6 ON GRETA IN SERIES 0"F SECXrONS OF THE 



be cut by a plane passing through that point and parallel to ABC; 

 hence its section will be an equilateral triangle with a side equal 

 to I AD'. In tig. 2 its vertices A^D', B^D', (\B' are indi- 



4 ■ '4 T 



cated as a, b, c. Similarly the sections of B^CBA', C B A B' , 



BAB C' are equilateral triangles with a side equal to | A B' . Their 



vertices are the points (J?j J', C^J, B^A'),{C^B', D^B', A^B'), 



4 4 T 1 ~i T 



{B^C, A,C', B,C'). 



4 4 4 



It has been shewn bv means of the tetrahedron ABCB' that S 



1 



passes through the points A-^B', C^B', and by the tetrahedron 



4 4 



ACBB' that it passes through the points ./j 7i', C^B' . But the edges 



4 4 



ABI, CB' are conunon to ABCB' and ACB' B' and the edges AB' , 

 CB' are common to ACBB' and ACB'B'. Hence we have four 



points on ACB' B' through which /S' passes, giving as section a 



rectangle with sides J and ^ of A B' indicated in hg. 2 by acde. 



There are also rectangular sections of the tetrahedra on the 

 other edges of AB C B. 



Again A B' , A B' , A C' are edges of the tetrahedron A B' C' B' 



and it has been shewn that S cuts them in the points A^B', 



A^B', A^C', giving as section of this tetrahedron an equilateral 



44" 

 triangle with side equal to \ A B' . Similarly the section (tig. 3) 



of each of the tetrahedra on the vertices B, C, and B will be 



an equilateral triangle with sides equal to \ AB'. 



In figui'e 2 are also indicated the sections of the IG-cell by 



spaces &\ )S parallel S and cutting yl B' in the points A^B', 



A^D'. The shapes are shewn in the figures 4 and 5. 

 T 

 A space S ^ parallel to S and })assing through B' wovdd also 



pass through A' B and C' , giving as the last shape of the series 

 a tetrahedron equal to ABCB oj)positely placed, this being the 

 bounding tetrahedron of the 16-cell op])osite to ABCB. 



The 24-cell, 



6. Let AB G B Tj i^(tig. G), an octahedron in a space 8 , be one 

 of the bounding solids of a 24-ccll. In this tigure there are 6 

 solids at each vertex, a conditon that will be satisfied if one be 



