tetraedri

Tetraedri is the Italian plural of tetraedro, and in English the term is tetrahedra. A tetraedro (tetrahedron) is a polyhedron with four triangular faces, six edges, and four vertices. It is the simplest three-dimensional polyhedron that is not a pyramid with a base polygon of more than three sides.

Regular tetraedri: all edges are equal and all faces are equilateral triangles. It is one of the Platonic solids and has full symmetry under the Td group. The dihedral angle between any two adjacent faces is arccos(1/3) ≈ 70.53 degrees.

For a regular tetrahedron with edge length a, the volume is V = a^3/(6√2), and the surface area is A = √3 a^2. The circumscribed radius is R = a√6/4, and the inradius is r = a√6/12. The height (distance from a vertex to the opposite face) is h = √(2/3) a.

In general (irregular) tetraedri, the four vertices a, b, c, d are not coplanar; the volume can be computed as V = |det(b−a, c−a, d−a)|/6. Irregular tetrahedra vary in edge lengths and face shapes but share the same connectivity, Euler characteristic V−E+F = 2.

Tetraedri appear in geometry, computer graphics, chemistry (sp3 hybridization and molecular geometry), and mesh generation, where they provide simple units for partitioning three-dimensional space.