6 



ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



prove the close relationship between the progéniture of the two 

 patriarchs, cube and octahedron, can be extended so as to make 

 the two families quite identical with each other; to that end we have 

 only to derive from each of the two, cube and octahedron, all 

 the expansion and contraction forms, the number of which amounts 

 in S n to 2" — 1. This important fact, which will be proved later 

 on, enables us to treat in the second and third sections the forms 

 with the symbols [a it a 2 , . . ., -|] and [a ± , a 2 , . . ., 0] successively, 

 without being obliged to postpone the study of the corresponding 

 nets built up by forms of both groups. 



In order to avoid fractions we will multiply the digits by two 

 in this section and the next one; under this circumstance the last 

 digit is unity or zero, the difference a k — a k + i of two unequal 

 adjacent digits is V^2 and the symbol represents a polytope with 

 edge 2. Moreover in order to simplify the symbols we will write 

 p for 1 -\-pV2 and put if possible \/2 outside the brackets, 

 substituting e. g. [11100]\/2 for [\/2, \/2, \/2, 0, 0]. 



49. For n = 2, 3, 4, 5 we 

 explained in the memoir of M' s 



n 



[1 1] =lh 



[PI] 



e lP*=l>8 



have successively in the symbols 

 Stott: x ) 



= 2. 









n - 



o 



— o. 









[111]= c 





[1'11] = e 2 



C=RCO 







[100] i/2= ce i C= 



[1T1] —e l C 



= w 



[2 , l'l'] = e l e 9 



c=tco 



[110JK2 ==<»! 



= 4. 



c=co 



[210] 1/2 = ^ 



e,C=lO 



[1 1 1 H = 



n 



[2'2'1'1] = 



e 1 e 2 C s 





[2110] y2 = 



ce l e. 2 C H 



[IT I'll = 



e \ ^8 



[2' l'l'l] = 



e i H c è 



[U10]j/2 = 



ce 1 C ó 



[2110] \/2 = 



ce 1 e s C 8 



riTi 11 = 



L J 



e 2 C s 



[Vim i] = 



H e 3 c s 



[1100] \/2 = 



C(J 2 °8 



[2100] \/2 = 



ce 2 e. 6 C H 



[I'l 1 1] = 



H c z 



[3' 2' l'l] = 



H H H C 6 



[1000] y2 = 



= 5. 



ce s C 8 



[3210] \/2 = 



c<\ e 2 % O s 



[11111] = 



^10 



[2'2T1 1] = 



n z 





[22100] ]/2 = 



ce. 2 e s C l0 



[IT l'l'l] = 



e \ ^10 



[2'1T1 1] = 



e 2 e é Çi 



[11110]j/2 = 



ce 1 O 10 



[21100] \/2 = 



ce, e± C 10 



[l'l'l'] 1] = 



e 2 Qo 



[2' l'l 1 1] = 



e S e i ^10 



[11100] j/2 = 



ce z C 10 



[21000] ]/2 = 



<*a e i Cio 



[l'l'l 1 11 = 



H c io 



[3'3'2'1'1] = 



e \ e 2 e 6 C \0 



[11000]j/2 = 



ce s C l0 



[33210] ]/2 = 



ce Y e % e% C 10 



[l'l 111] = 



e i C 10 



[3'2'2Tl] = 



e l e Z e * C lQ 



[10000]i/2 = 



ce i <?10 



[32210] V% = 



ce 1 e i e 4t C 10 



[2'2'2T11 = e } 



e 2 ^10 



[3'2'1T1] = 



ei e s e * c \o 



[22210]K^ = 



ce ie . 2 C l0 



[32110]K2 = 



ce^e+Cw 



[2'2'1T1] = e } 



% C \Q 



[3' 2' l'l ]] = 



h ^ % r io 



[22110]j/2 = 



ce 1 e i r i0 



[32100] \/2 = 



ce Z e Z e i C IQ 



[2TlTl] = e 2 



H ^io 



[4'3'2TH = 



e l e 2 e S e 4, C 10 



[21110] J/2 — 



^3^0. 



[43210] ]/2 = 



ce 1 e 2 e i e 4! C l0 



l ) For the deduction of the e and c symbols from the symbol of coordinates compare 

 the part 7) of this section; here />„ means: p u turned 45° about the centre. 



In Table IV added at the end of this memoir are put on record for n = 3, 4, 5, the 

 different polyhedra and polytopes deduced from the measure polytope. Of this table the first 

 column contains the symbols of deduction of the polytope from measure polytope and cross 

 polytope — with the first of which we are concerned in this section only — and the third the 

 symbol of coordinates. The second and the following columns will be explained farther on. 





