42 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



b) In the second place we prove \hs\i' each net admits a frame 

 symbol, and that the frames of all possible nets are similar. 



If a rectilinear transnational motion of the net over a distance d 

 brings any polytope (P) of it into coincidence with its repetition (P) (1) 

 and therefore the net with itself, a rectilinear translation al motion 

 of the net over a p times larger distance pd in the same direction, 

 p being any integer, will bring (P) into coincidence with an other of 

 its repetitions (7 J ) (/0 and therefore also the net with itself. This is 

 self evident if we consider the motion over pd as the result of p 

 motions d in the same direction executed one after another. In other 

 terms: the frame of any set of constituents of any simplex net proper 

 must be characterized by the property of containing the point C (p) 

 determined by the vector equation CC (p) =p . (7C (4) as soon as it con- 

 tains the centres C and C (1) and/; is any integer. Now let d it d 2 , . . . , 

 d n +i w ith the condition 2^ = represent the frame coordinates of C 

 with respect to any centre C of the frame, and let us consider 

 d { , do, . . ., d n , — i. e. all these integers, d n + i alone excepted — , as 

 the rectangular coordinates of a point flying in an other space jS' u bea- 

 ring the system of coordinates O (X i , X 2 , . . . ,X n ). Then to each point C, 

 C (, \ C^ofthe frame correspond points V, V iX \ V {p) of S' n and the vector 

 equation CC (p) =p. CV7 (1) includes the vector equation YV^ v>i: =p. W y] \ 

 i. e. there is a correspondence one to one between the centres C of the 

 frame and the images Vm S' n , the points V in S' n being characterized 

 by the property of having integer coordinates x ± , œ 2 ,. . . ,a? n . But all 

 the points V with integer coordinates form evidently the vertices of a 

 net of measure polytopes with edge unity; so the system of images /^is 

 either the total system of vertices of this net of measure polytopes or a 

 portion of it, containing always the origin O corresponding to the centre 

 C and partaking of the geometrical property of containing the point 

 F { ' n , determined by the vector equation W Cp) =p. VF' if it contains 

 V, V' and p is integer. In this form it is immediately evident — 

 in connection with the equivalence of the different coordinates — 

 that the portion can only be a system of vertices, the coordinates of 

 which are integers admitting a common factor r, i. e. the set of 

 vertices of a net of measure polytopes with edge r. So Ave have shown 

 now that the system of points V of JS' n must be {rb i ,rb. 1 ,. . .,rb n ), 

 where the n quantities b t are arbitrary integers, whilst r is a definite 

 integer. From this result it follows immediately that the frame of 

 the centres C admits the frame symbol 



(rb l3 rb 2 , rb s , . . .,rb n , rb n + i ), F) 



the arbitrary integers b t satisfying the condition Hb t = 0. The 



