26 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



i. e. the number of units of the edge of the extended simplex, is 

 the extension number £, the truncation numbers r, which — ■ as 

 S itself — will prove to be always integer, are simply the nume- 

 rators of the truncation fractions with the extension number £ as 

 common denominator. 



17. If we indicate the truncation numbers corresponding succes- 

 sively to a truncation at a vertex, an edge, a face,... by 

 To , T\, T 2 , . • • , the theorem holds : 



Theorem V. "Let (m , m lt m 2 , . . ., 0) be the zero symbol of the 

 polytope ; then the sum m = Hm, of the digits is the extension 

 number e s and the truncation numbers r , T i3 T 2 , . . . are repre- 

 sented by the forms 



T = m — m Q , Ti = m — m — m i9 r 2 = w — m — n^ — m %9 . . . ' 



Proof. By the extension £ the simplex of coordinates 8 (i) (n -j- 1) = 



n 



(1 00 : . . 0) of 8 n is changed into the concentric simplex 8( £ )(n-\- 1) 



n 



(f 00. . .0) with edges £. Then the space # n _ 4 represented in the 

 latter case by œ ± = contains a limit (/) H _i of the considered 

 polytope (P) n forming a part of the limiting simplex jS( £ ) (n) = 



n — 1 n 



{£ 00. . .0) of (s 00. . .0), at which limit of the highest order of 



dimensions this S( e )(n-\-l) is not sliced off. If now we go back 



to true coordinate values the last digits in the two symbols of 



(P) n and /S( £ ) (n -f- 1) must still be the same, which will be the 



case, if we have to subtract from nought in both cases the same 



.„ 2% — 1 , £ — 1 1 • T 1 



amount, l. e. it — — — and — — - are equal, l. e. it we have 



n -\- 1 n-f-l 



£ = 2/y/i = in. 



From what we remarked in the preceding article it follows that 



the truncation fraction of the extended simplex S 0i,) (n -j- 1) at any 



limiting # (m) (/£-|-- 1) can be derived from the mutual position of three 



parallel spaces 8 n _ i normal to the line joining the centre of S (m) (&-\- 1) 



to the centre of the opposite S (>n) (n — k)\ of these three spaces one 



passes through 8^ m) {k -f- 1), an other through the opposite S ' (n — fr), 



whilst the third is the truncating space lying between these two. 



For, if as in the preceding article P is any vertex of 8 {m) (k~\~ 1), 



Q any vertex of 8 im) (n — k) and B the point of intersection 



of PQ with the third of these three parallel spaces, which may 



be represented by 8 (i) n _ iy 8 (2) n _ i} # (3) n _ l9 according to definition 



