Geodesic
Some Applications of the Formula for the Volume of an~Octahedron
Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 27-43.

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We introduce the notions of combinatorial, metric and spatial symmetries of a polyhedron. In the case of symmetric octahedra, we present explicit forms of canonical polynomials for determining their volumes. 
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R. V. Galiulin; S. N. Mikhalev; I. Kh. Sabitov. Some Applications of the Formula for the Volume of an~Octahedron. Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 27-43. http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a3/

[1] Sabitov I. Kh., “Obobschennaya formula Gerona–Tartalya i nekotorye ee sledstviya”, Matem. sb., 189:10 (1998), 105–134 | MR | Zbl

[2] Astrelin A. V., Sabitov I. Kh., “Minimalnyi po stepeni mnogochlen dlya ob'ema oktaedra”, UMN, 50:5 (1995), 245–246 | MR | Zbl

[3] Galiulin R. V., Lektsii po geometricheskim osnovam kristallografii, Izd-vo Chelyabinskogo universiteta, Chelyabinsk, 1989