DERIVED FROM THE REGULAR POLYTOPES. 45 



difficulty underlying this method. So, in the case of truncation of 

 an octahedron (fig. 16) at the edge BC, it makes a difference 

 whether we choose BA or BC' as the edge on which we deter- 

 mine the amount of truncation. For if we move the truncating 

 plane (through BC normal to O M, where M is the midpoint of 

 BC) parallel to itself until 1 it passes through O it contains the 

 other extremity A of the edge BA, while it bisects the edge BC' . 

 This difficulty can be overcome by stipulating that the edge to be 

 chosen may not contain a vertex opposite to one of the vertices 

 of the limit at which the truncation takes place. But this implies 

 always that we measure quite as w T ell on the line MO joining the 

 centre of that limit to the centre of the polytope. So if the trun- 



. MB 



eating space cuts MO in P the amount of truncation is ——. Now 



° l MO 



PO 



the complement — — of this quantity can be deduced immediately 



from the symbol of coordinates [a i9 a 2 ,. . . , a n ~\ of the cross polytope 

 form considered. If we suppose that the truncation takes place at the limit 



n — 1 n 



(l)k-i °f th e corresponding extended cross polytope [1 , 0, . . . , 0]S^ 



i 



lying in the space represented by x x -\- w 2 ~\- . . . -f- w k = constant 



PO 



it is immediately evident that — — is equal to the quotient 



k 



of the sum 2^ of the first k digits of the symbol of the trun- 



l 



cated polytope by the corresponding sum of the extended cross 



k 



polytope, i. e. by 2^. So from — — = — - we deduce: 



i MO ^ 



l 



n 



amount of truncation = — — - === -J- — . 



MO 5, 



l 

 We illustrate this theory by the example [3' 3' f 1 1] for which 



we have determined above the extension number. Here we find 



moreover 



5 5 5 



2>, = 4 + 6 V2 9 Sflf = 3 + 3 V/2, ta { = 2 + V2, a 5 = 1 



2 3 4 



and therefore 



4 + 6V2 3 + 3V2 2 + V2 1 



5 + 9 V"2 5 -)- 9 V2 5 + 9 V2 ' 5 + 9 V2 



