38 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



the net N(p à ) is decomposed into the different constituents of 

 the set of triangle A ± A 2 A 3 . In the same way the symbol 

 (Jj { -f~ 1 , b. 1 -\~\, b ó -\~ 0) characterizes the other set of triangles 

 under the condition 2 b t = — I. 1 ) 



24. In the first place we remark that the net of triangles admits 

 a net symbol with only integer digits and we examine now to 

 what extent this property is a general one. 



If we choose for simplex of coordinates jS (1) (n ~\- 1) a simplex 

 with respect to which a definite poly tope of the net — let us call 

 it the central polytope (P)° of the net — can be represented by 

 its zero symbol and we restrict ourselves to the cases in which 

 the constituents of the net are exclusively forms derived from the 

 simplex, we can easily prove that the coordinates of all the vertices 

 of the net must be integer. To that end we call any polytope of 

 the net "orientated'* with respect to the simplex of coordinates, if 

 a translational motion of the polytope which brings its centre into 

 coincidence with the centre of that simplex gives it a position in 

 which it is represented by its zero symbol or by the reverse ; this 

 definition enables us to state the following lemma: 



"If two polytopes of the net in S n have a limiting n — 1- 

 dimensional polytope in common, they are both orientated with 

 respect to the simplex of coordinates, as soon as this is the case 

 with one of them". 



We remark — in order to prove this lemma first — that, if 

 two polytopes derived from /# (1) (u -\~ 1) have a limiting n — 1- 

 dimensional polytope in common, this limit has either with respect 

 to both the same import or its import with respect to the one is 

 complementary to that with respect to the other. For, according to 

 the last two lines of art. 13, any two limits (l) n _ ± , represente das 

 to their imports by g k and g k ., are prismotopes (P k ; P n _ k _ i ) and 

 (P k >; P n _ /C '_ 1 ), and these prismotopes cannot coincide, unless we have 

 either /•' = l- or Jc =n — k — 1 . 



This remark leads to a proof of the lemma in the following way. 

 Let (P) a n and (P)][ be the two given polytopes and (P) n _ 4 their 

 common n — 1 -dimensional limit lying in the space S%\. Let the 

 S (i) (n -\- 1), from which (P)'' t can be derived by means of the operations 

 e : . and c, be our simplex of coordinates; then Pty is not only 

 orientated with respect to that simplex but also concentric with it. 



l ) As soon as the idea of splitting up the digits of the symbol into two parts, an 

 immovable- part and a permutable one, had presented itself, the analytical deduction of 

 the nets of simplex extraction was within grasp. 



