2 4 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



But this case corresponds to the simplex 8(2 P ) of space /S>_ d . 

 Cross ' poly topes. The cross polytope is characterized by the pro- 

 perty of having all its diagonals of the same length (=J/2 times 

 an edge) and passing through the same point. So in order to 

 represent a cross polytope the symbol of coordinates of (P) n can 

 contain end digits only, for the supposition of three different digits 

 as in (210) leads inevitably to three different distances. Let us 

 suppose the two end digits are 1 and 0. Then we have to take 

 in at least two of each in order to create the possibility of inter- 

 changing two pairs of digits; this gives us the octahedron, the 

 diagonals of which are the joins of the pairs of vertices repre- 

 sented by 



1100 ) 1010 ) 10011 



0011 ) * 0101 j' 0110) ' 



and pass therefore through the point \>\y\>\- Finally in the 



n — 1 n — 1 



cases (11 00 ... 0) and (11 ... 100) we have to deal also with 

 polytopes admitting only diagonals = ]/" 2 times an edge, but 

 here these diagonals do not pass through the same point (centre). 

 For in the case of (11000) the centre is the point all the coor- 

 dinates of which are -| and this point lies not on the diagonal 

 joining the points 11000 and 00110, etc. 



C. Extension number and truncation integers and fractions. 



15. "How can all the new polytopes {abc . . .) found analyti- 

 cally be deduced geometrically from the regular simplex?" 



As we remarked in the introduction the new polytopes have been 

 discovered geometrically by M 1S . A. Boole Stott; we will consider 

 her method thoroughly under D. Here we wish to indicate first 

 that the answer to this question can also be given by the theorem: 



Theorem IV. "The new polytopes, all with edges of length unity, 

 can be found by means of a regular extension of the regular sim- 

 plex of coordinates followed by a regular truncation, either at the 

 vertices alone, or at the vertices and the edges, or at the vertices, 

 edges and faces, etc." 



Proof. This theorem is an immediate consequence of that given 

 in art. 6 (theorem II) about the equality of the non vanishing 

 coefficients c t of the coordinates x- t in the equation c { x { -\- c 2 x 2 -\~ 

 . . . =p of a limiting space S n _ i of the neAv polytope deduced 

 from the simplex 8(n-\- 1) of S n . So in treating in art. 7 the 

 example (32110) we found that the limiting spaces 



