64 



ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



I 

 II 



III 



IV 



V 



VI 



VII 



I 



e 



"! 



1 



Ill 



2 



j 



1 



IV 



2 



( 



12 



V 



12 







Ver- 



mstituents 



tex 



e i 



H 



gap 







1 



e 2 



2 



— 



*1 



— 







e i 



02 



1 



e e 2 







01 



e i e 2 



2 







e e i 



02 



01 



— 



Constituents 



T 







— T 



T 



tT 



— tT 



tT 



— tT 



— T 



CO 







CO 



tT 



to 



— tT 



CO 



tT 



to 



to 



— tT 



CO 



to 



to 



to 



Ver- 

 tex 



gap 



— T 

 T 

 O 

 CO 



— tT 

 tT 



According to this table the principal net N{1\ 0) can be transformed 

 into the net 3 7/ - either by applying to O and — 7 7 the operations e and 

 e i or by applying to T and O the operations e ± and e 2 ; these two trans- 

 formations are of the same kind; as they pass into each other by inter- 

 changing the two sets of tetrahedra and at the same time the two sets of 

 four non adjacent faces of each octahedron in contact with them. Whilst 

 each of the two nets S JU and 3 V can be deduced in one way only, 

 there are three manners of deduction of net 3 IV ; of these the first 

 stands by itself and the second and the third pass into each other 

 by the indicated interchange of the two sets of tetrahedra, etc. 



The table for # 4 is the following 



Ver- 



Constituents 



1 



2 



3 



1 



3 



12 



2 



13 



23 



12 



13 



23 



123 



123 







2 

 3 



02 







23 



03 



2 



3 



023 



02 



03 



23 



023 



e 2 

 1 

 3 





 13 

 01 



3 



1 

 03 







13 



013 



01 



03 



013 



2 







1 



2 



12 







02 



1 



01 



02 



12 



012 



01 



012 



tex 

 gap 



e-s 



e 

 e l 



e 2 



^e 3 



e e x 



e \ e 3 



e o e 2 

 e±e 2 



e e i e 3 



e e 2 c 3 



e \ e 2 e B 



Cq c x g 2 



Constituents 



e o e 2 

 e e \ 



e e^ e 2 



^o ^2 



e e \ e 3 



e e 1 e 3 

 e e*_ e 2 



e e i e 3 



e e i e 3 



e o e i&2 e s 

 e ^i c 2 e 3 



e i &2 



~~ e e 2 



e i 



Cq 6^ 6 2 



e e 1 



e e ± e 3 

 e i e 2 



e e^e 2 e 3 



6q 6^ 6 2 



e e x e 3 

 e§e^e 2 



e e l e 2 e 3 



e 2 



6^ 6 2 



— e e ± 

 — e x 



e e 2 



e e i e 2 



6q 6\ 6 2 



e e x 



e i e 2 

 e o e \ e s 



e e 2 



— 6q G\ 6 2 



e e x e 2 e 3 



&0 e \ e 2 



e e i e 2 e 3 



e. 



e e ± 



— e 



e e 3 



€q ^2 



— e e i 

 e e i e 2 



e o e 3 

 -e§e^e 3 



e o e 2 

 e e \ e 3 



— e Q e x e 3 



e e i e 2 



e e \ e 2 e 3 



e e x e 3 



e e \ e 2 e 3 



Ver- 

 tex 



gap 



— e Q 



e x 



— e x 

 e Q e 3 



-e e i 

 e e i 



e Q e 2 



e e \ e 3 

 -e e i e 3 



' ^0 ^1 ^2 

 &0 6\ ^2 



