DERIVED FROM THE REGULAR POLYTOPES. 35 



/• _^_ i digits of the zero symbol of the given form. Of this general 

 result theorem X considers the special case k zero. 



Now if we apply the contraction c = c to the simplex of coor- 



n 



dinates (100...0) itself we find the point with the zero symbol 



n + 1 



(000. . .0) i. e. the centre 0. This result is geometrically evident: 

 if we brins: the vertices nearer to the centre so as to annihilate 

 the separating edges the result is a single point. In this point of 

 view the inverse operation e can be considered as corresponding 

 to the generation of the simplex starting from a point. 



Remark. By introducing the operation e the contraction symbol 

 c can be shunted out. So, if jS ([) (n -]- 1) represents the point 

 which is to become the /S (l) (n 4- 1) by applying the operation e , 

 we can replace c e h e t /S' (1) (n -\- 1 ) by e k e t /% (1) (n -{- I), but this 

 implies that we write e e k e l /S (1) (n -\- 1) f° r e k e i^ w { n ~h !)• 



This new notation will prove to be preferable in the case of 

 the nets (see under E the art. 30 at the end of page 57). 



E. Nets of poly topes. 



22. As to recent literature about space fillings or nets we may 

 mention À. Andreim's "Sulle reti di poliedri regolari e semiregolari 

 e sulle corrispondenti reti correlative" (Roma, 1905), two papers 

 of mine ("Fourdimensional nets and their sections by spaces" and 

 "The sections of the net of measure polytopes M n of space Sp u 

 with a space Sp n _ i normal to a diagonal", Proceedings of Amsterdam, 

 vol. X, pp. 536, 688) and the memoir of M rs . Stott quoted 

 several times. 



We exclude what may be called a prismatic net, i. e. a net 

 in S n obtained by prismatizing a net of S n _ i in a new direction, 

 and divide the remaining uniform nets derived from the simplex 

 into two groups x ): 1°. pure nets with only one (central symmetric) 

 constituent and 2°. mixed nets either with one non central sym- 

 metric constituent in two opposite positions or with constituents 

 of different kind. If we restrict ourselves to the plane the first 

 group consists of the hexagon net only, while the second is represented 

 e. g. by the triangle net and the net of hexagons and triangles; 

 if we proceed to ordinary space the first group contains the tO 

 net only, Avhile the second is represented e. g. by the net of T 

 and O. 



l ) This division — of no fundamental importance in itself — is introduced here, 

 merely in order to smooth the way leading to the analytical representation of the nets. 



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