The volume of a compact hyperbolic antiprism

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We consider a compact hyperbolic antiprism. It is a convex polyhedron with 2n vertices in the hyperbolic space H3. This polyhedron has a symmetry group S2n generated by a mirror-rotational symmetry of order 2n, i.e. rotation to the angle π/n followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in H3. 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.

Original languageEnglish
Article number1842010
Number of pages12
JournalJournal of Knot Theory and its Ramifications
Issue number13
Publication statusPublished - Nov 2018


  • Compact hyperbolic antiprism
  • hyperbolic volume
  • rotation followed by reflection
  • symmetry group S 2 n
  • symmetry group S n
  • symmetry group S-2n


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