TETRAHEDRON (Gr. riepa-, four, Ebpa, face or base)

This article appears in Volume V26, Page 671 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: TAV-THE

TETRAHEDRON (Gr. riepa-, four, Ebpa, face or 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

REGULAR

tetrahedron. This is one of the Platonic solids, and is treated in the

article

ARTICLE (from Lat. articulus, a joint)

POLYHEDRON

POLYHEDRON (Gr. rain, many, ESpa, a
base
- BASE
)

, as is also the derived Archimedean solid named the " truncated tetrahedron "; in addition, the

regular

REGULAR

tetrahedron has important crystallographic relations, being the hemihedral form of the regular

octahedron

OCTAHEDRON (Gr. 6K-rw, eight, Mpa,
base
- BASE
)

and consequently a form of the cubic

system

SYSTEM

. The bisphenoids (the hemihedral forms of the tetragonal and rhombic bipyramids), and the trigonal pyramid of the hexagonal

system

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

CONSTANT, BENJAMIN (1845-1902)

, invariably holds. This system is of much service in following out mathematical,

physical

PHYSICAL

and chemical problems in which it is necessary to represent four variables.
Related to the tetrahedron are two

spheres

SPHERES, MUSIC OF THE

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

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.

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