G ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



in congruent lozenges including a rhombohedron ; for this is the 

 solid bounded by three pairs of parallel planes admitting of an 

 axis of period three. 



We deduce the exact form of the lozenges limiting the section 

 from another source. If H x H 2 H 3 Ii A H b H & (fig. 2) is the hexagon // 6 

 situated in the plane t, OX x and OX 2 are two axes of symmetry of 

 //q normal to each other and OX 3 is the axis of the rhombohedron 

 bearing the endpoints A, A' at equal distances from 0, then the 

 planes containing the lozenges are 



A{H X H 2 ), A(B S II 4 ), A{H b H & ), 

 A\H,H b ), A\H,H X ), A\H 2 H Z ). 



Now the line of intersection AD of the planes connecting A 

 with H X H 2 and H^H± will meet x in the point of intersection P 

 of H X H 2 and H^H^, etc. So the edges through A and A' are 

 found by joining A to the vertices of the triangle PQB and A' 

 to the vertices of the triangle P'Q'B'. Then the figure is com- 

 pleted by drawing through the pairs of opposite vertices (I/ 1 , ff±), 

 (II 2 , H b ), (JI 3 , i/ 6 ) of // 6 lines respectively parallel to AP, AQ, AR. 



Now from PH 2 = \PQ we deduce DP = \ AP , etc.; so the 

 three points B, D, F project themselves on the axis O A in the same 

 point K for which OK = \ O A (in accordance with the relation 

 OB = ^ O A in fig. 1). Likewise C,K, G project themselves in the 

 same point L for which OL = I OA' = OK, whilst O is the 

 projection of the six points H x , H 2 , . . . , H§ on the axis. So we 

 find BH X = H X C, the projections KO, OL of these segments of 

 the same line on the axis being equal, i.e. the vertices H x , H 2 , 

 . . ., B 6 of //g are the midpoints of the sides of the skew hexagon 

 BCDEFG. This gives us the relation BD = 2 H X H 2 . So the diago- 

 nals of the lozenges crossing the axis A A' , or as we will say the 

 transverse diagonals, are equal to \/lZ , the edge of the eightcell 

 being unity, for in this unit we have H X H 2 = \ }/2. *) 



For the other diagonals intersecting the axis, the axial diagonals, 

 we find 



A C= | AN = |- ]/ 6>Z^-P ON 2 = 



Now in fig. 1 we have B'A = ±OA; so AC is the hypotenuse 

 of a rectangular triangle with the kathetsc B'A and jB'C=2|/ / 3. 



') "We henceforward suppose the side of // 6 to be 3 cm ; then PQ = 3 V 2 cm. and 

 pp' = 3 V/"6 cm. 



