14 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 



B F G H, denoted by the symbol c e x 8 where 8 is the square 

 A B C D. 



In figure 10/5 is shown a cube transformed by the e 1 operation, 

 i. e. an e x C; the triangles of vertex import are brought nearer to 

 the centre by the c operation and the result is a CO whose sym- 

 bol is now c e t C. 



Again, the tCO may be considered in two ways. It may be 

 deduced from either the octahedron or the cube (compare Figs, la 

 and lb), so it may be denoted by e 1 e 2 C or e 1 e 2 0. Though the 

 identity of these results may be expressed in the form of an 

 equation: e x e 2 C = e x e 2 0, it must still be borne in mind that the 

 imports are different. Let each of these symbols be preceded by c Q . 

 What are the results? If the tCO has been derived from the cube, 

 the hexagons are of vertex import; if, on the other hand, it has 

 been derived from the octahedron, the octagons are of vertex import. 

 Thus the symbol c e 1 e 2 C indicates that the hexagons, and the 

 symbol c e 1 e 2 O that the octagons, are the subject of the inverse 

 operation whence c Q e i e 2 C = tO (Fig. lc), c e i e 2 O — W (Fig. la). 

 But the octagons correspond to the faces of the cube and the 

 hexagons to the faces of the octahedron, so that c Q e ± e 2 C = c 2 e ± e 2 0, 



21. An example will show the combination of inverse operations. 

 The tCO derived from a cube (Fig. \\a) may be reduced to an 

 octahedron by moving the squares and the hexagons nearer to the 

 centre; the tCO derived from an octahedron (Fig. 11$) may be 

 reduced to a cube by moving the squares and the octagons nearer 

 to the centre. 



These operations are denoted respectively by the equations 



c c i e. v e 2 C = O , c c 1 e x e 2 O = C. 



22. In figure 5 are shown, the limiting bodies of an e 2 C s . If 

 the octahedra of vertex import be made the subject of the inverse 

 operation, the following changes will take place: each F 3 , sepa- 

 rating two neighbouring octahedra, is reduced to two coincident 

 triangles. This annihilates the edges of the prism parallel to the axis. 

 But these are the edges of the original C s in the new positions due to 

 expansion and if these be annihilated each BCO will be reduced 

 to an octahedron. Thus the new body is a £ 24 , eight of whose 

 limiting bodies are compressed BCO, while sixteen are of vertex 

 import in the expansion e 2 C 8 . 



As in the enumeration of the polytopes and the nets given in 

 the three Tables only the c appears, c has been replaced by c. 



