GEOMETKICAL DEDUCTION OF SEMIKEGULAR ETC. 19 



A shaded face A^B^ common to two BCO (in Fig. 19) is the 

 new position of a face ABC common to two octahedra in (7 24 ; 

 A^B X , A 2 B 2 , A B B 3 are three new positions of an edge AB of the 

 C 2k , and the two positions A X B X , A 2 B 2 in the BCO are identical 

 with the two positions A V B X , A 2 B 2 in the prism. Again, the vertices 

 of the CO are the 12 positions taken by a vertex A of the C 2k 

 of which four A x A 2 A k A 5 are identical with four A x A 2 A k A§ in 

 the BCO. 



In the net of (7 24 an edge is common to four and a vertex to 

 3.2 faces (members of the subject), so that the edge gap is defined 

 by four positions of an edge and the vertex gap by 32 positions 

 of a vertex. The limiting bodies surrounding these two gaps may 

 be found in the following manner. Four (7 24 meet in an edge and 

 eight in a vertex of the net (7 24 . In each, the edge is changed 

 into a P 3 and the vertex into a CO. Thus among the limiting 

 bodies surrounding the edge and vertex gaps there must be four 

 P 3 in parallel positions in the former and 8 CO in the latter. 



Now in the original net two adjacent C 2k , let us say M & N, 

 have a common octahedron, or it may be said that two octahedra, 

 limiting bodies of two adjacent C 2 \, coincide. So in the transformed 

 net two adjacent e 2 C 2!k have an BCO (transformed octahedron) in 

 common; or it may be said that two BCO, limiting bodies of two 

 adjacent e 2 C lk , M & N, coincide. 



Thus the BCO (Fig. 19tt) represents two coincident limiting 

 bodies, one belonging to M and the other to N. In each the face 

 {A X B V , A 2 B 2 ) is in contact with a P 3 and these two P 3 can have no 

 other point in common, or else the polytopes M and N, having 

 already one common limiting body, an BCO, would coincide. 



Thus two adjacent B 3 surrounding the edge gap have a square 

 face in common. It remains now to seek a poiytope which satisfies 

 the following conditions. It must be determined by four parallel 

 positions of an edge and have amongst its limiting bodies four 

 parallel P 3 of which any adjacent two have a square face in common. 

 A fourdimensional prism on a tetraheclral base is the only body 

 which satisfies these conditions, so that the limiting bodies are 4 B 3 , 

 2T (Fig. 19/3). 



Each of the tetrahedra is determined by its vertices i. e. four 

 positions assumed by the end point of an edge of the net C 2k and 

 is therefore of vertex import. 



As 16 edges meet in a vertex of the net C 2k , there are 16 of these 

 tetrahedra surrounding the vertex gap. 



The limiting bodies of the poiytope which must fill the vertex 



