1 8 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 



is changed into an BCO. Four of these are shewn in Fig. 1G 

 after having been adjusted so that a face which was common to 

 two cubes becomes common to two BCO. 



This adjustment leaves edge gaps and vertex gaps. 



As an edge belougs to four and a vertex to twelve faces (mem- 

 bers of the subject) the edge gap is defined by four new parallel 

 positions of an edge and the vertex gap by twelve new positions 

 of a vertex. Therefore the first is filled by a square prism (a cube) 

 and the second by a CO. In the CO the triangles are supplied by 

 triangular faces of the eight BCO (expanded cubes) and the squares 

 by the bases of the six prisms (expanded edges) surrounding the 

 gap. Thus the net of cubes is changed by the e 2 transformation into 

 a net e 2 NC with the three constituents BCO, <7and<70(A. 20) l ). 



The e 2 expansion may be applied to a net N(0,T) of O and 

 T by taking either the group of O or the group of T as inde- 

 pendent variable, and the faces of that group as subject. Whichever 

 group is chosen, its faces in their original position define the net 

 N(0,T), in their final position the new net. Thus if the e 2 expansion 

 be applied to the O each O is changed into an BCO (Fig. Sô) whose 

 triangular faces are in contact with the untransformed tetrahedra. 

 The vertices of each O are now changed into squares (Fig. 35) 

 and as six octahedra meet in a vertex of N(0,T) the vertex gap 

 is a cube. Thus the new net e 2 N(0,T) has three constituents 

 BCO, C, T (Fig. 17) (A. 19). ~ 



In figure 18 is shewn the result e i N(0,T) == e v N(0,T) of the e i 

 expansion applied either to the octahedra or to the tetrahedra of 

 the net (0,T). 



31. In fourdimensional space an example is given of the e 2 

 expansion e 2 NC 2V Each C 2% is changed into an e 2 C 2i limited by 

 24 BCO, 96 P 3 , 24 CO (see rule, art. 9 and Fig. 19t). 2 ) 



The BCO are transformed octahedra, the P 3 are expanded edges, 

 and the CO expanded vertices. When the transformed (7 24 are 

 adjusted so that an octahedron which in the regular net is common 

 to two C 2i is changed into an BCO common to two e 2 C 2k , there 

 are edge gaps and vertex gaps. 



In order to facilitate the determination of these gaps it will be 

 well to state clearly the manner in which the three kinds of limi- 

 ting bodies are mutually arranged in the e 2 C 24 . 



') This means Fig. 20 in Andreini's memoir quoted in art. 1. In order to facilitate 

 comparison a table of threedimensional nets is given on plate III. 



2 ) Here and in the following figures ?r means "principal'' constituent, while a, /3, etc. 

 stand for the polytopes filling the vertex gap, the edge gap, etc. 



