ANALYTICAL TREATMENT UF THE POLYTOPES BEGULARL"* 



C. The characteristic numbers of the G 24 -family . 



L 24. The determination of the characteristic numbers bears direct 



relation to the results of the researches of Mrs. Stott contained 

 in the list of the limiting bodies (Table I) and the incidences 

 (Table III). We obtain the number of vertices of the new geo- 

 metrical creations from the table of incidences, that of the limiting 

 bodies and limiting faces from the list of the limiting bodies; 

 finally the number of edges is deduced from Euler's theorem 1 ). 

 We begin by applying the principle of expansion and then 

 proceed to the contraction. 



125. Forms e 1 C 24 , e 2 C' 24 , e. A ([ >4 . As the circnm polyhedron of 

 (' 24 is the cube, it is evident, that 8 edges, 1.2 limiting faces, 

 limiting octahedra meet in a vertex. Accordingly the limiting 

 bodies of vertex import of e x (' 24 , e 2 C 24 , e 3 C 24 are respectively 

 C, CO, 0. So the number of the vertices is 8, 12 and 6 times 

 24, viz. 192, 288, 144 2 ). 



We deduce from the limiting bodies (24 t(J and 24 C for e y 6 24 ; 

 2êRC0, 96 P 3 and 24 CO for e 2 <7 24 ; 24 0, 90 1\ , 96 P 3 , 

 24 for e 3 C 24 ) for the numbers of limiting bodies and faces 

 consecutively 48, 240 for e x C 24 ■ 144, 720 for^C 24 ; 240, 672 

 for e 3 C 24 . So the complete result is 



{ 24 



( 24 , 



96 , 



96 , 



24); 



e i C 2i 



(192 , 



384 , 



240 , 



48); 



''■1 -24 



(288 , 



864 , 



720 , 



144); 



H C 24 



(144 , 



576 , 



67.2 , 



240) . 



126. Forms e x e 2 C' 24 , e x e 3 C 24 , e 2 e 3 C' 24 , e x e 2 e 3 6 24 . The number 

 of vertices of e x e 2 6 24 is 6/ = 576 ; of e x e 3 C 24 24 r = 576 ; 

 of e 2 e 3 C 24 24 r = 576. For e x e 2 e 3 C 24 we obtain 48 r = 1152. * 

 For the limiting bodies and faces the numbers are respectively 

 144, 720 for e x e 2 C 24 ; 240, 1104 for e { e s C 24 and e 2 e 3 C 24 and 

 240, 1392 for e 1 e 2 e 3 C 2i . The complete result is 



') We remember, that the author gave in his „Mehrdimensionale Geometrie" II, art. 

 19 — 20 p. 60 — <',4, the formula of Euler's theorem for it dimensions, and that his 

 symbol is (e, k, /, r) for the number of vertices (e), edges (k), faces (/') and limiting 

 bodies (»■). 



2 ) Another shorter expression is: for eiC 2i twice the number k of C^\ for e^C^ thrice 

 the number /'; for <' : ;'<l'i s '^ tines r\ as we sec by considering the vertices of e\C<^ as 

 extremities of the displaced edges of '.' L >|. etc 



