( 489 ) 



980 bounding bodies consist of 420 octahedra and 560 tetrahedra 



the 448 four-dimensional bounding pol j topes of 336 poly topes 



2 1 



- (M^) and 112 polytopes - (iij), i. e. of 336 tive-cells truncated as 



far as lialf of the edges, found under n = 6, and 112 five-cells. 

 The 112 five-dimensional bounding polytopes are as far as one half 



is concerned - (il/g) = (20, 90, 120, 60, 12) already found above, 

 as far as the other half is concerned - (J/J = (15, 60, 80,- 45, 12) 



Ö 



bounded by six five-cells truncated as far as half the length of the 



edges and six five-cells. Finally the sixteen six-dimensional bounding 



3 

 polytopes are as - (J/^) polytopes (35, 210, 350, 245, 54, 84) 



bounded by seven (20, 90, 120, 60, 12) and seven (15, 60, 80,45, 12) '). 

 From this all we easily deduce the following general laws : 

 "The vertices of the section are vertices of il/n for even 7i, for odd 7i 



they are centres of edges oïAfn; they are always of the same kind ")." 



1 

 "The common length of the edges is 1/2 for even n and — |/2 



for odd n; they are always of the same kind')." 



"The faces are triangles for even n, hexagons and (smaller) triangles ^) 

 for odd n." 



"The bounding bodies are octahedra and tetrahedra for even n, 

 truncated tetrahedra and (smaller) tetrahedra for odd n'. 



"The four-dimensional bounding polyhedra are five-cells truncated 

 as far as halfway the edges and five-cells for even 7i, five-cells 



1) If we had set to work, when enumerating the results, in that sense inversely 

 that with each new value of n of the bounding polytopes with the greatest 

 number of dimensions we had descended to the vertices, we should have furnished 

 a geometrical variation of the well known nursery-booic : "the house that Jack built". 

 However with two differences. Wfien descending from every one round higher of the 

 ladder we pass every other time again the same stadia and the ladder is a Jacob's 

 ladder with an infinite number of rounds, 



') That is, in each vertex as many edges meet in the same way, etc. 



^) The cases n — odd seem to be an excephon to this, as there are for the 

 truncated tetrahedra two kinds of edges, namely : sections of two hexagonal faces 

 and sections of an hexagonal and a triangular face. However, this is only appa- 

 rently. For, for each edge we find that in the section itself always again the 

 number of faces passing through it of each of the two sorts is steadfast, thus 

 for w = 5 two hexagonal faces and one triangular one. 



•*) We do not mention here, that for w = 3 only an hexagon appears. Neither 

 that of the bounding bodies the tetrahedra do not appear for w =:= 4, etc. 



