44 A NALYTICAL TREATMENT OF THE POLYTOPE3 REGULARLY 



At first sight it may seem that the introduction of the common 

 factor r, by means of which the frame is only enlarged but not 

 changed in form, is of no avail, as the scale of the diagrams is of 

 no importance whatever. But then one overlooks the fact that the, 

 frame is enlarged, while the central polytope (P)° remains unaltered. 

 So in the case of the central triangle A x A 2 A s in the plane : if 

 we take r= 1 we have to deal with the net N(p 3 ) of fig. 5, 

 whilst the supposition r=2 gives the vertices of the net N(p^,p Q ) 

 by means of the triangle A x A 2 A 3 and its equally orientated 

 repetitions (fig. 9). 



This simple example shows in the first place the influence of 

 the period r. But on the other hand it gives a glimpse of the 

 fact that with a given central polytope not all integer values of r 

 lead to existing nets. So the supposition r == 3 brings already the 

 central triangle A x A 2 A 3 and its repetitions too far apart. x ) 



26. We pursue our investigation in the direction of the last 

 sentence of the preceding article, entering into details about the 

 relationship between the period r and the largest digit q x of the 

 zero symbol (q it q 2 , . . ., r/ n , 0) of the central polytope (P)°. 



If we call any repetition (P) of the central polytope {P)° corres- 

 ponding with it in orientation adjacent to it, if the distance 

 between their centres C, and C is as small as possible 2 ), i. e. if 

 the coordinates of C can be deduced from the equal coordinates 



x t = ~y~ of C by altering only one pair of coordinates by addi- 

 tion and subtraction of only one time r, we find: 



"The central polytope and one of its adjacent repetitions overlap 

 for r <C 0[, whilst for r = q x they are in contact and for r > q x 

 free from each other". 



Of these three cases of relationship between r and q x we consider 

 first the case r = q u then the two cases r ^ q x at a time. 



Case r = q x . The two adjacent polytopes represented by 



(</i,</2,^,- • •,</„,<)), (r + q lf — r-\-q 2> q 3t . . .,</ u ,0) 



have all the vertices 



*) Application of the case r = 3 to the triangle A X A 2 A 3 gives one of the two sets 

 of triangles of the net of fig. 8, already discarded for two different reasons. 



a ) This is the case if the image F of C (compare the preceding article under b) lies 

 on an axis OÀ"; at distance r from 0. 



