tetraedre

Tetraedre, or tetrahedron in English, is a polyhedron with four triangular faces, four vertices and six edges. It is the simplest non-degenerate polyhedron and can be viewed as a triangular pyramid: a pyramid with a triangular base and three side faces.

In the regular tetrahedron all faces are equilateral triangles of equal edge length a. The dihedral angle between any two adjacent faces is arccos(1/3) ≈ 70.53 degrees. The volume is V = a^3/(6√2) and the surface area is A = √3 a^2. A common way to describe its geometry places its vertices at equal distances from the center.

For a general tetrahedron with vertices at position vectors, the volume can be computed as V = (1/6) |a · (b × c)|, where a, b, c are edge vectors from a common vertex. The dual polyhedron of a tetrahedron is another tetrahedron, and the tetrahedron is self-dual in the sense of dual polyhedra. Its Euler characteristic satisfies V − E + F = 2, with V = 4 vertices, E = 6 edges and F = 4 faces.

Symmetry: the rotational symmetry group of a regular tetrahedron is isomorphic to A4 and has 12 elements; the full symmetry group including reflections has 24 elements. A regular tetrahedron can be inscribed in a sphere, with the circumradius R related to edge length by R = a√6/4.

Applications and occurrences: in chemistry the methane molecule CH4 adopts a tetrahedral arrangement around carbon; in computer graphics and finite element methods, tetrahedra are used for mesh generation and 3D modeling due to simple connectivity and favorable numerical properties. The tetrahedron is one of the five Platonic solids.