oktahédra

Oktahédra is a polyhedron characterized by eight triangular faces, six vertices, and twelve edges. It is one of the Platonic solids and is commonly known as the regular octahedron; it is the dual polyhedron of the cube.

The name derives from Greek okta- meaning eight and hedra meaning surface or face.

In a regular oktahédra, all faces are congruent equilateral triangles, and four faces meet at each vertex. Its Schläfli symbol is {3,4}, and it has octahedral symmetry. The rotational symmetry group is isomorphic to the permutation group S4 and has order 24; the full symmetry group, including reflections, has order 48. The vertex coordinates can be taken as (±1,0,0), (0,±1,0), and (0,0,±1) in a suitable scale, placing its center at the origin. The oktahédra is the dual of the cube; its faces correspond to the cube’s vertices and vice versa.

Occurrences and applications: In chemistry and crystallography, octahedral coordination describes many metal complexes where a central atom is surrounded by six ligands at the vertices of an octahedron. In geometry and modeling, octahedra appear in tilings and in the construction of certain 3D meshes. In gaming culture, octahedra are used as dice, notably the eight-sided die (d8), which utilizes an octahedral shape.

Related shapes include the cube, tetrahedron, and dodecahedron.