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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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/

[1] N.V. Abrosimov, E.S. Kudina, A.D. Mednykh, “On the volume of a hyperbolic octahedron with $\overline{3}$-symmetry”, Proceedings of the Steklov Institute of Mathematics, 288 (2015), 1–9 | DOI | MR | Zbl

[2] N. Abrosimov, B. Vuong, “The volume of a compact hyperbolic antiprism”, Journal of Knot Theory and Its Ramifications, 27:13 (2018), 1842010 | DOI | MR | Zbl

[3] N.V. Abrosimov, Vuong Huu Bao, “The volume of a hyperbolic tetrahedron with symmetry group $S_4$”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 23, no. 4, 2017, 7–17 | DOI | MR

[4] D.V. Alekseevskii, E.B. Vinberg, A.S. Solodovnikov, “Geometry of spaces of constant curvature”, Geometry II: Spaces of Constant Curvature, Encyclopedia of Mathematical Sciences, 29, Springer, 1993 | MR | Zbl

[5] W.P. Thurston, The Geometry and Topology of 3-Manifolds, Lecture Notes, Princeton University, 1980

[6] A. Yu. Vesnin, On Volumes of Some Hyperbolic 3-Manifolds, Lecture Notes Series, 30, Seoul National University, 1996 | MR | Zbl

[7] A. Yu. Vesnin, A.D. Mednykh, “Hyperbolic volumes of Fibonacci manifolds”, Siberian Mathematical Journal, 36:2 (1995), 235–245 | DOI | MR | Zbl