DERIVED FROM THE REGULAR POLYTOPES. 3 



117. The polyhedron ce 1 /is similar to the convex polyhedron 

 whose vertices are the middle-points of the edges of the original 

 icosahedron. This polyhedron must be enlarged in such a ratio that 

 the length of its edges becomes equal to that of the edges of the 

 original icosahedron. Now the distance of the middle-points of two 

 adjacent edges of an icosahedron is half the length of an edge, 

 and the coordinates of the middle-point of an edge are obtained 

 by dividing the sum of the corresponding coordinates of the two 

 terminal-points by 2. Hence the coordinates of the vertices of ce x 1 

 are obtained by simply adding the corresponding coordinates of the 

 terminal-points of all the edges of / '). 



In this manner two symbols of coordinates of ce l I are found 

 corresponding with the two types of edges of I with respect to the 

 system of coordinates. VVc may determine only one vertex of each 

 type and deduce from it a coordinate-symbol according to the rule 

 of pentagonal hemiedry. 



The computation of the coordinate-symbols now becomes : 



/ 2 , 1 -f e , 



I — 2 , 1 + e , 



""1 



I 



1 * 



2 



, 1 



+ * 



, o 





" . ~» «»~ vu , 





I 



, 





2 



, ] v<> 







ce l 



T 



[2 , 



3 



-f* 



,!'+«]: 



2. 



■ 24 „ ; 



Total ... 30 vertices. 



The total number of vertices may be controlled by means of 

 the table of art. 115. 



118. The polyhedron ce 2 I is similar to the convex polyhedron 

 whose vertices are the centres of the faces of /. This polyhedron 

 must be enlarged in such a ratio, that the distance between two 

 adjacent vertices becomes equal to the length of the edges of the 

 icosahedron, i. e. = 4. 



The coordinates of the centre of a face of / are obtained by 

 adding the corresponding coordinates of the vertices of the face 



') The same rule holds to find the cei-form of any polytope whose faces arc triangles. 



More generally the coordinates of the vertices of the ce n -form of a polytope, whose 

 limits /„ , ! are simplexes, are obtained by adding the corresponding coordinates of the 

 vertices of all the limits 7 of the polytope. 



L* 



