74 AX ALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



opposite to the simplex replacing the opposite vertex of the measure 

 poly tope. So in this respect HM 2n presents analogy to measure 

 polytope and cross polytope, whilst HM 2n+i imitates the simplex. 

 We still remark that the case n = 2 is exceptional in this sense 

 that the corresponding HM 2 is a line, i. e. a diagonal of the square, 

 instead of a form of two dimensions; as we shall see this remark 

 is essential in the theory of the nets derived from the half measure 

 polytopes. 



91. It is easy to prove that the half measure polytopes partake 

 of the two properties characterizing the semiregular polytopes con- 

 sidered in the preceding sections, i. e. that all the vertices are of 

 the same kind and all the edges of the same length, here 2 '\/2. 

 Indeed we already solved incidentally in art. 47 the more general 

 question : 



"Under what circumstances will the symbols 



represent the vertices of polytopes in JS ni all the edges of which 



have the same length, say 2\/2"? 



For the length 2V 7 2 of the edges the solution takes the form of 

 Theorem LVI. "The symbol + £ [ a i> a 2> - • •> fl n] f° r which 



represents a polytope admitting the required properties under the 

 conditions: a n _^ i = a n = 1 and the difference between any two unequal 

 adjacent digits equal to 2". 

 So we find 



in £3 the two forms f [111], £[311], 



» fl* „ four „ £[1111], £[3311], 4 [3111], £[5311], 

 ,. #5 ,, eight „ i[lllll],i[33311],i[331il],i[31111] 5 



£[553 11], £[53311], i[53111], £[75311], 



etc., which are represented in the following table by other sym- 

 bols referring to T, C ±6 and HM\ these symbols will be explained 

 later on. x ) 



n = 3 

 £ [1 1 1] = T = ILV 3 , £[311] = tT = e 2 HM,. 



n = 4 

 f [1111]= C i6 = RMA £[3111]= ce 2 C i6 = e 6 KM k 

 £ [3311] = e, C i6 = e 2 EM^ £[5311] = ce, e 2 C iQ = e 2 e 3 HM k 



l ) We remark here that the symbols e before HM n are related to the limits of M w 



