8 ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



the equal portions of the rhombohedron lying outside the eightcell 

 are regular triangular pyramids. In other words : the distance of 

 the endplanes to the plane ir must exceed that of the vertices of 

 the skew hexagon BCBEFG to that selfsame plane, the sides of 

 that hexagon not being cut by the endplanes. 



In fig. 3 (in which we lay down the different results) the faces 

 of the section are constructed for the two cases of this group we 

 wish to consider, the cases where the endpoint W of the line- 

 segment / situated on PP' is the point W 3 halfway between P 

 and Pj and where this point JV 4 coincides with P v The figure 

 shows the two lozenges A 3 B 3 C 3 D 3 and A 4 B 4 C 4 B 4 , obtained in the 

 way described. By means of the point W 3 where the normal in 

 W 3 on A 3 B 3 cuts A 3 C 3 the line F 3 F 3 normal to A 3 C 3 is found, 

 giving us in its turn the truncated lozenge B 3 C 3 D 3 F 3 F 3 ; the sec- 

 tion is limited by six of these truncated lozenges and two equila- 

 teral triangles with F 3 F 3 as side. In repeating this construction 

 for <P = 60°, where JV 4 coincides with P l we find for TV 4 ' the 

 centre of the lozenge A 4 B 4 C 4 D 4 , i.e. the truncated lozenge be- 

 comes an equilateral triangle B 4 C 4 .D 4 , the section itself is an octahe- 

 dron. This is as it should be : in the case Cp = 00° the projection 

 A A' (fig. 1) is normal to the diagonal P'Q of the eightcell and 

 therefore S 3 (t) is normal to that line. 



G. The case 60° << <J> •< 90. Here the rhombohedron is 

 truncated at both ends by planes normal to the axis AA' bearing 

 hexagonal sections. 



In fig. 3 the sections with the side-cubes are constructed for 

 the two cases determined by the relations P 2 fV b = 3 W 6 P 1 and 

 P W % = 2 W 6 P 1 . The results, obtained in the indicated way , are 

 the isosceles trapezia F 5 F 5 G b H 5 and F G F Q G 6 H 6 ; the corresponding 

 endplanes bear equiangular hexagons with the alternate sides equal 

 to E h F b , G 5 II 5 and to F e F 6 , G^H & . The manner in which the 

 midpoints JV 5 ' , JV e ' and W h ' , 7V 6 ' of the lines FF and GR on 

 the macrodiagonal AC are deduced from 1V b , W Q and JV b , IV 6 

 is indicated in the figure. 



In the extreme case <$ = 90° the section becomes a hexagonal 

 prism bounded by six rectangles with sides of 3 ]/2 cm. and 3 cm. 

 and by two regular hexagons with sides of 3 cm. 



7. After this explanation we think it will be evident to the reader 

 that fig. 3 enables us to make cardboard models of the seven 

 sections considered. *) But this figure can teach us quite as well 



') A more expeditious method will be given farther on. 



