ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



I. SECTIONS OF A SINGLE EIGHTCELL BY ANY 



CENTRAL SPACE NORMAL TO THE PLANE CONTAINING 



TWO OPPOSITE EDGES. 



2. Let O be the centre of the eightcell C 8 and PQ one of the 

 two edges situated in ir '. Let 11 be the midpoint of PQ and C 6 (3> 

 the cube of intersection of C 8 with the space through OP normal 

 to PQ. Then OP is a diagonal of C 6 (3) and the plane of that space, 

 bisecting that diagonal normally, is the fixed plane %; it cuts C 6 (3) 

 in a regular hexagon. So this hexagon is situated in the boundary 

 of the solid that forms the intersection of C 8 by any space con- 

 taining 7T, i. e. by any central space normal to the plane w' through 

 O and PQ. This hexagon will be indicated by // 6 . 



3. We will try to smooth the way to an exact knowledge of 

 the sections in question by considering the projection of C 8 on the 

 plane t'. It consists (fig. 1) of a rectangle with sides PQ = 1 and 

 PP' = J/3 (where the length of the edge of C 8 is unity) divided 

 by two parallels P l Q 1 and P 2 Q 2 to PQ into three equal rectangles. 

 We indicate successively the projections of the 16 vertices, the 32 

 edges, the 24 faces and the 8 limiting cubes. 



The vertices are 



(P + 3 P x -f 3 P 2 + P') + ( Q + 3 Q, + 3 Q 2 -f- Q'). 



The edges are 



(3 PP, + G P X P 2 + 3 P 2 P') + (3 QQ 1 + Q l Q 2 + 3 Q 2 Q') 



+ (7 J Q+3P 1 Q 1 +3P 2 Q 2 + P'Q / ). 



The faces are 



(3 PP X P 2 -f 3 P,P 2 P') + (3 QQ X Q 2 -f 3 Q l Q 2 Q') 



-\-(SPP,Q i Q + 6P 1 P 2 Q 2 Q 1 + SP 2 P'Q'Q 2 ). 



The limiting bodies are 



PP X P 2 P' + QQ^l^Q' + 3PP 1 P 2 Q 2 Q l Q^3P 1 P 2 P'Q'Q 2 Q 1 . 



We call the cubes projecting themselves in the lines PP,P 2 P' 

 and QQ 1 Q 2 Q' the upper and lower cubes, the six other limiting 

 bodies the side-cubes. 



Any line T1\T 2 T' parallel to PP X P 2 P' is the projection of a cube 



