7 6 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



In his dissertation — which is about to appear — M r . E. L. Elte 

 has created an artificial system 1 ) according to which it is possible 

 to count the degree of regularity of the partially regular polytopes 

 deduced from the regular polytopes by regular truncation. In this 

 system the regularity of such a polytope is expressed by a fraction, 

 the denominator of which is equal to the number of dimensions, 

 while each group of limiting elements as vertices, edges, faces, etc. may 

 contribute a unit to the numerator. With the exception of the group 

 of vertices 2 ) every group of limiting elements has this unit subdivided 

 into two halves, one half for equality of form, the other half for 

 equality of position with respect to the surroundings ; moreover only 

 successive contributions count, beginning at the vertices. So in the 

 case of tT the contributions of vertices and edges are 1 , \ and 



1 _L 1 1 



the degree of regularity is — ~- - = — and this is the case with all 



the Archimedian semiregular polyhedra, except CO and ID, where 

 the dihedral angles on the edges are equal aud the degree of regu- 



, . .1 + 1 2 

 larity is — - - = -. 



Of the two halves corresponding to equality of form and to equality 

 of position with respect to the surroundings the first needs no expla- 

 nation, while the second may seem rather difficult to grasp. But this 

 second half also will become clear, if we indicate it as follows. Equality 

 of vertices means that the figures formed by the systems of edges 

 concurring in the different vertices (vertex polyangles) are congruent, 

 equality of edges means that the edges have the same length (first 

 1) and that the figures formed by the systems of intersecting lines 

 of the faces passing through the different edges with spaces JS n _ i 

 normal to the edges (edge polyangles) are congruent (second -Jr) , 

 equality of faces means that the faces are congruent (first |-) and 

 that the figures formed by the systems of intersecting lines of the 

 limiting threedimensional spaces passing through the faces with 

 spaces S n _ 2 normal to the faces (face polyangles) are congruent 

 (second •*-), etc. 



So we Avili be able to determine the regularity fraction of a 

 given polytope derived from the simplex in the scale of M r . Elte, 

 if Ave have found the different subgroups of each of the limiting 



') "We can only give a glimpse of the system here. For more particulars we must 

 refer to the dissertation written in English. 



2 ) If we count from the other side (see the next page) we must say: "with the excep- 

 tion of the group of limits {l) n —i , \ etc. 



