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arXiv · 1807.08297

The volume of a compact hyperbolic antiprism

Nikolay Abrosimov·Bao Vuong
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Abstract

We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space $\mathbb{H}^3$. This polyhedron has a symmetry group $S_{2n}$ 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 polyhedra in $\mathbb{H}^3$. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.

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Nikolay Abrosimov, Bao Vuong. 2018-07-22. The volume of a compact hyperbolic antiprism. https://arxiv.org/abs/1807.08297

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