7 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



n + 1 n + 1 



pattern vertex satisfy the equation Ha L œ t = ^a 2 t — 1 ; as this equa- 



i i 



tion represents a space 8 n _ i normal to the line from the centre 

 of the coordinate simplex to the pattern vertex *), the first part of 

 the theorem is proved. 



From the regularity of the considered polytope it can be deduced 

 that the distance OP' from the centre O to the space 8 n _ l c ontaining 

 the vertices V i adjacent to any vertex V does not change with 

 that vertex. So all the vertices of the considered polytope are trans- 

 formed into the spaces 8 n+i containing their adjacent vertices by 

 means of an inversion with respect to the spherical space with O 

 as centre and \f OP. OP' as radius. 



38. If we use the symbol JSf (?i-\-\Y i) introduced in art. 21 we 

 have : 



Theorem XXIII. "The two polytopes 



e a e b e c . . . e r e s e t S [n -\- 1) (1) , e a . e v e & . . . e r , e s . e v S (n-\- 1) (4) 



are equal and concentric, but of opposite orientation, if and only 

 if we have generally 



a-\-t' = b-\-s' = c-\-r = .... =r-\- c' = s-\-b' = t-\-d = n — 1'" 



<( For a = d,b = b' ,c = c, . . . , r = r', s = s\ t = t' the poly- 

 tope in which the two given ones coincide is central symmetric, if 

 and only if we have 



a-\-t = b-\-$-=c-\-r= ... = n — 1 



under which conditions there may be an unpaired middle expansion 

 £ n _! for n odd". 



2 



This theorem gives in analytical form the results published in a 

 joint paper of M rs Stott and myself 2 ), already quoted on page 17, 

 as far as the simplex offspring is concerned ; for the supposition 

 that the reciprocal polytopes A and A' mentioned in art. 3 of that 

 paper are e Q 8 () (>-|-l) (1) and e n _ ± 8 (n-\-l) (i \ i. e two concentric 

 and equal simplexes /S(#-|-l) (1) of opposite orientation, specializes 

 the general results found there to the simplex theorem stated just 

 now here. To prove the latter analytically we have only to write 

 out the result of the operations e a e b e c . . ,e r e 8 e t and e a .e b .e c .. . .e r .e s .e t . 



*) Compare " Nieuw Archief voor Wiskunde", vol. IX, p. 140, remark I. 

 2 ) Reciprocity in connection with semiregular polytopes and nets, " Proceedings of 

 the Academy of Amsterdam", September, 1910. 



