DERIVED FROM THE REGULAR POLYTOPES. 17 



From the examples given in the art. 51 and 52 it is clear that 

 in the enumeration of the limits of the highest number of dimen- 

 sions we proceed from h = n — 1 to h == ; this principle has 

 been followed too in column five of Table IV. 



C. Extension number and truncation integers and fractions. 



56. Theorem XXXI. "The new polytopes, all with half edges 

 of length unity, can be found by means of a regular extension of 

 the measure poly tope 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." 



This theorem is an immediate consequence of that given in art. 50 

 (theorem XXIX) about the equality of the absolute value of the 

 non vanishing coefficients c i of the coordinates œ i in the equation 

 + c i x x + g 2 % 2 + . . . = p of a limiting space 8 n _ t of the polytope. 

 As to the proof we can refer to art. 15. 



The extension number is always equal to the largest digit of the 

 symbol of coordinates. So, if in the case [2'1 / 1] of tCO of three- 

 dimensional space the cube [1 1 1] with edge 2 is extended to the 

 cube [2' 2' 2' with edge 2 (1 -{- 2\/2) it is precisely large enough 

 to enable us to deduce [2'1'1] from it by truncation; for the 

 limit of face import lies in the space + œ L = 2'. Likewise in the 

 case [\/2, V/2, 0,0] of C 2!ic in S k , which symbol winds up in zero, 

 we have to extend the eightcell [1111] to [V / 2, V2, V2, V2] 

 by multiplying its linear dimensions by \/2, etc, 



The manner in which the amount of truncation is measured 

 most easily can be explained as follows. If the measure polytope 



M n (2) = [11. . .1] of S n with centre O is extended to M n (2£) 



n 



= [££...£], S being the extension number, and this extended 

 M (2s) is truncated at a /--dimensional limit M / . (2f) with centre M 

 by a space /S > n _ 1 normal to O M cutting in R any edge PQ of 



PR 



M,} 2s) one end point P of which belongs to M k C2s) , then -z— - is 



■L hi 



considered as the "truncation fraction". Now, as we will prove 

 immediately, PR is always a multiple of \/2 with half the edge of 

 M (2) as unit, whether the symbol of coordinates of the polytope 

 deduced from M n {2e) by truncation terminates in unity or in zero; 

 so, in the relation PR = qV2 the multiplicator q which is 

 integer may be called the "truncation integer". So the truncation 



Verb. Kon. Akad. v. Wetenseh. I e Ser-tie Dl XI No. 5. E 2 



