GEOMETRICAL DEDUCTION OF SEMIKEGTJLAR ETC. ( J 



bodies, e. g. to C 8 and C lQ , also to C l20 and C 600 , must produce 

 the same result; while applied to a self reciprocal form it pro- 

 duces a polytope whose limiting bodies of vertex import are like 

 the original limiting bodies and of the same number, and whose 

 limiting bodies of face import are of the same number and kind 

 as those of edge import. 



Thus as in the C 8 each face belongs to two, each edge to three, 

 and each vertex to four cubes, it follows that in the expansion 

 each face takes two, each edge three, and each vertex four posi- 

 tions. The e s C 8 is therefore limited by 8 cubes of body import 

 (cubes of the original C 8 ), 24 P 4 of face import, 32 P 3 of edge 

 import, and 16 tetrahedra of vertex import (Fig. 60). In the (7 16 

 each face belongs to two, each edge to four, each vertex to eight 

 tetrahedra, so in the expansion each face takes two, each edge 

 four, and each vertex eight positions and the e 3 C 1Q is limited by 

 16 tetrahedra of body import, 32 P 3 of face import, 24 P 4 of 

 edge import, and 8 cubes of vertex import (Fig. 6$). These two 

 polytopes are alike except that the imports are reciprocal. 



11. Generally there are four groups of limiting bodies: 



1 st : polyhedra of body import like the limiting bodies of the 

 original cell, 



2 nd : prisms of face import defined by their bases (two positions 

 of each face of the original cell), 



3 rd : prisms of edge import defined by their edges parallel to 

 the axis (as many positions of an edge as there are bodies meeting 

 in an edge of the original cell), 



4 th : polyhedra of vertex import having as many vertices as there 

 are bodies meeting in a vertex of the original cell. 



So in e 3 C b there are 10 T, 20 P 3 ; in C u there are 48 0, 192 P 3 . 



This expansion of a C 120 and a C 600 (reciprocal cells) can easily 

 be determined. 



12. Mule. The rule for the <? 3 expansion of a regular polytope 

 P of fourdimensional space is as follows: 



The limiting bodies of P are moved apart (untransformed). 



The faces are replaced by prisms whose bases are parallel posi- 

 tions of a face of P. The edges are replaced by prisms each having 

 as many edges parallel to the axis as there are limiting bodies 

 meeting in an edge of P. Each vertex is replaced by a regular 

 polyhedron, the number of whose vertices is equal to the number 

 of limiting bodies meeting in a vertex of P. 



