tetraedr

Tetraedr is not a standard term in English geometry. In most contexts, the four-faced polyhedron is called a tetrahedron. When encountered, tetraedr is often a misspelling, a truncated form from other languages, or a brand name. The tetrahedron is the simplest of the Platonic solids, consisting of four triangular faces, six edges, and four vertices.

In its regular form all faces are equilateral triangles and all edges have equal length a. The volume is V = a^3/(6√2) and the surface area is A = √3 a^2. The dihedral angle between any two adjacent faces is arccos(1/3), about 70.5288 degrees. A regular tetrahedron is self-dual, meaning its dual polyhedron is another tetrahedron of the same shape. A common way to embed a regular tetrahedron in 3D space is to place its vertices at (1,1,1), (1,-1,-1), (-1,1,-1), and (-1,-1,1), which yields a symmetric configuration on the unit sphere.

Symmetry and group structure are notable: the rotational symmetry group is isomorphic to A4 with 12 elements, and the full symmetry group including reflections is isomorphic to S4 with 24 elements. Tetrahedra appear in various applications, including chemistry (the tetrahedral geometry of methane, CH4), computer graphics (tetrahedral meshes for 3D modeling), and dice design (four-sided dice shaped as regular tetrahedra).

Etymology and terminology reflect its Greek roots: tetra- means four and hedron means face or seat. In other languages, the corresponding term may differ (for example, Tetraeder in German, tétraèdre in French, tetraedro in Spanish). The form “tetraedr” may occur as a truncation or misspelling in some texts.