GEOMETBICAL DEDUCTION OF SEMIREGULAtt ETC. 21 



The members of each group are in triangular contact with mem- 

 bers of the same and in square contact with members of the other 

 group. 



This polytope, called a simplotope, is a special case of a group 

 of polytopes called prismotopes l ). 



Two kinds of limiting bodies surrounding the edge gap have 

 now been found, i. e. square prisms due to the transformed (7 16 

 (Fig. 22y) and 1 D 3 due to the expanded face (Fig. 22/3); there are 

 six of the former and eight of the latter, since six C iQ and eight 

 faces meet in an edge of NC i6 . As the axes of these 14 prisms 

 are parallel, the body must be a fourdimensional prism whose base 

 is a CO of vertex import (since its vertices are the 12 positions 

 taken by the end point of an edge). 



The vertex gap is surrounded by cubes (jr) and CO (/3), and 

 there are 24 of each since 24 C [G and 24 edges meet in a vertex 

 of NC l6 . 



Thus there are four constituents in the new net e% NC. l6 : e 3 C i6 , 

 prismotope (3 ; 3), P co ; and a polytope e £ C iQ limited by 24 C, 24 CO. 



The manner in which these different bodies are in contact is 

 indicated by the imports in the drawings and by the vertical lines. 



35. Two examples are given in order to show how a second 

 operation may be applied to the result of a single expansion 

 (Figs. 24 & 25). 



Let it be desired to apply the e { expansion to the net obtained 

 above. Here those constituents taking the place of edges in the 

 original NC m are the subject and must be moved unchanged into 

 new positions. Thus the edge gap in the new net is like that in 

 the e 3 expansion (compare Figs. 22/3 & 24/3). 



Moreover those limiting bodies of edge import in the transformed 

 C i6 and in the prismotope (face gap) must also remain unchanged 

 (compare the parts 7T and y of Fig. 22 and Fig. 24). 



The tetrahedra (Fig. 227r) are transformed by the e i expansion 

 into tT (Fig. 24tt). 



A careful examination of the manner in which the P 3 of face 

 import and the cube of vertex import in the same polytope (tt) are 

 in contact with the tetrahedra will show in what manner they must 

 be changed (see Fig. 24 w). From these may be traced the changes 

 in the face gap (y) and vertex gap (#). 



36. If it be desired to apply the e. z expansion to e 3 NC [6 the 



*) Compare the foot note 3 ) in art. 1. 



