DERIVED FROM THE REGULAR POLYTOPES. 



31 



Example I. The two nets with 

 form admit the constituents : 



ff i0 ..[ö'^'s'a'Vyyri] 



[5' 4 4 3' 3' 2' 2' 2' 1' l] as ground- 



's 

 9i 



9* 



'5 

 <?4 



'a 



'2 



'l 



[4'3'3'2'2'2Tl],[10]Y/2 

 [3 / 3 , 2'2 , 2 , l , l],[110]V / 2 

 [3'2'2'2'1'1],[2 1 10]\/2 



2'2'2T1 

 2'2Tl], 



[2'ri] 



[1],[4 



,[221 10]V2 

 3 2 2 11 0]V/2 



,[33221 10]\/2 

 3 3 3 2 2 11 0]V2 

 3 3 3 2 2 1] 0l\/2 



...[5 433322110; 



N/2 



'io 



'9 

 '8 

 '7 

 '6 

 '5 

 '4 

 '3 



9* 



"s'^'s's'^'a'ri] 



'4' 4' 3' 3' 2' 2' 2' ri] , [î] 



[4' 8' 3' 2' 2' 2' ri] , [ri] 



[3' 3' 2' 2' 2' ri] , [rri] 



[V 2' 2' 2' v 1] , [2'rri] 



= 2'2'2'ri] , [2 , 2 / r ri] 

 V2'ri] , [s' 2' 2' r ri] 



[2 ; ri] , [3' 3' 2' 2' r ri] 



Tl] , [8' 8' 8' 2' 2' 1' 1' 1] 



= i] , [4' 3' 3' s' 2' 2' r r 1] 



. .. [5' 4' 8' 8' 8' 2' 2' l' l'I] 



Example IL The two nets with [5 443322210] V2 as ground- 

 form admit the constituents : 



y 10 . . [54433222 10] V2 



's 



Si 



9e 



'5 

 '4 

 '3 

 fhl 



90 



[43322210^2 , [1 

 [3322210]V2 , [11 

 [322210]V2 , [211 



[22210]V / 2 , [2211 

 [2210]\/2 , [32211 

 [210]\/2 , [332211 

 [10]V2 , [3332211 



[543332211 







'10 • • 





'9- • • 



0]N/2 



'8- • • 



0] 



V2 



'7- • • 



o z 



^2 



>e- ■ • 



0] 



V/2 



'5- • • 



o] 



V2 



j/4- • • 



0]V2 



'3- • • 



0]V/2 



92" • 



o] 



V2 



#>• ' ■ 



..[5 4433222 10]\/2 

 [1],[4 4 3 3 2 2 2 10]V2 



[l'I]. 



rn 



4 3 3 2 2 2 l()]N/2 



,[33222 10]V/2 



2 , 1 / 1'1] , [3 2 2 2 10]\/2 

 '2' 2' l' l'I], [2 2 2 10] V2 

 = 3'2'2Tl'l],[2 2 10]V2 

 '3' 8' 2' 2' r l'I], [2 10]V/^ 

 [3'3'3'2'2'lTl],[10]\/2 



[5'4'3'3'3'2'2Tri] 



The nets of measure polytope extraction of the spaces JS> S , # 4 , S 5 

 are put on record in the Tables V and VI. The first column of 

 these tables is concerned with the "name" of the net; it contains 

 the system of operators e k and c which are to precede the general 

 symbot N(M n 2 ) in order to obtain the symbol of the net. This 

 system of operators is in close connection with the consideration of 

 the net of S n as a simple polytope of S n + i ; for a = a i it is equal 

 to the system of operators characterizing the groundform, for 

 a — a l -\~~ 1 it consists of latter system completed by e n . So of the 

 three parts into which each of the three cases n=3, ^ = 4, /* — 5 

 has been subdivided, the first contains the nets (e, c), the second the 

 nets (e, e), the third the nets (c, c). Therefore the question rises where 

 the nets (c, e) are to be found. 



The algorithm indicated in our last theorem immediately shows 



