Geodesic
Complexification of Tetrahedron and Pseudo-Projective Transformations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 4, pp. 1-7 
V. I. Arnol'd. Complexification of Tetrahedron and Pseudo-Projective Transformations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 4, pp. 1-7. http://geodesic.mathdoc.fr/item/FAA_2001_35_4_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that octahedron is the complex version of tetrahedron in the following sense. The symmetry group of tetrahedron, $A_3$, can be regarded as the group of projective transformations of the space $\mathbb{R}\mathbb{P}^2$ that preserve a quadruple of points. This group can be extended to the group of transformations of the space $\mathbb{РЎ}\mathbb{P}^2$ that preserve a quadruple of points and take complex lines into complex ones. This group turns out to be the symmetry group $B_3$ of octahedron.

[1] Arnold V. I., “Tainstvennye matematicheskie troitsy”, Studencheskie chteniya MKNMU, vyp. 1 (Lektsiya 21 maya 1997 g.), MTsNMO, M., 2000, 4–16

[2] Arnold V. I., “Polymathematics: is mathematics a single science or a set of arts?”, Mathematics: Frontiers and Perspectives, eds. Arnold V. I., Atiyah M. F., Lax P., Mazur B., IMU, Amer. Math. Soc., 2000, 403–416 | MR