oktaedral

Oktaedral, or octahedral, describes a polyhedron with eight faces that are triangles. The regular octahedron has eight equilateral triangular faces, six vertices, and twelve edges. It is one of the five Platonic solids and has Schläfli symbol {3,4}, with four faces meeting at each vertex.

Construction and symmetry: A regular octahedron is dual to the cube. It can be viewed as two square pyramids joined at their bases. A standard realization places vertices at (±1,0,0), (0,±1,0), and (0,0,±1). It possesses 24 rotational symmetries and 48 symmetries in total if reflections are included, with the full symmetry group denoted Oh.

Applications and occurrences: In chemistry, octahedral coordination geometry describes six ligands arranged around a central atom, common in transition metal complexes and related minerals (for example SF6 or [Co(NH3)6]3+). In crystallography and mineralogy, octahedra appear in various structures, including perovskites where a central cation sits inside an octahedral array of anions. In mathematics and gaming, the octahedron is used as the D8 die.

Other notes: The term oktaedral is used interchangeably with octahedral in many contexts, reflecting the same geometry. The octahedron shares a dual relationship with the cube and is part of discussions of polyhedral symmetry and Platonic solids.