Geodesic
The volume of a spherical antiprism with $S_{2n}$ symmetry
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1165-1179

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We consider a spherical antiprism. It is a convex polyhedron with $2n$ vertices in the spherical space $\mathbb{S}^3$. This polyhedron has a group of symmetries $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation to the angle $\pi/n$ followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedron in $\mathbb{S}^3$. Then we find relations between its dihedral angles and edge lengths in the form of cosine rules through a property of a spherical isosceles trapezoid. Finally, we obtain an explicit integral formula for the volume of a spherical antiprism in terms of the edge lengths.
Keywords: spherical antiprism, spherical volume, symmetry group $S_{2n}$, rotation followed by reflection, spherical isosceles trapezoid. 
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     title = {The volume of a spherical antiprism with $S_{2n}$ symmetry},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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     publisher = {mathdoc},
     volume = {18},
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     year = {2021},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a31/}
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N. Abrosimov; B. Vuong. The volume of a spherical antiprism with $S_{2n}$ symmetry. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1165-1179. http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a31/