The tetrahedron is a particular case.
Related to the tetrahedron are two spheres which have received much attention.
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.
"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.
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.
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.
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.
The tetrahedron is a particular case.
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.
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.
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.
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.
The names of these five solids are: (r) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 20 equilateral triangles.
These solids played an important part in the geometry of the Pythagoreans, and in their cosmology symbolized the five elements: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube), universe or ether (dodecahedron).
The truncated tetrahedron is formed by truncating the vertices of a regular tetrahedron so as to leave the original faces hexagons.
We may also note that of the Archimedean solids: the truncated tetrahedron, truncated cube, and truncated dodecahedron, are the reciprocals of the crystal forms triakistetrahedron, triakisoctahedron and triakisicosahedron.
Since the tetrahedron is the hemihedral form of the octahedron, and the octahedron and cube are reciprocal, we may term these two latter solids " reciprocal holohedra " of the tetrahedron.
The equilateral triangle is the basis of the tetrahedron, octahedron and icosahedron.'
It is readily seen that the tetrahedron is its own reciprocal, i.e.
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.