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This article is cited in 12 scientific papers (total in 12 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.
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
Linking options:
https://www.mathnet.ru/eng/mzm8472https://doi.org/10.4213/mzm8472 https://www.mathnet.ru/eng/mzm/v86/i2/p190
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Abstract page: | 905 | Full-text PDF : | 200 | References: | 59 | First page: | 15 |
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