The volume of a compact hyperbolic antiprism

Abstract

We consider a compact hyperbolic antiprism. It is a convex polyhedron with [Formula: see text] vertices in the hyperbolic space [Formula: see text]. This polyhedron has a symmetry group [Formula: see text] generated by a mirror-rotational symmetry of order [Formula: see text], i.e. rotation to the angle [Formula: see text] followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in [Formula: see text]. 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.