DERIVED EROM THE REGULAR POLYTOPES. 



79 



under the headings (/) , (l\, . . • ,(/) 5 ; so the number 11 on the 

 lines of the nets 5 VI a , 5 y/ & under (7) 4 indicates that in these equal 

 nets of space S b the constituents admit together eleven differently 

 shaped limits (7) 4 . What is taken from Table III — and what 

 is self evident — is incribed in small type. The other numbers — 

 inscribed in heavy type — , of which only two correspond to faces, have 

 been found separately. We treat here two of these cases in detail. 

 Case 4 /7/ . Here the constituent (11000) has only one kind of 

 edge. Does this imply that the net has only one kind of edge? The 

 example of the net 3 IV where the CO admits also only one kind 

 of edge, whilst Andreini rightly mentions the fact (see his treatise , 

 p. 32 under n°. 21) that of the five edges concurring in a vertex 

 one is common to 2 tT and 2 tO and each of the four others to 

 tT, tO, CO, must prevent us from jumping too rashly to this con- 

 clusion. So we investigate this point and examine if, e. g. in the 

 case of the constituent (21100) with two kinds of edges, (21)100 

 and 21(10)0 these two edges are different with respect to the net 

 or not. So we enumerate first the different limits (/) 4 to which the 

 vertex 21100 is common. They are 



^2 • • 



■ (2, 



1 , 



-■- 3 



, 



) 



-e 2 . . 



■ (2, 



1 , 



-i- 3 



-2 + 2, 



) 



- e 2 . . 



(2, 



1 , 



A ? 



,- 



-2 + 2) 



ce y . » 



■ (2, 



1 , 



-*- 3 



-2 + 2,- 



-2 + 2) 



e s • • ■ 



(2,- 



-2 + 3, 



A 3 



-2 + 2,- 



-2 + 2) 



e Z - ' 



(2, 



1 ,"• 



-2 + 3,- 



-2 + 2,- 



-2 + 2) 



ce,[ . . , 



(2,- 



-2 + 3,- 



- 2 + 3 , - 



-2 + 2,- 



-2 + 2) 



é?2 . • 



(3,- 



- 2 + 3 , ■ 



-2 + 3,- 



- 2 + 2 , - 



-4 + 4) 



é?2 • . . 



(2,- 



-2 + 3,- 



-2 + 3,- 



- 4 + 4 , - 



-2 + 2) 



■e. 2 . . . 



(8,- 



-2 + 3,- 



-2+3,- 



-4+4,- 



— 4+4) 



1 



2 

 3 



4 



Starting from (2 11 00) we have indicated in this list of ten 

 polytopes first the two polytopes deduced from (21100) by varying 

 the form of one of the digits 1, 1, 0, 0, then the only poly tope 

 obtained by varying two of the digits, etc., see the curved brackets 

 and the numbers 1, 2, 3, 4 at the right. As we can augment all the 

 interchangeable parts by the same integer provided that we diminish 

 all the unmovable parts by the same amount, we find in this manner 

 all the polytopes to which the chosen vertex 21100 is common, 

 though we leave the first digit 2 alone. 



If we denote the ten polytopes of the list by (P\, (P) 2 , . . . ,(P) 10 



