8 GEOMETRICAL DEDUCTION OP SEMIREGULAE ETC. 



three new positions which are the three parallel edges of a right 

 prism on an equilateral triangular base. 



This manner of determining the prism (expanded edge) bears the 

 most direct relation to the particular expansion under consideration, 

 namely that in which the faces are the subject; but it could have 

 been determined otherwise. Thus in a C 8 three cubes meet in an 

 edge and as each is changed into an RCO, its edges are changed 

 into squares, so that instead of three coincident edges there are 

 now three squares, the side faces of a right prism. 



Again in the C 8 each vertex belongs to six faces and therefore 

 must assume six positions. From this it is evident that the body 

 taking the place in the new polytope of the vertex in the C Q has 

 six vertices and it remains to determine its faces. 



In figure 5 are represented, in their true relative positions as 

 far as threedimensional space will allow, two of the four RCO 

 and two of the four P 3 which have taken the place of the four 

 cubes and the four edges meeting in a vertex of the C 8 . It shews 

 that each RCO supplies a triangular face and each prism a trian- 

 gular face — all equilateral — to the body that takes the place 

 of the vertex of the C 8 . This body therefore is a regular octa- 

 hedron, four of whose faces are in contact with RCO and four 

 with P 3 . 



The new polytope then, e 2 C 8 , is limited by 8 RCO, 32 P s of 

 edge import, 16 O of vertex import. 



9. Rule. The rule for the e 2 expansion of a regular fourdimen- 

 sional polytope P may be stated thus: 



The limiting bodies of P are transformed by the e 2 expansion. 

 The edges are expanded into prisms each having as many edges 

 parallel to the axis as there are faces meeting in an edge of P. 

 The vertices are expanded into bodies having two groups of faces, 

 one kind of edge, and as many vertices as there are faces meeting 

 in a vertex of P. One group of faces is supplied by the bases of 

 the prisms of edge import and of these the number is equal to 

 the number of edges meeting in a vertex of P; the other is supplied 

 by the expanded vertices of the transformed limiting bodies, of 

 which the number is equal to the number of limiting bodies meeting 

 in a vertex of P. 



PJœamples of the e% expansion. 



10. Here the limiting bodies are the subject; and it is at once 

 evident that this expansion applied to reciprocal fourdimensional 



