1 2 ON THE SECTIONS OF A. BLOCK OF EIGHTCELLS 



C 3 D 3 F 3 W 3 , P x Q 2 D 4 , F 5 G 5 W\ W 5 ' , F 6 G & W e W % of the six side-faces 

 already given in fig. 3 and moreover in P 1 JF 1 W 1 'Q l even the half 

 of the side-face P 1 P 2 Q 2 Q l of the hexagonal prism of case VII. 



It goes nearly without saying that also by this new simple 

 method the shape and size of all the faces can be found. So in 

 the case V the endplanes of the section are equiangular hexagons 

 with alternately equal sides E b F b , G b H b . And by constructing the 

 unknown katheta of a right-angled triangle , the hypotenuse of 

 which is the segment of l b ' within PP 2 Q 2 Q while one katheta is 

 the difference of the distances of the sides E b F b , G b H b to the 

 centre of the hexagon, we find the distance of the endplanes, which 

 will enable us to make a drawing of the corresponding solid in 

 parallel perspective, etc. 



II. SECTIONS OF THE BLOCK OF EIGHTCELLS. 



10. Before entering upon the subject of the sections it will be 

 well to say a few words about the block itself. We suppose 81 

 eightcells to be built up into a rectangular block having three 

 along each edge and , in the fourdimensional space containing the 

 block that is itself an eightcell of three times the size, we imagine 

 a plane nr' passing through two opposite edges of this large block 

 and therefore containing its centre , which may still be indicated by 

 0. Now the block is to be cut by a space S 3 (r) normal to x' 

 in O and containing therefore the plane t in O normal to ir' . This 

 plane x cuts the large block in a regular hexagon , the sides of 

 which are three times the side of // 6 ; we will call it // 6 '. 



Now in the initial position of the intersecting space S 3 (t) , the 

 case cp = , the section evidently consists of a block of 27 cubes, 

 central sections of 27 of the 81 eightcells, forming the "middle 

 layer" of the block, while the other two layers lie beyond S 3 {ir) 

 in opposite directions (parallel to the two edges situated in vr') which 

 may be called "above" and "below" or "plus" and "minus". This 

 section of a very simple character is represented in fig. 6 *) in 

 parallel perspective , in the same manner as fig. 2 ; but in order 

 not to obtain too large figures in future we will suppose the sides 

 of /i 6 to be 1 cm. , those of // 6 ' 3 cm. As fig. 6 shows, the cubes 

 of this section have been numbered in a definite way 1,2,3,..., 



*) The shading of the fig». 6, 7, 10 will be explained later on. 



