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.
"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.
Related to the tetrahedron are two spheres which have received much attention.
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.
This usually has the form of a tetrahedron, with its points base occupying the surface of the body of the axis and its apex pointing towards the interior.
In the stem, segments are successively cut off from the sides of the tetrahedron, and b~ their subsequent division the body of the stem is produced.
In the root exactly the same thing occurs, but segments are cut off alsc from the base of the tetrahedron, and by the division of thes~ the root-cap is formed (fig.
The " tetrahedral theory " brought forward by Lowthian Green,' that the form of the earth is a spheroid based on a regular tetrahedron, is more serviceable, because it accounts for three very interesting facts of the terrestrial plan - (1) the antipodal position of continents and ocean basins; (2) the tri angular outline of the continents; and (3) the excess of sea in the southern hemisphere.
Gomberg's triphenyl-methyl play no part in what follows), it is readily seen that the simplest hydrocarbon has the formula CH 4, named methane, in which the hydrogen atoms are of equal value, and which may be pictured as placed at the vertices of a tetrahedron, the carbon atom occupying the centre.
Assuming the four valencies of the carbon atom to be directed from the centre of a regular tetrahedron towards its four corners, the angle at which they meet.
(The stereo-chemistry of carbon compounds has led to the spatial representation of a carbon atom as being situated at the centre of a tetrahedron, the four valencies being directed towards the apices; see above, and Isomerism.) A form based on Kekule's formula consists in taking three pairs of tetrahedra, each pair having a side in common, and joining them up along the sides of a regular hexagon by means of their apices.
We may therefore regard the nitrogen atoms as occupying the centres of a cubic space lattice composed of iodine atoms, between which the hydrogen atoms are distributed on the tetrahedron face normals.
If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form (a+13+ - y+5) P = aA+sB +yC +SD: a, a, y, b are, in fact, equivalent to a set of homogeneous coordinates of P. For constructions in a fixed plane three points of reference are sufficient.
Thus the C.P. of a rectangle or parallelogram with a side in the surface is at a of the depth of the lower side; of a triangle with a vertex in the surface and base horizontal is 4 of the depth of the base; but if the base is in the surface, the C.P. is at half the depth of the vertex; as on the faces of a tetrahedron, with one edge in the surface.
The tetrahedron is a particular case.
Then the tetrahedron may be regarded as the difference of a wedge with parallel ends, one of the edges being R, and a pyramid whose base is a parallelogram, one side of the parallelogram being S (see fig.
For a tetrahedron, two of whose opposite edges are AB and CD, we require the area of the section by a plane parallel to AB and CD.
Philolaus, connecting these ideas, held that the elementary nature of bodies depends on their form, and assigned the tetrahedron to fire, the octahedron to air, the icosahedron to water, and the cube to earth; the dodecahedron he assigned to a fifth element, aether, or, as some think, to the universe (see Plut.
If this be so the form of the diamond is really the tetrahedron (and the various figures derived symmetrically from it) and not the octadehron.
If the grooves be left out of account, the large faces which have replaced each tetrahedron corner then make up a figure which has the aspect of a simple octahedron.
R shows a combination of two tetrahedra, in which the four faces of one tetrahedron are larger than the four faces of the other; further, the two sets of faces differ in surface FIG.
2, is a combination of the rhombic dodecahedron with a three-faced tetrahedron y (311); the six faces meeting in each triad axis are often rounded together into low conical forms. The crystals are frequently twinned, the twin-axis coinciding with a triad axis; a rhombic dodecahedron so twinned (fig.
The right-hand expression is six times the volume of the tetrahedron of which the lines AA, BB representihg the forces are opposite edges; and we infer that, in whatever way the wrench be resolved into two forces, the volume of this tetrahedron is invariable.
Again, if G be the mass-centre of four particles a, $, 7, situate at the vertices of a tetrahedron ABCD, we find a: ~ :~: tet GBCD: tetUGCDA: tetGDAB: tetGABC, and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space.
If a+$+y+~=O, G is at infinity; if a = fi =~ =~, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if a: ~: 7: = M3CD: z~CDA: ~DAB: L~ABC, G is at the centre of the inscribed sphere.
As particular cases: the mass-centre of a uniform thin triangular plate coincides with that of three equal particles at the corners; and that of a uniform solid tetrahedron coincides with that of four equal particles at the vertices.
For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R2=j a2, if a be the length of an edge.
If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.
If six films of the same liquid meet in a point the corresponding tetrahedron is a regular tetrahedron, and each film, where it meets the others, has an angle whose cosine is - i.