94 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



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integers a t satisfying the condition X a t odd is the point 2^,2 a.>, 2 a 3 , 



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and for S a i odd this centre itself is a vertex of the net, i. e. 

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these CO overlap. 



Case {tO, tT). — As we have p = 3 the point 2, 0, 0, originally 

 common to the central O and an other in vertex contact with it, 

 is carried away from the origin to thrice the distance and arrives 

 at 6, 0, 0. So with respect to this centre of a new constituent as new 

 origin the original net symbol becomesfO^ — l)+4, 6« 2 +2, 6« 3 +0], 

 2 a L even, i. e. [6 a { + 4, 6 a 2 + 2, 6 a d + 0], 2 a t odd. Now the 

 supposition a A = — 1 , a 2 = a z = gives the square — 2 [2, 0] and 

 so the six suppositions a it a 2 , a 3 = [10 0] give the six squares of 

 the [2, 2, 0], i. e. of the CO. The eight triangles of this CO are fur- 

 nished by tl , four of each group. So by putting a x = 0, a 2 ===== — 1 , 

 # 3 = — - 1 in the net symbol of the group of positive tT we get 

 2 [3 — (— 3 , - — 3 — (— 1 , — 3 -\- 1], i. e. reduced to the new origin 

 6, 0, the symmetrical form | [-3 + 3, -3-f 1,-3 + 1], 

 the triangle (0, — 2, — 2) of which is a face of the CO found 

 above. 



Case (ECO, T). — Here we have p = 1 + \/2 and the centre 

 of the new constituent becomes 2 (1 + \/2), 0,0. So the net 

 symbol with respect to that new origin is 



[2(l + j/2)ff 1 +2 + K2, 2(l + |/2) « 3 + |/2 5 2(1+^2)^ + ^/2], S fl x odd. 



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Here the six suppositions a x> a 2 , a s = [100] give the six 

 limiting squares of the cube [\/2, \/2, \/2]. 



Case (WO, tT). - Rerep = 3 + V% and therefore 2(3 + V2), 0, 

 is the new origin, leading to the new form 



[2^3+^2)^+4 + ^/2, 2(3 + |/2) * 3 + 2 + J/2, 2 (3+J/3) ^ + ^2], 2 a t odd 



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of the net symbol. Here the same suppositions give the six limiting 

 octagons of the W represented by [2 + V2, 2 + V2, V2]. By 

 putting a x = , a 2 = — 1 , # 3 = — 1 in the net symbol of 

 the group of positive tT we get here 



J [8 + V2 + 3, - (3 + V2) + 1, - (3 + Vi) + 1], 

 or with respect to the new origin 



i [_ (3 + v/2) + 3, — (3 + V/2) + 1, — (3 + V/2) + 1], 

 the triangle (— V2, — 2— V2.,— -2 — V 7 2) of which is a face of the /6'. 



Remark. The /> introduced above is not to be confounded with 

 the extension number of the octahedron group which according 



