DERIVED FROM THE REGULAR POLYTOPES. 25 



VV U VV 2 , VF Z we have to replace the digits of (1+1/2,1 + V/2, 1) 

 by their complements to a = 2 + V/2, giving (1,1, 1 + V/2), 

 i.e. (1 + V/2, 1,1). In order to multiply the last triangle round 

 the new origin V we have to return to square brackets. So 

 [1 + \/2, 1,1] is the symbol of the new constituent ECO. We 

 repeat that the digits of this new symbol are the complements to 

 # — 2 -j- V/2 of the digits of the "groundform" tf (7 taken in inversée! order. 



In the case of the edge A A' and the cube derived from it we 

 have to assume V x , the centre of the cube, as new origin, and 

 V* V, V x V' 2 , V x V\ as new axes. Thereby œ± = [1], # 2 = 1 + V/2, 

 a» 8 = 1 -j- \/2 is transformed into a? 4 = [1], af 2 = 1 , #'3 = 1 ; so by 

 multiplication we get # 4 == [1], o? 2 , a? 3 = [1 , 1] or shorter [1], [1,1], 

 which in this special case may be combined to a? 4 , a/ 2 > «p'3 = [1 > ! » 1] 

 or shorter [1,1, 1], the cube. 



Finally the face À ABB' . . .represented by or ± , x 2 = [1 -)- V/2, 1], 

 #? 3 = 1 -J- V/2 passes by multiplication into a? d , x 2 = \\ + V/2, 1], 



a?. 



= [1] or shorter [1 + V/2, 1] [1]. 



So if we arrange the constituents in the order g^ g>i,g\,g§ °f 

 decreasing import we get 



y s = [l + ^2,1+^2, 1] 

 y 2 = [l+\/2, 1] [1] 



* = [ 1 1 [1 '!](' 

 ^ = [1+^2, 1 ,1]! 



the first and the last being semiregular polyhedra deduced from 

 the cube, whilst the intermediate ones appear as prisms. We remark 

 that the pairs of syllables of the symbols of g 2 and g i can be derived 

 from the symbols of g 3 and g by taking for g 2 the last two digits 

 of ^3 and the last digit of g , for g x the last digit of g 3 and the 

 last two digits of g . 



Now it is obvious that in the general case of the polytope (P) n 

 of JS n represented by \ji A ,a 2 ,. . . ,# M _i,# n ] the introduced multipli- 

 cation of the limits of different import, which multiplication can 

 be performed for any value of the constant a, leads in general to 

 n + 1 constituents g n ,g n -i, . -,^1,^0» represented by 



yo = 



a* 



<*2 



5 a -l •> a i > ) a n—2i a n-\> a n\ 



> a B •> a k j } a ,t-\-> a n\ \_ a a \\ 



ffn—k L^/,+1 j a lc+2 » > a nj \_ a a k> a a k— 1 > • • • > ^ ö lJ 



j7l = [#n] [« «n-1 ï « «n-2» > « «2> « a \\ 



g = \a a ni a ct H _ i , a — a n _ 2 , , ci — a 2 , a — a ± j 



