44 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



Of these seven polyhedra concur, on account of the reasons given 

 above, in the pattern vertex in the indicated order: 



\tl\ 4P 6 , 2 P 3 , 2 CO, 



2P 4 , 2P 4 , ICO. 



So we find: 



(tT 3 CO 4P 6 4P 4 , 2 p; 



= SO tT + 240 CO -f 320 P 6 -f 480 P 4 -f 320 P 3 , 



i. e. 1440 limiting polyhedra. 



Finally the limiting polytopes split up into four groups: 



(32110), (321)[10], (82) [110], [2110] 



and so we find: 



32 ^#(5), 80 (6; 4), 40 P c0 , 10c^e 3 C 8 , 



i.e. 162 limiting polyhedra. 



So the result is (960, 3360, 3680, 1440, 162) in accordance 

 with the law of Euler. 



With respect to the import we have still to add that we pass to 

 the complementary import, if a polytope of the measure polytope 

 family is regarded as a polytope of cross polytope descent. So in 

 the first of the two examples where the cross polytope import has 

 been indicated the result is complementary to that registered in 

 Table IV read from left to right. 



C. Extension number and truncation integers and fractions. 



74. Theorem XLV. "The new polytopes, all with edges of length 

 unity, can be found by means of a regular extension of the cross polytope 

 followed by a regular truncation, either at the vertices alone, or at the 

 vertices and the edges, or at the vertices, edges and faces, etc.' ; 



Por the proof we refer to the art 8 . 15 and 56. 



Mere the limit (l) n _ i of the highest import, i. e. g n _ it corresponds 

 to the equation œ ± -f- œ 2 -\- . . . -\- œ n = constant. So the extension 

 number is the sum of the digits of the new polytope divided by 

 the sum of the digits of the cross polytope, i. e. by \/2. So the 

 extension number of [3'3'2'ri] is 5 -f 9 V2 divided by V2 y 



5 



i. e. 9 + -Y/2. 



Ai 



We can stick here to the method of measuring the amount of 

 the different truncations on the edges. But we must point out a 



