4 ANALYTICAL TREATMENT OF THE POLYTOPES REGULAKLY 



This simple result, in close connection with the new deduction 

 of formula 1), shows us that we shall have to enlarge the scope 

 of our symbol of coordinates in order to find something new. 



47. We remember that the symbols [■£, ■§-, |-] and [-J- \/2, 0, 0] 

 represent the coordinates of the vertices of cube and octahedron 

 Avith edge unity, if the square brackets indicate that all the per- 

 mutations of the values they include must be taken, each value 

 being affected successively either by the positive or by the negative 

 sign. Moreovor |- j~|-, -J-, -J] and — i Ci~> "ib "i"] can represent in the 

 same way the two tetrahedra, the vertices of which form together 

 the vertices of the cube [4-, -J-, |-], if by the coefficient -| we indi- 

 cate the vertices with an even , by the coefficient — -^ the vertices 

 with an odd number of negative coordinates. 



In connection with this we amplify the question of art. 1 as 

 follows: "Under what circumstances will the symbols 



[a u a 2 , . . . , a n ] , + 1 [a ± , a 2 , . . . , a n ] 



represent the vertices of poly topes in S n , all the edges of which 

 have the same length, say unity?'' 



The answer to this question runs as follows: 



Theorem XXVIII. "If the values a lt a,, . . . , a n are arranged in 

 decreasing order, a p being the smallest non vanishing one, and if 

 a,., a,. + 4 represent any couple of adjacent unequal ones, we must have 

 in the case of the first symbol \a { , a 2 , . . . , « ; J 



eitlier p = n , a n = ^ , a k — a k + 1 == -J- \/2 | 

 or p < n , a p = 1 \/2, a,. ■ — a k + ± = -| V2 j ' 



in the case of the second symbol + \ \_ a \i a ^ • • • > a n\ 



p = n,a n _ i = a n = ±\ / 2,a k — a k + 1 = ±V2." 



Proof. The part of the proof concerned with the common value 

 | v 2 of the difference a,. — % + i of two unequal adjacent digits 

 is the same as that o-iven in art. 1. So we have to add onlv a 

 few words about the values of a n in the case of the first and of 

 a n _j i and a n in the case of the second symbol. 



Symbol \_a. { , a. 2 , . . . , a n ~]. Tn the supposition a,. — ^ fc + 1 = \ V2 

 the length of the edge of the polytope is unity. Therefore the 

 distance 2a n between the points 



-L . . . co-j (',,, CL-) Ctj , X-) === - c?9, ... 



">6 ... X\ — — - it f tl"2 ~~~ 0-\i ™3 ~~ "'■->> ... 



which are transformed into each other by inverting the sign of 



