GEOMETRICAL DEDUCTION OF SEMIREGUIAR ETC. 7 



sion here changes each octahedron into a tO (Fig. 25) and each 

 vertex into a cube (of vertex import) whose vertices are the eight 

 new positions taken by a vertex of C 24 . 



In a similar manner the e 1 expansions of C 120 and (7 600 may 

 be determined. 



6. Hide. These examples lead to the general rule for the e x 

 expansion of a regular fourdimensional polytope P. The limiting- 

 bodies of P are transformed by the e l expansion and the vertices 

 expanded into regular polyhedra each having as many vertices as 

 there are edges meeting in a vertex of P. 



Examples of the e 2 expansion. 



7. As the faces are the subject in this expansion there can be 

 no application to a single polygon in twodimensional space. 



The e 2 expansion of a cube, an BCO, is shewn in figure 3a-, 

 there are three groups of faces: 



1 st : squares corresponding to the faces of the original cube 



55 55 55 55 ^U^VyO ,, ,, ,, ,, 



3 rd : triangles „ „ „ vertices „ „ „ „ 



In this expansion of any regular polyhedron the faces of the 

 first group are like those of the original polyhedron; the faces of 

 the second group are always squares, since they are determined by 

 the two new positions of an edge of the original polyhedron; those 

 of the third group are triangles, squares or pentagons according 

 as a vertex of the original polyhedron belongs to three, four or 

 five faces. 



As the cube and the octahedron are reciprocal bodies, the num- 

 ber of vertices lying in a face of one being equal to the number 

 of faces meeting in a vertex of the other, it follows that the e 2 

 expansion of the octahedron is also an BCO (Fig. So). 



Again the tetrahedron is self reciprocal, the number of vertices 

 lying in a face being equal to the number of faces meeting in a 

 vertex; so in the e 2 expansion the faces of vertex import are, 

 like the faces of the tetrahedron, equilateral triangles (Fig. 4). 



The e 2 expansion of the icosahedron and dodecahedron, which 

 are reciprocal bodies, is an BID. 



8. The e 2 expansion of the C 8 transforms each cube into an 

 BCO and, as in the C Q each face is common to two cubes, so 

 those faces in the RCO which are faces of the cubes in new posi- 

 tions must now be common to two BCO. In the C 8 each edge 

 belongs to three faces, so in the new polytope each edge takes 



