DERIVED FROM THE REGULAR POLYTOPES. 73 



Section IV : 



POLYTOPES AND NETS DERIVED FROM THE HALF MEASURE POLYTOPE. 



A. Tli e symbol of coordinates. 



90. Several times we have commemorated the fact that the 

 eight vertices of a cube can be split up into, two groups of four 

 points, the vertices of two regular tetrahedra , and that Avith respect 

 to the cube the vertices of each group may be said to be non 

 adjacent, i. e. not connected by an edge of the cube - see e. g. 

 the introduction of section I and the foot note of art. 4. l ) Also 

 that the sixteen vertices of an eightcell can be split up into two 

 groups of eight non adjacent points, the vertices of two regular 

 sixteencells (compare e. g. art. 78). So in general the 2 n vertices of 

 the measure polytope M n of space S n can be split up into two 

 groups of 2 n_1 non adjacent points, but the polytopes of which 

 these groups of points are the vertices are not regular for 

 n > 4. So in the case n = 5 there are two different kinds of limits 

 (/) 4 , viz. cells C iQ forming what remains of the limiting eightcells 

 of M and simplexes #(5) replacing the vanished vertices of J/ 5 . 

 In relation with their generation we call these new polytopes half 

 measure polytopes and we investigate in this section these polytopes 

 and the nets which can be derived from them. 



In the cases [111] and [1111] of cube and eightcell we have 

 represented the two half measure polytopes by the symbols +-^[111] 



// 



and +-J-[1H1] respectively. Likewise we indicate by +-J£H. • -1] 

 the two half measure polytopes into which M n can be decomposed 



n 



according to the vertices, where -\- i[l 1 . . . 1] includes all the vertices 



of which an even number of coordinates is negative and — 2" [11 * ■ • ^] 

 all the vertices of which an odd number of coordinates is negative. 

 These symbols immediately reveal a difference in character between 

 the half measure polytopes of 8 2n and 8 2n+i which we will represent 

 for short by HM 2n and HM 2n+ i> In the case of HM 2n the polytope 

 admits a centre of symmetry, as the reversion of the signs of all 

 the coordinates of anv vertex furnishes an other vertex of the same 

 group; on the contrary in the case of HM. ln+A every vertex is 



l ) Tlie result mentioned contains a numerical error; it ought to be replaced by 

 (i (7 + |/2) ," J (3 + 1/2) , * (|/2 — 1) , I (5 + 3j/2)) , 

 ft(7 + V2); K3-I/2), i(l + K2), i(5+ >/2)), 

 see „Wiskundige Opgaven", vol. XI, problem 96. 



