Geodesic
The Volume of the Lambert Cube in Spherical Space
Matematičeskie zametki, Tome 86 (2009) no. 2, pp. 190-201.

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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 $$ \Delta(\alpha,\theta)=\Lambda(\alpha+\theta)-\Lambda(\alpha-\theta). $$
Mots-clés : Lambert cube
Keywords: spherical space, hyperbolic space, Lobachevskii function, Schläfli formula. 
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D. A. Derevnin; A. D. Mednykh. The Volume of the Lambert Cube in Spherical Space. Matematičeskie zametki, Tome 86 (2009) no. 2, pp. 190-201. http://geodesic.mathdoc.fr/item/MZM_2009_86_2_a4/

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