If we make a wire cage, in the form of a regular tetrahedron, and dip it into soap-solution, then when we withdraw it we see that to each one of the six edges of the tetrahedron, i.e. to each one of the six wires which constitute the little cage, a film has attached itself; and these six films meet internally at a point, and constitute in every respect the symmetrical figure which we have just been describing. In short, the system of films we have hereby automatically produced is precisely the system of partition-walls which exist in our tetrahedral aggregation of four spherical bubbles:ãprecisely the same, that is to say, in the neighbourhood of the meeting-point, and only differing in that we have made the wires of our tetrahedron straight, instead of imitating the circular arcs which actually form the intersections of our bubbles. This detail we can easily introduce in our wire model if we please.
Let us look for a moment at the geometry of our figure. Let o (Fig. 120) be the centre of the tetrahedron, i.e. the centre of symmetry where our films meet; and let oa, ob, oc, od, be lines drawn to the four corners of the tetrahedron. Produce ao to meet the base in p; then apd is a right-angled triangle. It is not difficult to prove that in such a figure, o (the centre of gravity of the system) {318} lies just three-quarters of the way between an apex, a, and a point, p, which is the centre of gravity of the opposite base. Therefore
Therefore
That is to say, the angle aod is just, as nearly as possible, 109ô¯ã₤28ÿ£¢ãýã₤16ÿ£¢ã°. This angle, then, of 109ô¯ã₤28ÿ£¢ãýã₤16ÿ£¢ã°, or very nearly 109 degrees and a half, is the angle at which, in this and every other solid system of liquid films, the edges of the partition-walls meet one another at a point. It is the fundamental angle in the solid geometry of our systems, just as 120ô¯ was the fundamental angle of