oktaedron

Oktaedron, better known in English as the regular octahedron, is a polyhedron with eight equilateral triangle faces, six vertices, and twelve edges. It is one of the five Platonic solids and is the dual polyhedron of the cube, meaning each vertex corresponds to a cube face and vice versa. An octahedron can be formed by joining two square pyramids along their bases or, equivalently, by connecting the centers of the six faces of a cube.

Geometric properties: In the regular form, four faces meet at each vertex. If the edge length is a, the vertices can be placed at (±a/√2, 0, 0), (0, ±a/√2, 0), (0, 0, ±a/√2). The circumradius (distance from the center to a vertex) is R = a/√2, and the inradius (distance from the center to a face) is r = a√6/6. The volume is V = (√2/3) a^3, and the surface area is A = 2√3 a^2. The dihedral angle between adjacent faces is arccos(-1/3) ≈ 109.47 degrees. If a cube of side length s is given, the octahedron formed by connecting the centers of the cube’s faces has edge length a = s/√2.

History and uses: The term derives from Greek and the shape appears in discussions of Platonic solids and crystallography. In chemistry, octahedral coordination describes a central atom surrounded by six ligands at the vertices of an octahedron. The octahedron is also a common eight-sided die (D8) in gaming.

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