28 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



D. Expansion and contraction symbols. 



18. If we compare the symbols containing the operators e L and 

 c of expansion and contraction introduced by M rs . Stott for the 

 offspring of the T= S(4) in S$ and the S{§) in S tii with the zero 

 symbol of these polyhedra and poly topes, we remark that all these 

 cases underlie certain general laws, up to now of an empirical 

 character. By proving these laws we will promote them to theorems, 

 the first of which can be stated as follows : 



Theorem VI. "The expansion e ki (k = 1; 2, 3, . . . , n — 1) applied 



to the S (l) (n -j- 1) of JS n changes the symbol of coordinates (1 00 . . . 0) 

 of that simplex into an other zero symbol which can be obtained 

 by adding a unit to the first k -j- 1 digits." 



Indeed this gives (compare the small table n = 4 under art. 3): 



^£(5) = (21000), e 2 JS(h) = (21100), % #(5) = (21110). 



Proof. The operation of expansion e,. consists in moving the 

 limiting S{k ~\- 1) of 8 {i) (n — |— 1) tö equal distances away from the 

 centre O of /$ {[) (n-\- 1), each S(k -f- 1) moving in the direction 

 of the line OM joining O to its centre M, these S(k -\- 1) "remai- 

 ning parallel to their original position , retaining their original size 

 and being moved over such a distance that the two new positions 

 of any vertex which was common to two adjacent limits (/),. in the 

 original 8 (l) (n -j- 1) shall be separated by the length of an edge". 1 ) 



Now let us consider (fig. 3) the plane through OM and any 

 vertex A of the S(k -f- 1) of which M is the centre; then, on 

 account of the regularity of jS {i) (n -f- 1), the angle AMO is a right 

 one. This plane will also contain the new position AM of AM. 

 What we have to do now is this: We select from the symbol of 



a 



coordinates (1 00 .. 0) the vertices of any limiting /S (1) {k -\- 1) , 

 calculate the coordinates of M , deduce from the coordinates of (J 

 and M those of M on the supposition that OM' : OM = À is 

 known. Then we have to determine the coordinates of A' by adding 

 to the coordinates of the vertex A chosen arbitrarily among the 

 k -\- 1 vertices of the S (i) (k -\- 1) the differences of the correspon- 

 ding coordinates of M' and M. Finally we have to determine À 

 by the stated condition that two new positions of the same vertex 

 A of 8^{n -j- 1) shall be separated by the length of an edge, 



*) Compare p. 5 of the memoir quoted of M>. Stott. 



