VII 



OF TETRAHEDRAL SYMMETRY 



497 



This is the centre of symmetry not only for our tetrahedron, but for 

 any close-packed tetrahedral aggregate of co-equal spheres ; we meet 

 with it over and over again, in a pile of cannon-balls, a froth of 

 soap-suds, a parenchyma of cells, or the interior of the honeycomb. 

 Moreover, in the actual demonstration by soap-films of this tetra- 

 hedral symmetry, we see reahsed all the main criteria laid down by 

 Plateau and by Lamarle for a system minimae areae: three films 

 and no more meet in an edge; four fluid edges and no more meet 

 in a point, just as three wire edges and one fluid edge met in a 

 point at each corner of the experimental figure. Lastly, the sym- 

 metry of the whole configuration is such that the three fluid films 



Vh 



178. A regular tetrahedron, with its centre of symmetry. 



meeting in an edge, or the four fluid edges meeting in a point, all 

 do so at co-equal angles. 



In the plane configuration we saw without more ado that the 

 angles of symmetry were the co-equal angles of 120°; but the four 

 co-equal angles between the four edges which meet at the centre of 

 our tetrahedron require a little more consideration. If in our figure 

 of a regular tetrahedron (Fig. 178) o be the centroid, and we produce 

 ao to p, the centre of the opposite side^ bed, it may be shewn that 

 the fine ap is so divided that ao = Sop and ao = bo = co = do. 

 For let four equal weights be put at the four corners of the tetra- 

 hedron, a, 6, c, d. The resultant of the three at b, c, d is equivalent 

 to 3 If at p, the centre of symmetry of the equilateral triangle. 



