22 GEOMETRICAL DEDUCTION OF SEMIREGULAE ETC. 



face gap remains unchanged (Figs 22y and 25y), as well as the 

 limiting body of face import in the e 3 C iG (tt). 



The tetrahedron (Fig. 227r) is changed by the e 2 expansion into 

 a CO (Fig. 25;r) and again the manner in which the other limiting 

 bodies of this poly tope are affected by the change can be traced 

 by an examination of the manner in which they are connected 

 with the tetrahedra. 



The changes in the edge and vertex gaps can also be traced 

 (compare Figs. 22 and 25). 



The polytope of vertex import in Fig. 25 is remarkable, as it is 

 limited by 48 semiregular polyhedra of the same kind. 



The <? 4 expansion. 



37. The e k expansion applied to a net of C s , C' i6 or (7 24 sepa- 

 rates the adjacent constituents by a distance equal to an edge. 

 Thus two neighbouring members of a block are separated by a 

 fourdimensional prism whose two opposite bases are the two limi- 

 ting bodies that coincided in the regular net. The net of C. ti so 

 treated results in another net of C s of different imports. 



The net of C i6 transformed by the <? 4 expansion leads to the 

 following result. The C.& are separated, so that instead of two 

 having a tetrahedron in common they are separated by a distance 

 equal to an edge. 



In other words the tetrahedron common to two adjacent C i6 

 has assumed two parallel positions, the bases of a fourdimensional 

 prism (Fig. 26<J). 



The side limiting bodies of this fourdimensional prism are four 

 P 3 (of face import). As three C iQ meet in a face in the net of 

 C\ 6 each face must assume three positions which define a prismo- 

 tope (3 ; 3) (Fig. 2 67). 



Again six C iG meet in an edge of the net, therefore each edge 

 takes six positions, i. e. the new positions are the side edges of 

 a fourdimensional prism on an octahedral base (/3). It may be seen 

 by (tt), (J), (7) and (/3) that only one of these four polytopes pos- 

 sesses a limiting body with vertex import, i. e. the one filling the 

 edge gap (/3), so that the vertex gap is surrounded by octahedra, 

 and as in the net of C i6 there are 24 edges meeting in a vertex 

 it follows that 24 octahedra surround the vertex gap; that is, it 

 is a (7 24 . This new net evidently may also be obtained by 

 applying the e k expansion to the net NC. rr 



