DERIVED FROM THE REGULAR POLYTOPES. 93 



contact by the edge AB with an other polyhedron congruent to it. 

 In other words: if the coordinates x>y of the centre M of AB 

 are p, the centres of the O group are represented by the frame 



3 



[2 a i p,2a 2 p, 2 a s p] under the conditions a it a 2 , « 3 integer and Sa i even, 



l 



i. e. 2p is the period of the net. So, as the p has in the four 

 cases successively the values 1 , 3 , 1 + V 2 , 3 + V 2 we find for 

 the four net symbols under the stated conditions 



1 . . [ 2 a x + 2 , 2 « 3 + , 2 ff 3 -f !, 



2 . . [ 6 «! -j- 4 , 6 cC + 2 , 6 ff 3 + Ï, 



3 .. [2 (1 + ^2)«! + 2 + */2, 2(l + j/2)4 + */2 , 2 (1 + >/2) % + yi\ 



4 . . [2 (3 + j/2) «! + 4 + K2, 2 (3 + j/2) « 3 + 2 + |/2, 2 (3 + j/2) *, + |/2]. 



Though we pursue the study of these threedimensional nets 

 merely from a didactic point of view it is not necessary to deduce 

 from these net symbols of the O group the net symbols of the 

 two T groups. All we want is to show how the third constituents 

 CO, C, tC can be found. Therefore we give here the net symbols 

 of the two T groups in the form: 



!..+£[ 2a 1 ±1+1, 2fl a ±L+l l 2flg±l+n, 



2..±i[ 3(2a 1 ±l) + 8, 3(2«3 + l) + l, 8.(2«8±1V+1]", 



3.. +i[(2« 1 ±l)(l + K2) + l ; (2^±l)(l + i/2) + l 5 (2^3 + l)(l + |/2) + l], 

 4 . . ± i[(8«i ± l)(3+J/2) + 3,(2a 2 ± l)(3 + j/2)+l,(2« 3 ± 1)(3 + K2)+1], 



where the double sign refers to the two groups + HM 3 and the 

 conditions about the a L and their sum remain the same. 



As the polyhedra of the O group remain in contact by faces 

 with those of the two T groups and by edges with each other we 

 have only to look out for new polyhedra filling vertex gaps which 

 make their appearance in the second, third and fourth cases on 

 account of the truncation of the polyhedra of the O group at the 

 vertices. Though all the vertices of these new constituents are con- 

 tained in the net, the second and the fourth cases show that it 

 may happen that some of the faces of these new bodies have to 

 be furnished by the polyhedra of the T groups. At any rate we 

 have to determine the new constituent by starting from an octahedron 

 vertex and deducing from the net symbol the vertices at minimum 

 distance from that point. 



We treat further each of the four cases by itself. 



Case (O, T). — In this case there is no third constituent. Never- 

 theless we deduce from the net symbol of group O given above 

 that the vertices of all the CO represented by [2 a i + 2, 2 a 2 + 2, 2 a~ 3 + 0] , 



3 



2 a t odd, are vertices of the net. But these CO are no constituents 



l 



of the net; for the centre of the CO corresponding to any set of 



