DERIVED FllOM THE REGULAR POLYTOPES. 41 



values only. We show that this property too is a general property 

 i. e. that all the different sets of constituents of any simplex net 

 proper admit the same frame symbol with integer coordinates. 



In discussing the number of the regular polyhedra in ordinary 

 space the plane nets N(p 3 ), N(p^), N(p 6 ) appear as polyhedra with 

 an infinite number of faces, unyielding as to this that their faces 

 remain in the same plane instead of bending round in three 

 dimensions. Of these regular polyhedra with an infinite number of 

 faces the centre is at infinity in the common direction of the 

 normals to their plane in the space of three dimensions which is 

 supposed to contain them and the anallagmatic *) rotations and 

 reflections of the regular polyhedra proper pass into translations and 

 reflections in the case of N(p 3 ), N(p^, N(p Q ). In the same way 

 each net of S n may be considered as an n -f- 1 -dimensional poly- 

 tope with an infinite number of limits (l) n which instead of ben- 

 ding round in # n + 1 fill a space 8 n . On account of this each net 

 must be transformable in itself by a translation al motion which 

 brings a constituent polytope of the net into coincidence with 

 any repetition of that constituent in the same orientation. By means 

 of this property we prove now the following general theorem: 



Theorem XL "Any possible simplex net proper admits a net 

 symbol and for all the different sets of constituents the same frame 

 symbol. Moreover the frames of all the possible nets of 8 n are 

 similar to each other." 



a) We show first that all the different frames of a net are 

 equipollent. 



Let (F) and (Q) with the centres C p and C q be any two polytopes 

 of different kinds of a simplex net proper having at least one 

 vertex V in common. Let (P') be any polytope of this net equipollent 

 to (P) and let (Q')> V' , C ' p , C' q represent the new positions of (Q), V, 

 C p , C q after a translational motion which brings (P) into coincidence 

 with (P) and therefore the net with itself. Then (Q ; ), V, C' p , C' q are 

 respectively a polytope of the net equipollent to (Q), a vertex of the 

 net homologous to V, the centre of (P'), the centre of (Q'). From 

 this we derive the equipollency of the three lines VV' , C p C' p> 

 C q C q , i.e. C P C' P and C q C' q are mutually equipollent as they are 

 both equipollent to VV'. So all the different frames of a net are 

 equipollent, i. e. each of these frames can be brought into coin- 

 cidence with any other of them by means of a translational motion. 



*) These rotations and reflections which transform a polytope in itself will be studied 

 in part G. 



