20 ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



It goes without saying that the different A, B, C, I) mentioned 

 in this table stand for different sections. 



16. Now we have still to solve this problem: "if we are given , 

 for any of the six cases TI, III,. . . ., VII considered, a bag con- 

 taining the set of models representing the corresponding sections 

 of the eightcells of the block, how can these pieces be put to- 

 gether so as to obtain the corresponding section of the block 

 itself?" 



This problem can be easily solved in the following way. We 

 begin by arranging the set of models into classes according to 

 their shape, subdivide these classes into groups according to the 

 last table of results and assign to each model of any group one 

 of the numbers of the eightcells belonging to that group, without 

 sign , with the sign -j~ or with the sign — according to the layer 

 the group belongs to. Then we inscribe on the faces of any model 

 originating from the upper cube and the lower cube of the corres- 

 ponding eightcell the letters d and d' , on the three or the six side- 

 faces originating from the side-cubes the letters a, b, c or a', b ', c' 

 or both triplets, taking care to put the right triplet at the right 

 place and to write the three constituents of each triplet in due 

 order of succession. After having prepared the work in this way 

 we have only to consult a table of contact, of which we gave 

 above only the part refering to the central eightcell 14, in order 

 to be able to bring to coincidence the faces of the models that 

 are to coincide, i. e. in order to be able to build up the section 

 of the block of eightcells in the Avay desired. 



The result of this solution is given in the six groups of figures 

 10 n , 10 m ,. . .10 VII corresponding to the six cases II, III,. . .VII. 

 Each of these groups consists of: 



1 st . one relative large figure in parallel perspective representing 

 the section of the block of eightcells built up of the pieces that 

 are sections of the separate eightcells, 



2 nd . two or more small figures in parallel perspective marked 

 A, B, C, D, U, representing these different pieces, 



3 rd . one or two plane sections of the solids of intersection 

 obtained with planes on which we will fix attention afterwards. 



17. It will be necessary now to say a word or two about the 

 shading and the colouring of the groups of fig. 10; we begin by 

 the shading. 



The shading of the figures 10 is based on the supposition that the 

 eightcells of the block are alternately black and white in the 

 manner which may be considered as the consequent fourdiinensional 



