DERIVED EKOM THE EEGULAE POLYTOPES. 25 



x v = 3 containing a limiting CO, 



x v -j- x. 2 z = o ,, „ „ 1 3, 



<V\ "I x -l \ <%3 === O 55 55 55 ^G) 



are respectively parallel to the spaces 



œ ± = 1 containing a vertex, 



x \ ~\~ x -i = 1 „ an edge, 



«^ -f- a? 2 -j- œ è = 1 ,, a face, 



cT-i -f- ^2 ~h t2 a ~h ^4 = 1 » » limiting body 



of the regular simplex, while they are normal to the line joining 

 the centre O of the simplex to the centre of that limiting element. 

 Moreover it is evident that all the spaces of the same group, say 

 x k -j- x t -f- œ m = 6, have the same distance from the corresponding- 

 spaces x k -j- x L ~\- x m = 1 , etc. 



16. The meaning of the expression "extension number" is clear 

 by itself: an extension to an amount £ transforms the simplex 

 jS {1) (n -j- 1) with edge unity of S n into a simplex S( s )(n-\- I) of 

 edge £ . But we have to define beforehand what we will understand 

 e. g. by a truncation A. If we split up the n -f- 1 vertices of the 

 extended simplex 8( s )(n -f- 1) into two groups of k-\- 1 and n — k 

 points (see fig. 2, where the case n = 6 , k== 3 is represented), 

 forming the vertices of regular simplexes S( e )(k-\- 1), S( e )(n- - k) 

 lying in spaces 8 ki S n _ k _ ±i and we cut S( e )(n-\- 1) by any space S n _ i 

 at the same time parallel to these spaces S k , S n _,._ l , i. e. normal 

 to the line joining the centres M , J\l' of S( e ) (k-\~ 1), S( s )(n — k) in a 

 certain point O , any edge PQ joining a vertex P of Si s )[k-\-\) to 

 a vertex Q of S( € )(n — k) will be cut in a certain point R for which 



the ratio — — is equal to -——-, and therefore independent from the 

 PQ A MM l 



choice of the vertices P, Q. This ratio is the "truncation fraction" of 



jS( s )(u~\-\) at the limiting jS( s ) (k-\- 1 ) by the truncating space and its 



07? 



complement -—. - to unity is the "truncation fraction" of S( e )(n^\--I) 



hit 



at the limiting j$( e )(n — k) opposite to JS( e )(k-\-l) by the same space. 

 But, if we like, we can use the term "truncation number" for 

 the number of units contained in the segment PR or QR, accor- 

 ding to the truncation being performed at the side of P or of Q. As 

 the number of units of the denominator of the truncation fraction, 



