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 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.
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.
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.
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.
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 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.
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 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.