Octahedra

An octahedron is a polyhedron with eight equilateral triangle faces, six vertices, and twelve edges. It is one of the Platonic solids and is the dual of the cube, sharing the same symmetry group Oh. The regular octahedron can be described as two square pyramids joined at their bases or as the three-dimensional cross polytope.

In a regular octahedron, all edges have the same length a. Its main measurements are: volume V = (1/3)√2 · a^3 and surface area A = 2√3 · a^2. The circumradius (distance from the center to a vertex) is R = a√2/2, and the inradius (distance from the center to a face) is r = a√6/6. The dihedral angle between adjacent faces is arccos(−1/3) ≈ 109.47 degrees. A standard coordinate model places the octahedron at the origin with vertices at (±1,0,0), (0,±1,0), and (0,0,±1), giving edge length √2 in that configuration.

Regular octahedra are the duals of cubes, meaning each vertex corresponds to a cube face and each face corresponds to a cube vertex. They appear in chemistry as octahedral coordination geometry around a central atom, with six ligands at the corners of an octahedron. In crystallography and mineralogy, octahedral sites describe similar coordination environments, while in gaming and design, octahedra are used as eight-faced dice. The shape also features in various applications due to its high symmetry and straightforward geometric relations.