32 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



This theorem enables us to find immediately the expansion sym- 

 bol of a polytope with given zero symbol. We show this by the 

 example (5443322210) of art. 12. 



In (5443322210) five unit intervals occur, viz. if we represent 

 the p 1h digit by d p between (d i} d.,), (d z ,d fi ), (d 5 ,d 6 ), (d s , d 9 ), (d 9 ,d iQ ). 

 Of these the first corresponds to the original unit interval of the 



simplex (1 00. . .0), whilst the others are introduced by the expan- 

 sion operations e 2 , e ki e ly e s . So we find e 2 e k e 1 e s S (1.0). 



Reversely it is quite as easy to find back the zero symbol of 

 e. 1 e fl e 1 e s S (10). As there are to be four unit intervals more than 

 the original one the zero symbol begins by 5 and 4, and has to 

 show a unit interval behind the third, the fifth, the eighth, the 

 ninth digit, etc. 



20. It is obvious that the system of expansion operations cannot 

 lead to a zero symbol with two or more equal largest digits. So 

 the system of the expansion forms is not complete as to the total 

 number of possible forms. But the scope of this incompleteness is 

 not so large as we might think at first. For, if the zero symbol 

 winds up in two or more zeros, the inversion indicated in art. 3 

 will bring about a new zero symbol with more than one largest 

 digit. Nevertheless, after this extension of the system of expansion 

 forms, still the forms with a zero symbol containing two or more 

 largest digits and two or more zeros are lacking. 



So it was desirable to have at hand a new geometrical operation 

 leading to forms with a zero symbol containing more than one 

 largest digit. This now is given us by M rs . Stott in the operation 

 of contraction ; but before we show this we may devote a single 

 word to the introduction of different kinds of contraction. 



The subject of the operation c of contraction of an expansion 

 form in S n is always a group of limiting elements of the same 

 import and of the highest order of dimensions available ; so we 

 designate the contraction c as a c , a c if a c 2 , etc. according to the 

 subject elements being of vertex import, of edge import, of face 

 import, etc. Moreover these limits of the same import can be sub- 

 ject of contraction, when and only when all their vertices form 

 together exactly all the vertices of the expansion form, each vertex 

 taken once; in this case any two of these limits are still separated 

 from each other by the distance of an edge at least and now the 

 operation of contraction consists merely in this that all these limits 

 undergo a parallel displacement, of the same amount, towards the 



