DERIVED FROM THÉ REGULAR POLYTOPES. 5 1 



each black M£ 2) is truncated in the same way so as to retain a 

 — i[l, 1, 1, 1], the interstitial spaces between these two sets of 

 inclined (7 16 (2K2) can be filled up by a third set of erect G m {?V^, and 

 we obtain a fourdiinensional net formed by three equally numerous 

 groups of cells £7 16 (2j/2) with the property that all the poly topes of 

 the same group are equipollent. Moreover we can transform the net 

 iV(J/ 4 (2) ) of alternate white and black polytopes into a net of regular 

 cells 6! 24 (2) by 'decomposing each white M£® into eight mutually 

 congruent pyramids with the centre of the polytope as common 

 vertex and the eight limiting cubes of the polytope as bases, and 

 uniting each of these white pyramids to the black measure polytope 

 with which it is in contact by its base 1 ). Now what concerns us 

 here is that by treating the new regular net N(C i6 ) in the same 

 way in which the net N (M^ has been treated we find several new 

 fourdiinensional nets; for these nets the reader may compare Table II 

 of the memoir of M rs . Stott quoted several times 2 ). 



Remark. In art. 64 we have seen that with respect to measure 

 polytope nets any net (c, e) is also a net (e, c). This particularity 

 does not present itself for the nets deduced from iV((7 I6 ). So here 

 we will have to distinguish four cases 3 ), i. e. (e, c), (e, e) , (c, c) and 

 moreover (c, e). 



79. We have seen that the vertices of the net N(M£ 2) ) can be 

 represented by the symbol [2 a l -\~ 1, 2 a 2 -j- 1, 2 a 3 -\- ] , 2 # 4 ~\~ 1] 

 where the a, are arbitrary integers. By considering the pointy = 1, 

 (i = \ } 2, 3, 4), as the new origin of parallel axes and omitting the 

 square brackets we get for the coordinates of these vertices 



Z (l^ , Z CC-2 , Z ci 3 , Z d^ . 



From this we deduce that the vertices of the net iV(C l6 (2ï/2)) can 

 be represented by the same coordinate values under addition of the 



4 



condition that 2 a h has a defined character of parity. If we choose 



l 



the condition "2 a, is even" we get for the three sets of (7 16 (2J/2) 



l 



the coordinate symbols : 



*) Compare p. 242 of vol. II of my textbook "Mehrdimensionale Geometrie" or 

 Proceedings of the Academy of Amsterdam, vol. X, p. 536, 537. 



2 ) In the part of that Table concerned with the nets deduced from A T (CieHh e P T of 

 the line with the number 28 ought to find a place in the same column in the line 

 with the number 27. Moreover we can add in the last column of the line 29 that 

 this net is equal to that of line 47. 



The fact that several nets of this part are equal to nets deduced from cell C04 will 

 be explained in part F of this section. 



3 ) In (e, c), etc. the first letter is related to C^g, the second to 624. 



4* 



