TETRAHEDRON (Gr. riepa-, four, Ebpa,

See also:

• FACE (from Lat. fades, derived either from facere, to make, or from a root fa-, meaning " appear "; cf. Gr. cbatvstv)

face or

See also:

• BASE

base)  , in geometry, a solid bounded by four triangular faces . It consequently has four vertices and six edges . If the faces be all equal equilateral triangles the solid is termed the " regular tetrahedron . This is one of the Platonic solids, and is treated in the article POLYHEDRON, as is also the derived Archimedean solid named the " truncated tetrahedron "; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system . The bisphenoids (the hemihedral forms of the tetragonal and rhombic bipyramids), and the trigonal pyramid of the hexagonal system, are examples of non-regular tetrahedra (see CRYSTALLOGRAPHY) . " Tetrahedral co-ordinates " are a system of quadriplanar co-ordinates, the fundamental planes being the faces of a tetrahedron, and the co-ordinates the perpendicular distances of the point from the faces, a positive sign being given if the point be between the face and the opposite vertex, and a negative sign if not . If (u, v, w, t) be the co-ordinates of any point, then the relation u+v+w+t=R, where R is a constant, invariably holds . This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables . Related to the tetrahedron are two spheres which have received much attention . The " twelve-point sphere," discovered by P . M . E .

Prouhet {1817-1867) in 1863, is somewhat analogous to the nine-point circle of a triangle . If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the

mid-points of the segments of the perpendiculars between the vertices and their common point of intersection . This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces .