740 A NOTE ON POLYHEDRA [ch. ix 



We know also that, having no triangular facets, the polyhedral 

 skeleton must possess trihedral corners; and these, moreover, must 

 (by equation 5) be eight in number, plus the number of pentagons, 

 plus twice the number of hexagons. The total, 



8 +/5+ 2/6= 8 + 12 + (2 X 8) = 36, 



is precisely the number of corners in the figure, all of them trihedral. 

 We also know (from equation 2) that the sum of its plane angles 



= 4 (36 - 2) X 90° =12,240°, 



which agrees with the sum of the angles of twelve pentagons and 

 eight hexagons. The configuration, then, is a possible one. 



So here and elsewhere an apparently infinite variety of form is 

 defined by mathematical laws and theorems, and hmited by the 

 properties of space and number. And the whole matter is a running 

 commentary on the cardinal fact that, under such foedera Natural 

 "as Lucretius recognised of old, there are things which are possible, 

 and things which are impossible, even to Nature herself. 



