14 ON THE SECTIONS OF A BLOCK OE EIGHTCELLS 



Here we mean that the a cube of eightcell 13 coincides with the 

 a cube of eightcell 1 1 , etc. 1 ). 



The section of the block of cubes by the plane t is shewn in 

 fig. 7, the letters a , b' , c , a' , b , c' refering to the limiting cubes 

 of the block itself. It will be seen that , as to their sections with 

 this plane , the intersected cubes can be arranged in three groups. 

 The first consists of 2, 4, 12, 14, 16, 24, 26 cut in regular 

 hexagons, the second of 1, 11, 13, 21, 23, 25 cut in equal 

 and similarly placed equilateral triangles , the third of 3 , 5 , 7 , 

 15, 17, 27 also cut in equal and similarly placed equilateral 

 triangles , oppositely placed with respect to those of the second 

 group. And the non-intersected cubes, eight in number, consist, 

 as to the distance of their centres from tt , of four different groups 



19; 10, 20, 22; 6, 8, 18; 9. 

 So we find seven groups in all ; they will reappear directly in a 

 more important point of view. 



12. Let us now consider the projection of the 81 eightcells 

 on to the plane t'. In the above adopted reduced scale the pro- 

 jection of the block (fig. S) is a rectangle with sides 3 |/2 cm. and 

 3 ]/6 cm. ; in this large rectangle the concentric similar and simi- 

 larly placed rectangle PP'Q'Q, the sides of which are J/2 cm. and 

 J/6 cm. , represents the projection of the central eightcell 14. By 

 producing PP' and QQ' the large rectangle is divided into three 

 strips , an upper one , a middle one and a lower one , which are 

 evidently the projections of the three sets of 27 eightcells, forming 

 respectively the plus layer, the middle layer and the minus layer. 

 Now the manner in which the 27 eightcells of the middle layer 

 project themselves on the middle strip P Q P ' Q ' Q Q can be deduced 

 from that in which the 27 cubes of the section by the initial position 

 of the space S 3 (t) project themselves on the line AA' which forms 

 the projection of S. 6 (7r) and is at the same time (fig. 6) a diagonal 

 of the block of the 27 cubes. So we have now to consider for a 

 moment the projection of the 27 cubes on the diagonal AA' . 



If we first direct our attention to the vertices and begin (fig. 6) 

 with those of the large cube we find as projections the two extre- 

 mities A, A' and two points dividing AA' into three equal parts. 

 If then we pass to the cubes 9, 14, 19 we see that the projec- 



') In an infinite block of eightcells the numbers of eightcells respectively in space, 

 plane, line and point contact with any eightcell considered as the central one are 

 equal to the numbers of limiting bodies, faces, edges, vertices of that eightcell, i.e. 

 8, 24, 32, 16. So we find, the central eightcell included, 81 eightcells in all, i.e. all 

 the eightcells of our block are wanted in order to include the central one entirely. 



