VII] OF TETRAHEDEAL SYMMETRY 317 



with six planes, meeting symmetrically in a point, and constituting 

 there four equal solid angles. 



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 : — precisely 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 sym- 

 metry where our films meet ; and let oa, oh, oc, od, be hues drawn to 

 the four corners of the tetrahedron. Produce ao to meet the base 

 in f, then a'pd is a right-angled triangle. It is not difficult to 

 prove that in such a figure, o (the centre of gravity of the system) 



