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[Volumes of non-Euclidean tetrahedra] Н. В. Абросимов |
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Аннотация: The talk will provide an overview of the latest results on finding exact formulas for calculating the volumes of hyperbolic tetrahedra. The classical formula of G. Sforza [1] expresses the volume of a general hyperbolic tetrahedron in terms of dihedral angles. Its modern proof is proposed in [2], where a version of the Sforza formula for the volume of a spherical tetrahedron is also given. A formula in terms of edge lengths was obtained in [3]. The known formulas for the volume of a general hyperbolic tetrahedron are complicated and cannot always be applied to calculate the volumes of more complex polyhedra. A natural question arises about finding simpler formulas for sufficiently wide families of hyperbolic tetrahedra. In the second part of the talk, we will consider hyperbolic tetrahedra of special types: ideal, biorthogonal, trirectangular, and their generalizations. The volume of an ideal and biorthogonal hyperbolic tetrahedron was known to N.I. Lobachevsky. We will present new formulas for calculating the volume and normalized volume of a hyperbolic trirectangular tetrahedron [4], as well as its generalization for a 4-parameter family of tetrahedra with one edge orthogonal to a face. The latter formulas can be used to calculate the volumes of more complex polyhedra in Lobachevsky space. At the end of the talk, we will present a new formula for calculating the volume of a spherical trirectangular tetrahedron [5]. The list of Coxeter's spherical tetrahedra was constructed in [6]. Coxeter showed that there are 11 types of such tetrahedra in S^3. We will show that exactly 5 of these types belong to the family of trirectangular tetrahedra. We will calculate their volumes to verify our formula. References: [1] Sforza G., Spazi metrico-proiettivi. Ricerche di Estensionimetria Integrale. Ser. 1907. III, VIII (Appendice). P.41–66. [2] Abrosimov N.V., Mednykh A.D., Volumes of polytopes in constant curvature spaces. Fields Inst. Commun. 2014. V.70. P.1–26. arXiv:1302.4919 [3] Abrosimov N., Vuong B., Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths. Journal of Knot Theory and Its Ramifications. 2021. V.30. No.10, 2140007. arXiv:2107.03004 [4] Abrosimov N., Stepanishchev S., The volume of a trirectangular hyperbolic tetrahedron. Siberian Electronic Mathematicsl Reports. 2023. V.20. No.1, P.275–284. [5] Abrosimov N., Bayzakova B., The volume of a spherical trirectangular tetrahedron. Siberian Electronic Mathematicsl Reports. 2025. V.22. No.1, P.892–904. [6] Coxeter H.S.M., Discrete groups generated by reflections. Ann. Math. 1934. V.35. P.588–621. Язык доклада: английский Website: https://us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09 |
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