4. N. V. Abrosimov, G. A. Baigonakova, “Giperbolicheskii oktaedr s $mmm$-simmetriei”, Sib. elektron. matem. izv., 10 (2013), 123–140  mathnet
    5. V. A. Krasnov, “On integral expressions for volumes of hyperbolic tetrahedra”, Journal of Mathematical Sciences, 211:4 (2015), 531–541  mathnet  crossref
    6. Kolpakov A., Mednykh A., Pashkevich M., “Volume Formula for a a"Currency Sign(2)-Symmetric Spherical Tetrahedron Through its Edge Lengths”, Ark. Mat., 51:1 (2013), 99–123  crossref  mathscinet  zmath  isi  scopus
    7. N. V. Abrosimov, E. S. Kudina, A. D. Mednykh, “On the volume of a hyperbolic octahedron with $\overline3$-symmetry”, Proc. Steklov Inst. Math., 288 (2015), 1–9  mathnet  crossref  crossref  isi  elib
    8. N. V. Abrosimov, E. S. Kudina, A. D. Mednykh, “Ob'em giperbolicheskogo geksaedra, dopuskayuschego $\overline{3}$-simmetriyu”, Sib. elektron. matem. izv., 13 (2016), 1150–1158  mathnet  crossref
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