4 S ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



As a poly tope with a zero symbol with h — ] zeros cannot be 

 a repetition of a polytope with a zero symbol with /; zeros, the 

 first process does not suit our aim; but the second may do so 

 under the conditions 



r — 1 =5 r l»fi f l l=5 r 2»j r 2— ] =S r 3»-»j r n-2— 1— ^n-l.^n-l— 1 = 1. 



i. e. in the case of the central polytope (r — 1, r — 2, . . .,2,1,0), 

 i. e. if we have in S n the case (n,n — 1 , . . .,2,1,0) with r = n -\~ 1. 

 It is indeed easy to prove that the particular case of the reap- 

 pearance of the original constituent presents itself, w r hen and only 

 when we have r = n -j- 1 and q i = n. For, according to the law 

 of theorem I, q^ = n exacts that the zero symbol contains no two 

 equal digits and under this circumstance the substitution of n -\- 1 

 for zero followed by the diminution of all the digits by unity 

 reproduces the original zero symbol. In art. 30 (page 57 at the top) 

 it will be shown that the suppositions r = n -\- 1 , q^ = ?i lead to 

 the unique self space filler of 8 n . 



But now that the manner in which we have to account for the 

 existing pure simplex nets is secured we have to revise our notion 

 of "constituent of the same kind", if we will keep the analytical 

 theory developed just now in touch with the geometrical facts. 

 According to that theory w T e are obliged to say that the plane 

 net JV(p 6 ) contains three different groups of hexagons, though 

 geometrically all the hexagons are equipollent to each other and 

 therefore of the same hind. For in the case n = 2 the suppositions 

 r = n-\-l, q x = n give rise to the net with the decomposing 

 symbol (3^ -j- 2, 3 b 2 -\- 1 , 3 b 3 -\- 0), 2^ = 0, corresponding 

 (fig. 10) to the set of hexagons a with a heavy lined circuit, 

 whilst the net contains two other groups of hexagons which 

 admit alternately thick and thin sides, one group b where the 

 horizontal thick side is below, an other group c where the hori- 

 zontal thick side is above. So, though we keep saying that the 

 hexagon is a self plane filler, we will consider N(p 6 ) from an 

 analytical point of view as admitting three different groups of 

 hexagons, using here henceforward the more precise term of "group 

 of constituents" in order to indicate a "set of equipollent poly- 

 tópes, the vertices of which form together all the vertices of the 

 net, each vertex taken once". Only under this extension of our 

 former hind of constituents by our now introduced group of con- 

 stituents the theorems XI and XII are generally true. If we follow 

 the interpretation of the net N{tO) as a net with one kind of 



