12 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 



combination of operations may be applied in the same way to the 

 cells of fonrdimensional space as one or two examples will show. 



17. Case e 1 e 2 C 8 . — The e 2 operation applied to a C 8 produces 

 a polytope limited by 8 BCO, 32 P 3 , 16 O (Fig. 5). The symbol 

 e 1 e 2 C 8 directs that the new subject of expansion comprising those 

 limiting bodies in e 2 C 8 which correspond to edges of C Q , i. e. the 

 32 P 3 , shall, themselves unchanged, be carried away from the 

 centre (of the e 2 C 8 ). 



These P 3 in their new positions define the polytope sought. This 

 movement changes the BCO and the 0. Each BCO was derived 

 from a cube by the e 2 expansion ; the new expansion e 1 carries 

 out the group of 12 squares (corresponding to the edges of the 

 cube), thereby producing a ICO (Fig. 76). In order to determine 

 the change in the octahedron of vertex import it is only necessary 

 to observe that four of its faces (those in contact with bases of 

 P 3 ) are still in contact with them and are only changed in position, 

 while the other four (those which were in contact with BCO) are 

 changed into hexagons in contact with tCO. Thus the octahedron 

 is changed into a tT. The effect on a single octahedron is the 

 same as if its alternate faces had been made the subject of expan- 

 sion (Fig. 8). 



18.. Case e 2 e s C 8 . — The result of applying the e s operation 

 to a C 8 is a polytope limited by cubes (original cubes of the C 8 ), 

 P 4 of face import, P 3 of edge import, and tetrahedra of vertex 

 import (Fig. 6a). The symbol e 2 directs that the square prisms of 

 face import shall be moved away from the centre of e% C 8 , they them- 

 selves remaining unchanged except in position. These in their new 

 positions define the new polytope and it only remains to determine 

 in what manner their movement has modified the remaining limi- 

 ting bodies of the e 3 C 8 . This can be seen at once in a drawing. 

 In figure 9# are shewn seven limiting bodies of the e 3 C 8 ; one is a 

 cube of the original C 8 , after having been separated by the <? 3 

 movement from the adjacent cubes; three are cubes of face import 

 interposed by the same movement between the cubes of the C 8 ; 

 three are P^ of edge import, their bases being faces of a tetra- 

 hedron of vertex import. The symbol e 2 directs that the cubes of 

 face import are to be moved out. The result is shewn in figure do; the 

 original cube is changed into an BCO, the F s into a P 6 and the T 

 into a tT. It is necessary to bear in mind that only one limiting 

 body of any polytope can be in threedimensional space at a time, 

 and in representing several at once in it there must be either distortion 

 of the limiting bodies or separation of faces and edges which ac- 



