tetrahedral

Tetrahedral is a term used in geometry, chemistry, crystallography, and mathematics to describe objects or arrangements that have four faces. In geometry, a regular tetrahedron has four triangular faces, four vertices, and six edges; its faces are equilateral triangles. The dihedral angle between any two adjacent faces is about 70.53 degrees. It is one of the five Platonic solids, and its rotational symmetry group has order 12 (including reflections, 24). For a tetrahedron with edge length a, the volume is a^3/(6√2) and the surface area is √3 a^2; the height is √(2/3) a, the circumradius is a√6/4, and the inradius is a√6/12.

In chemistry, tetrahedral describes a molecular geometry in which a central atom forms bonds to four substituents arranged at about 109.5 degrees, typical of sp3 hybridization. Methane (CH4) is the archetype. Real molecules may deviate from the ideal angle due to lone pairs or ligand effects; the term also applies to tetrahedrally coordinated metal centers in many compounds and minerals.

In crystallography and mineralogy, silicate minerals commonly feature SiO4 tetrahedra, with silicon at the center and oxygens at the corners. These tetrahedra link to form chains, sheets, or three-dimensional frameworks. Some compounds exhibit tetrahedral coordination around metal ions in structures such as the zinc blende motif.

In number theory, tetrahedral numbers count spheres arranged in a tetrahedral pyramid: Tn = n(n+1)(n+2)/6.

Tetrahedral concepts also appear in computational fields, where tetrahedral meshes are used in finite element analysis and 3D graphics. The name derives from the Greek tetra-, four, and hedron, face.