Volumes of polyhedra in non-Euclidean spaces of constant curvature

V. A. Krasnov

Contemporary Mathematics. Fundamental Directions, Algebra, Geometry, and Topology, Tome 66 (2020) no. 4, pp. 558-679.

Résumé

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for $3$-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute). In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in $3$-dimensional spherical and hyperbolic spaces of constant curvature $K=1$ and $K=-1$, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincaré model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

Bibliographie

[1] N. V. Abrosimov, “To the solution of Seidel's problem on the volumes of hyperbolic tetrahedra”, Siberian Electron. Math. Bull., 6 (2009), 211–218 (in Russian) | MR | Zbl

[2] N. V. Abrosimov, “Seidel's problem on the volume of a non-Euclidean tetrahedron”, Rep. Russ. Acad. Sci., 435:1 (2010), 7–10 (in Russian) | Zbl

[3] N. V. Abrosimov, G. A. Baygonakova, “Hyperbolic octahedron with $\mathrm{mmm}$-symmetry”, Siberian Math. J., 10 (2013), 123–140 (in Russian) | Zbl

[4] N. V. Abrosimov, B. Vuong Huu, “The volume of a hyperbolic tetrahedron with symmetry group $S_4$”, Proc. Inst. Math. Mech. Ural Branch Russ. Acad. Sci., 23, no. 4, 2017, 2–17 (in Russian)

[5] N. V. Abrosimov, M. Godoy-Molina, A. D. Mednykh, “On the volume of spherical octahedron with symmetries”, Contemp. Math. Appl., 60, 2008, 3–12 (in Russian)

[6] N. V. Abrosimov, E. S. Kudina, A. D. Mednykh, “On the volume of a hyperbolic octahedron with $\bar{3}$-symmetry”, Proc. Math. Inst. Russ. Acad. Sci., 288, 2015, 7–15 (in Russian) | Zbl

[7] N. V. Abrosimov, E. S. Kudina, A. D. Mednykh, “Volume of a hyperbolic hexahedron with $\bar{3}$-symmetry”, Siberian Electron. Math. Bull., 13 (2016), 1150–1158 (in Russian) | MR | Zbl

[8] E. M. Andreev, “On convex polyhedra in Lobachevsky spaces”, Math. Digest, 81 (1970), 445–478 (in Russian) | Zbl

[9] E. M. Andreev, “On convex polyhedra of finite volume in Lobachevsky space”, Math. Digest, 83 (1970), 256–260 (in Russian) | Zbl

[10] D. V. Alekseevskiy, E. B. Vinberg, A. S. Solodovnikov, “Geometry of spaces of constant curvature”, Totals Sci. Tech. Contemp. Probl. Math. Fundam. Directions, 29, 1988, 1–146 (in Russian)

[11] G. A. Baygonakova, M. Godoy-Molina, A. D. Mednykh, “On geometric properties of hyperbolic octahedron with $\mathrm{mmm}$-symmetry”, Bull. Kemerovo State Univ., 47:3/1 (2011), 13–18 (in Russian)

[12] V. M. Bukhshtaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Pak, “Cohomological rigidity of manifolds defined by 3-dimensional polyhedra”, Progr. Math. Sci., 434:2 (2017), 3–66 (in Russian)

[13] V. M. Bukhshtaber, T. E. Panov, Toric Actions in Topology and Combinatorics, MTsNMO, M., 2004 (in Russian) | MR

[14] A. Yu. Vesnin, “Volumes of 3-dimensional hyperbolic Labelle's manifolds”, Math. Notes, 64:1 (1998), 17–23 (in Russian) | MR | Zbl

[15] A. Yu. Vesnin, “Rectangular polyhedra and three-dimensional hyperbolic manifolds”, Progr. Math. Sci., 434:2 (2017), 147–190 (in Russian)

[16] E. B. Vinberg, “Volumes of non-Euclidean polyhedra”, Progr. Math. Sci., 290:2 (1993), 17–46 (in Russian) | MR | Zbl

[17] A. A. Gayfullin, “Embedded flexible spherical cross-polytopes with variable volumes”, Proc. Math. Inst. Russ. Acad. Sci., 288, 2015, 67–94 (in Russian) | Zbl

[18] A. A. Gayfullin, “Analytical continuation of the volume and the bellows hypothesis in Lobachevsky spaces”, Math. Digest, 206:11 (2015), 61–112 (in Russian) | MR | Zbl

[19] R. V. Galiulin, S. N. Mikhalev, I. Kh. Sabitov, “Some applications of the octahedron volume formula”, Math. Notes, 76:1 (2004), 27–43 (in Russian) | MR | Zbl

[20] D. A. Derevnin, A. D. Mednykh, “Volume of the Lambert spherical cube”, Math. Notes, 86:2 (2009), 190–201 (in Russian) | MR | Zbl

[21] D. A. Derevnin, A. D. Mednykh, M. G. Pashkevich, “Ob'em simmetrichnogo tetraedra v giperbolicheskom i sfericheskom prostranstvakh”, Siberian Math. J., 45:5 (2004), 1022–1031 (in Russian) | MR | Zbl

[22] A. A. Kolpakov, A. D. Mednykh, M. G. Pashkevich, “Volume formula for a $\mathbb{Z}_2$-symmetric tetrahedron”, Siberian Math. J., 52:3 (2011), 577–594 (in Russian)

[23] V. A. Krasnov, “On integral expressions for volumes of hyperbolic tetrahedra”, Contemp. Math. Fundam. Directions, 49, 2013, 89–98 (in Russian)

[24] V. A. Krasnov, “On the volume of hyperbolic octahedra with nontrivial symmetry”, Contemp. Math. Fundam. Directions, 51, 2013, 74–86 (in Russian)

[25] V. A. Krasnov, “Non-Euclidean octahedra with $\mathrm{mm2}$-symmetry”, Math. Notes, 99:1 (2016), 145–148 (in Russian) | MR

[26] V. A. Krasnov, “On volumes of hyperbolic simplexes”, Math. Notes, 106:6 (2019), 866–880 (in Russian) | Zbl

[27] V. A. Krasnov, “On application of the modern proof of the Sforza formula to computation of the volumes of hyperbolic tetrahedra of a special form”, Contemp. Math. Fundam. Directions, 65, no. 4, 2019, 623–634 (in Russian) | MR

[28] N. I. Lobachevskiy, “Imaginary Geometry”, Complete Set of Works, v. 3, M.–Leningrad, 1949 (in Russian)

[29] F. V. Petrov, “Inscribed quadrilaterals and trapezoids in absolute geometry”, Mathematical Education, 13, MTsNMO, M., 2009, 149–154 (in Russian)

[30] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series. Elementary Functions, Nauka, M., 1981 (in Russian) | MR

[31] I. Kh. Sabitov, “Volume of a polyhedron as a function of its metrics”, Fundam. Appl. Math., 2:4 (1996), 1235–1246 (in Russian) | MR | Zbl

[32] I. Kh. Sabitov, “Volumes of polihedra”, Mathematical Education, 21, MTsNMO, M., 2009 (in Russian)

[33] I. Kh. Sabitov, “On one method for computation of volumes of bodies”, Siberian Electron. Math. Bull., 2013, no. 10, 615–626 (in Russian) | MR | Zbl

[34] I. Kh. Sabitov, “Hyperbolic tetrahedron: computing volume with application to the proof of the Schläfli formula”, Model. Anal. Inform. Syst., 20:6 (2013), 149–161 (in Russian)

[35] D. Yu. Sokolova, “On the area of a trapezoid on the Lobachevsky plane”, Siberian Electron. Math. Bull., 9 (2012), 256–260 (in Russian) | MR | Zbl

[36] Abrosimov N. V., Mednykh A. D., “Volumes of polytopes in spaces of constant curvature”, Fields Inst. Commun., 70:1 (2014), 1–26 | MR | Zbl

[37] Alexandrov V. A., “An example of a flexible polyhedron with nonconstant volume in the spherical space”, Beitr. Algebra Geom., 38:1 (1997), 11–18 | MR | Zbl

[38] Bilinski S., “Zur Begründung der elementaren Inhaltslehre in der hyperbolischen Ebene”, Math. Ann., 180 (1969), 256–268 | DOI | MR | Zbl

[39] Bolyai J., “Appendix. The theory of space”, Janos Bolyai, Budapest, 1987

[40] Cho Yu., Kim H., “On the volume formula for hyperbolic tetrahedra”, Discrete Comput. Geom., 22 (1999), 347–366 | DOI | MR | Zbl

[41] Connely R., Sabitov I., Walz A., “The bellows conjecture”, Contrib. Algebra Geom., 38:1 (1997), 1–10 | MR

[42] Coxeter H. S. M., “The functions of Schläfli and Lobatschefsky”, Quarterly J. Math. Oxford, 6 (1935), 13–29 | DOI

[43] Coxeter H. S. M., Greitzer S. L., Geometry Revisited, The Mathematical Association of America, Washington, 1967 | MR | Zbl

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

[45] Diaz R., “A characterization of Gram matrices of polytopes”, Discrete Comput. Geom., 21:4 (1999), 581–601 | DOI | MR | Zbl

[46] Dickinson W., Salmassi M., “The right right triangle on the sphere”, College Math. J., 39:1 (2008), 24–33 | DOI | MR | Zbl

[47] Gaifullin A. A., “Sabitov polynomials for polyhedra in four dimensions”, Adv. Math., 252 (2014), 586–611 | DOI | MR | Zbl

[48] Gaifullin A. A., “Generalization of Sabitov's theorem to polyhedra of arbitrary dimensions”, Discrete Comput. Geom., 52:2 (2014), 195–220 | DOI | MR | Zbl

[49] Gaifullin A. A., “The bellows conjecture for small flexible polyhedra in non-Euclidean spaces”, Mosc. Math. J., 17:2 (2017), 269–290 | DOI | MR | Zbl

[50] Guo R., Sönmez N., “Cyclic polygons in classical geometry”, C. R. Acad. Bulgare Sci., 64:2 (2011), 185–194 | MR | Zbl

[51] Kellerhals R., “On the volume of hyperbolic polyhedra”, Math. Ann., 285:4 (1989), 541–569 | DOI | MR | Zbl

[52] Kneser H., “Der Simplexinhalt in der nichteuklidischen Geometrie”, Deutsche Math., 1 (1936), 337–340

[53] MClelland J. W., Preston T., A Treatise on Spherical Trigonometry with Application to Spherical Geometry and Numerous Examples, v. II, Macmillian and Co, London, 1886

[54] A. D. Mednykh, “Brahmagupta formula for cyclic quadrilaterals in the hyperbolic plane”, Sib. Elektron. Mat. Izv., 9 (2012), 247–255 | MR | Zbl

[55] Milnor J., “Hyperbolic geometry: the first 150 years”, Bull. Am. Math. Soc., 6:1 (1982), 307–332 | DOI | MR

[56] Mohanty Y., “The Regge symmetry is a scissors congruence in hyperbolic space”, Algebr. Geom. Topol., 3 (2003), 1–31 | DOI | MR | Zbl

[57] Murakami J., “The volume formulas for a spherical tetrahedron”, Acta Math. Vietnam, 33:3 (2018), 219–253 | MR

[58] Murakami J., Ushijima A., “A volume formula for hyperbolic tetrahedra in terms of edge lengths”, J. Geom., 83:1-2 (2005), 153–163 | DOI | MR | Zbl

[59] Murakami J., Yano M., “On the volume of a hyperbolic and spherical tetrahedron”, Commun. Anal. Geom., 13 (2005), 379–400 | DOI | MR | Zbl

[60] Murakami H., Yokota Y., Volume Conjecture for Knots, Springer, Singapore, 2018 | MR | Zbl

[61] Schläfli L., “Theorie der vielfachen Kontinuität”, Gesammelte mathematische Abhandlungen, Birkhäuser, Basel, 1950 | DOI | MR

[62] Sforza G., “Spazi metrico-proiettivi”, Ric. Esten. Different. Ser., 8:3 (1906), 3–66

[63] Thurston W., “Three-dimensional manifold, Kleinian groups and hyperbolic geometry”, Bull. Am. Math. Soc. (N.S.), 6:3 (1982), 357–381 | DOI | MR

[64] Ushijima A., “A volume formula for generalized hyperbolic tetrahedra”, Non-Euclid. Geom., 581 (2006), 249–265 | DOI | MR | Zbl

[65] Valentine J. E., “An analogue of Ptolemy's theorem and its converse in hyperbolic geometry”, Pacific J. Math., 34 (1970), 817–825 | DOI | MR | Zbl

[66] Wimmer L., “Cyclic polygons in non-Euclidean geometry”, Elem. Math., 66 (2011), 74–82 | DOI | MR | Zbl