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arXiv · 2110.12265

The volume of a spherical antiprism

Nikolay Abrosimov·Bao Vuong
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Abstract

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.

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Nikolay Abrosimov, Bao Vuong. 2021-10-23. The volume of a spherical antiprism. https://arxiv.org/abs/2110.12265

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