GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 11 



the operation of expansion applied to a set of limits has the effect 

 of separating any two adjacent members, it follows that the first 

 group can, the second group cannot, be made the subject of ex- 

 pansion . 



For instance in e x C (Fig. la) the triangles cannot be moved 

 away from the centre without increasing the length of the edges 

 joining them, but the octagons may be moved away from the centre 

 until the edge A B common to two has assumed two new positions 

 A ' B' , A" B" which are the opposite sides of a square. The new 

 positions of the octagons define a polyhedron having the required 

 kind of semiregularity. l ) 



15. This double operation may be denoted by the symbol e 2 e 1 C 

 where it is understood that the faces forming the subject of 

 the e 2 expansion are only those which have taken the place of 

 faces in the original cube. Similarly the interpretation of the sym- 

 bol e l e 2 C is that the e 2 expansion is applied to a cube and that 

 the subject of further expansion is composed of those faces which 

 have taken the place of edges in the original cube. This is shewn 

 in Fig. lb where the group of 12 squares (corresponding to the 

 edges of the original cube) form the subject of expansion. These 

 two figures la and lb show that 



e x e 2 C = e 2 e 1 C = tCO 



and it is evident that the order in which the operations are applied 

 to any regular polyhedron is indifferent, for the two operations 

 could have been carried out simultaneously. 



In Fig. 1c is shewn the result of the double operation e 2 e x O 

 applied to an octahedron. This is also a tCO. 



If the double operation be' applied to a I and an D the result 

 in both cases will be a tID. 



This body and the tCO are incapable of further expansion. 



16. Thus it appears that three expansions can be applied to 

 the cube, octahedron, dodecahedron, icosahedron, namely e 1> e 2 , 

 e x e 2 . But more can be done with the tetrahedron owing to the 

 fact that it is self reciprocal. 



Fig. Id and le show respectively the result of the e 2 e 1 and 

 the e x e 2 expansion applied to a tetrahedron , and the result in both 

 cases is a tO which can be further expanded into a tCO (Fig. 7c?). 

 Thus the self reciprocity of the tetrahedron allows an expansion 

 which cannot be carried out in the other four polyhedra. The 



l ) Here trie group of octagons may be called the „independent" variable, while the 

 triangles, which are transformed into hexagons, are the „dependent" ones. 



