( 488 ) 



3 

 Each of the ten bounding bodies is as - {M,) — compare in the first 



o 



diagram the section with a space perpendicular to ae in the point 

 in the middle between c and d — a (12, 18, 8) bounded by four 



- (31,) and four - (M,), i. e. by four of the hexagons and four of 

 2 6 



the triangles, and therefore a tetrahedron truncated regularly at the 



vertices, i. e. the first of the equiangular semi-regular (Archimedian) 



bodies. 



Case ?i = 6. Out of the third of the diagrams we read that 

 the section is a (20, 90, 120, 60, 12). All the edges have a 

 length 1/2, all the faces are triangles. The bounding bodies are for 



one half (30) as - {M,) octahedra, for the other half (15 + 15) as 



1 2 



- (.17.) tetrahedra. The twelve bounding pol v topes are as - (M^) - 

 4 " 5 



compare now again the second diagram — polytopes (10, 30, 30, 10) 



bounded by five of the octahedra and five of the tetrahedra, which 



can be regarded as regular five-cells, regularly truncated at the 



vertices as far as half of the edges, so as to lose all the original 



edges by this truncation. 



Case n = l. We arrive at a (140, 420, 490, 280, 84, 14). 



The length of the edges is - 1/2. The 490 faces consist of 210 



hexagons and 280 triangles, the 280 bounding bodies of 210 trun- 

 cated tetrahedra and 70 tetrahedra, the 84 four-dimensional bounding 



polytopes of 42 polytopes - (il/^) = (30, 60, 40, 10) found already 



z 



3 

 under n = 5 and 42 polytopes — {M,) = (20, 40, 30, 10) bounded by 



five truncated tetrahedra and five tetrahedra — regular five-cells 



truncated at the vertices as far as a third of the edges. The 



5 

 14 five-dimensional bounding polytopes are as — {MJ polytopes 



1 A 



(60, 150, 140, 60, 12) bounded by six (30, 60, 40, 10) and six 

 (20, 40, 30, 10). 



Case n = 8. Here a (70, 560, 1120, 980, 448, 112, 16) is the 

 result. The length of the edges is K2, all faces are triangles. The 



