DERIVED FROM THE REGULAR POLYTOPIES. 37 



vei'ticcs we get all the vertices of the net and each vertex once. 



In other words: the system of the centres of either of the two sets 



of triangles is equipollent to the system of vertices of the net, i.e. 



if we move all the vertices of the net in the direction A v O over 



that distance it passes into the system of the centres of the triangles 



corresponding in orientation with A { A 2 A 3 , whilst we get the system 



of the centres of the other set of triangles by a motion over the 



same distance in opposite direction. So, as the three coordinates of 



any vertex of the net are found by adding to the coordinates 1, 0, 



of A l three integers with a sum zero, and the true coordinates of 



the centres O and O x are \, \, \ and ■ — 3-> f > f the symbols 



{b { + !, b 2 -f- \,b z -\- \) and (b x — -J-, ö. z + f, b s -f |) represent the 



centres of the two sets of triangles under the condition that the 



three b t are integers with sum zero. In both cases the three integers 



b L with sum zero indicate what is to be added to the coordinates 



of any centre of each set in order to obtain the whole set; as we 



call the two sets of centres the "frames" of the two kinds of 



triangles, we call the system of differences b i} b 2 , b 3 the "frame 



coordinates" and (b lt b 2 , b s ),'£,b i = the "frame symbol" of both 



sets of triangles. 



Recapitulating we find the following result for N(p 3 ): 



Net symbol (a ± , a. ls a 3 ), 2 a = 1. 



Set of triangles ( Symbol of constituent A X A 2 A 3 (1,0, 0), 



A i A 2 A 3 j Frame (0 4 + |, b 2 4- •$■/, b s + |), £ b == 0. 



Other set of ( Symbol of O l 2 3 . . (— |, f , f), 



triangles | Frame .... {b x — i , b 2 ~-\- | , b è -f- -| ), 2 b = 0. 

 Frame symbol (b i} b 2 , b 3 ), 2 $ = 0. 



Here O t 2 3 represents a central triangle oppositely orientated 

 to A x A 2 A 3 . We may still remark that the second frame may be 

 written in the more symmetrical form (^ -|- J- , b 2 -j- -| , b 3 -\~ -|) , 

 S b i = — 1 , or if one likes {b x — |-, b 2 — J-, b 3 — -|), 2 b t = 2. 



But it is much simpler here to decompose the net into the 

 repetitions of the two triangular constituents by introducing a new 

 symbol still, the symbol (/^-j-1, # 2 + 0, ^3+0), obtained by 

 addition of the corresponding digits of the frame symbol and the 

 symbol of the constituent A V A 2 A 3 , the heavy round brackets 

 meaning that only the parts of the digits written in heavy type 

 are to interchange places, whilst the arbitrary integers b l satisfy 

 the condition Hb i = 0. For each system of values of the b t satis- 

 fying the condition stated the symbol represents a definite triangular 

 constituent of the set to which A l A 2 A 3 belongs; so by this symbol 



