Geodesic
The volume of a hyperbolic antipodal octahedron
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 949-958.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the hyperbolic antipodal octahedron. It is an octahedron with antipodal symmetry in the hyperbolic space $\mathbb{H}^3$. We establish necessary and sufficient conditions for the existence of such an octahedron in $\mathbb{H}^3$. By dividing the octahedron into appropriate tetrahedra we obtain an explicit integral formula for the volume of the hyperbolic antipodal octahedron.
Keywords: hyperbolic octahedron, hyperbolic volume, antipodal symmetry, hyperbolic tetrahedron, integral formula. 
@article{SEMR_2022_19_2_a24,
     author = {B. Vuong},
     title = {The volume of a hyperbolic antipodal octahedron},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {949--958},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a24/}
}
TY  - JOUR
AU  - B. Vuong
TI  - The volume of a hyperbolic antipodal octahedron
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2022
SP  - 949
EP  - 958
VL  - 19
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a24/
LA  - en
ID  - SEMR_2022_19_2_a24
ER  - 
%0 Journal Article
%A B. Vuong
%T The volume of a hyperbolic antipodal octahedron
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2022
%P 949-958
%V 19
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a24/
%G en
%F SEMR_2022_19_2_a24
B. Vuong. The volume of a hyperbolic antipodal octahedron. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 949-958. http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a24/

[1] E.M. Andreev, “On convex polyhedra in Lobachevskii spaces”, Math. USSR, Sb., 10:3 (1970), 413–440 | DOI | MR | Zbl

[2] G.D. Mostow, “Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms”, Publ. Math. IHES, 34 (1968), 53–104 | DOI | MR | Zbl

[3] G.D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, Princeton University Press, Princeton, 1973 | MR | Zbl

[4] C. Petronio, D. Heard, E. Pervova, “Combinatorial and geometric methods in topology”, Milan J. Math., 76 (2008), 69–92 | DOI | MR | Zbl

[5] D. Heard, E. Pervova, C. Petronio, “The 191 orientable octahedral manifolds”, Exp. Math., 17:4 (2008), 473–486 | DOI | MR | Zbl

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

[7] N.V. Abrosimov, G.A. Baigonakova, “Hyperbolic octahedron with $mmm$-symmetry”, Sib. Èlektron. Mat. Izv., 10 (2013), 123–140 | MR | Zbl

[8] N.V. Abrosimov, M. Godoy-Molina, A.D. Mednykh, “On the volume of a spherical octahedron with symmetries”, J. Math. Sci., New York, 161:1 (2009), 1–10 | DOI | MR | Zbl

[9] D.V. Alekseevskii, E.B. Vinberg, A.S. Solodovnikov, “Geometry of spaces of constant curvature”, Geometry II: Spaces of constant curvature, Encycl. Math. Sci., 29, Springer-Verlag, Berlin, 1993 | MR | Zbl

[10] G. Sforza, “Ricerche di estensionimetria negli spazi metrico-projettivi”, Modena Mem. Acc., Ser. III, 8, Appendice (1907), 21–66 | Zbl

[11] N. Abrosimov, A. Mednykh, “Area and volume in non-Euclidean geometry”, Eighteen essays in non-Euclidean geometry, IRMA Lect. Math. Theor. Phys., 29, eds. Alberge Vincent (ed.) et al., European Mathematical Society (EMS), Zürich, 2019, 151–189 | DOI | MR | Zbl

[12] D.A. Derevnin, A.D. Mednykh, “A formula for the volume of a hyperbolic tetrahedron”, Russ. Math. Surv., 60:2 (2005), 346–348 | DOI | MR | Zbl

[13] N. Abrosimov, B. Vuong, “Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths”, J. Knot Theory Ramifications, 30:10 (2021), 2140007 | DOI | MR | Zbl