90 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



E. Nets of poly topes. 



101. Let us consider the net N(M n 2 ) and suppose that it is 

 composed of alternate white and black M n (2) , so that any two Mj^ 

 with a common limiting M%% differ in colour. Let us imagine 

 that each white iJ/ n (2) is split up into an inscribed positive 



n 



JIM n (= -]- A [11 . . 1]) and 2 n_1 pyramids on regular sim- 

 plexes S{?i)(%V%) the vertex edges of which have a length 2 and 

 meet at right angles, and that in the same way each black M n is 



split up into an inscribed negative HM n {= — ^[ll..l]) and 

 2" _1 pyramids. Then it is clear that a space filling of 8 n is formed 

 by three groups of polytopes, two groups of HM n , i. e. a group 

 of positive ones and a group of negative ones, and one group of 



n — 1 



cross polytopes [200 . . 0], each of which has for centre a vertex 

 of the net N(M 2 ) not belonging to an HM n and is generated by 

 the addition of 2 n of the equal pyramids. This net, which may be 

 represented by the symbol -ZV(+ HM n , Cr n ), forms our starting 

 point here. It is our aim to deduce from this simple net several 

 other ones the constituents of which are forms derived from the 

 regular polytopes and /impd., partaking with each other of the proper- 

 ties of admitting one kind of vertices and one length of edge, by 

 considering in the application of the expansion operations either the 

 two sets of half measure polytopes as independent and the set of 

 cross polytopes as dependent variables, or reversely. 



Any HM n of the original net N(+ HM ni Cr n ) is limited by 

 HM n _ i of (l) n _ i import and by simplexes S(?i) of truncation import; 

 by each HM n _ x it is in contact with an HM n of the other kind, 

 by each 8(n) with a Cr n . We now follow two polytopes HM n , 

 Cr n in S(n) contact through any group of expansion operations 

 leading to a new net, by which operations HM n and its S{n) pass 

 into (P) M and (Q) n _ 4 and likewise Cr n and its 8(n) into (P) n and 

 (Q)'n-i- Then it is evident that (Q),^ and (Q)' M _! must coincide, 

 as the application of the operation e n with respect to the group 

 of Cr n origin on one hand and the group of HM n origin on the 

 other would lead to a net with two different kinds of vertices, 

 those of the group of Or n origin and those of the group of HM n 

 origin. This coincidence dominates the hmpd. nets, as it creates a 

 very close relation between the two chief constituents. If we denote 

 by the symbol e n HM n the separation of the two groups of HM n 

 from each other by the intercalation of prisms on their original 



