Analytical treatment of the polylopes regularly derived 

 from the regular polylopes. 



Section V: Polytopf.s derived prom the extra tolytopes. 

 A. Introduction. 



114. We remember, that in the general introduction of this 

 series of memoirs a classification in five sections was given. According 

 to the author's intention the fifth section should have to deal with 

 the extra regular polybedra and polytopes. Hence the object of 

 this section will be to investigate the polyhedra deduced from the 

 icosahedron (or dodecahedron) and the four/dimension a] polytopes 

 deduced from C' u and from C 6()0 (or C l20 )- 



As to the nets that can be deduced from (7 24 , since these nets 

 can be deduced at the same time from the cross polytope C 16 , 

 we refer to Section HI, A', art 78—84 and Table VII. 



Now a difficulty is arising in consequence of the want of a 

 general theorem relating to the symbols used for the coordinates, 

 as we are compelled to use more than one symbol for the same 

 polytope. Obviously it is a priori not impossible to find a method 

 which allows us to find the condition necessary for the symbolical 

 representation of a polytope by two, three or even more symbols. 

 Yet, as a satisfactory result of these attempts is dubious, we prefer 

 a method which follows closely the geometrical operations, by means 

 of which the polytopes are defined.* 



B. The Icosahedron- family . 



115. The following table contains the symbols of the polyhedra 

 of this family considered as deduced from the Icosahedron, /, and 

 from the Dodecahedron, D, the symbols used by Mrs. Stott and 



the characteristic numbers (numbers of vertices, edges and faces): 



Verband. der Kon. Akad. v. Wetensch. I- Sectie Dl. XII N 3 . 2. B 1 



