24 ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



passes into that with the plane — x by a rotation through 180° 

 about the centre of the figure, succeeded by the substitution of 

 28 — k for any number k; as this simple rule holds in general, 

 we give in the groups 10 m , 10 IV , . . ., 10 V n the section with the 

 plane -{-* x only. 



If B represents a rhombohedrical section the solids A, B, C 

 satisfy the relations B = A = B -\- C; A and Caie always white, 

 B is always black. 



Case III. The small triangle marked e represents the endplane 

 below, invisible in the principal figure, as seen from within the 

 section, i.e. from above; in the four following cases the polygons 

 marked e must be interpreted in the same way. 



Here we have B = A -\- 2 C = 2 B ; A and C are white in 

 the middle layer and black elsewhere, while B is black in the middle 

 layer and white elsewhere. 



Case IV. Here the relation holds B — O -4- 2 T; the octahedra 

 are white, the tetrahedra black. 



This case is by far the most remarkable one; it solves the 

 question: "how to divide an octahedron (3) with edges equal to 

 three units into octahedra (i) and tetrahedra T (]) with edges 

 unity?" If we place the (/ 3) with one of its diagonals vertically, 

 the solution can be «nven as follows. Divide the vertical diagonal 

 into six equal parts. Cut the octahedron (3) by five horizontal 

 planes passing through the points of division. Divide the square 

 of the middle section into nine and the squares of the adjacent 

 sections into four squares equal to the squares of the extreme 

 sections. Then these 1 -(- 4 -|- 9 + 4 -\- 1 = 19 equal squares 



') In connexion with the space-filling properties of octahedra 00) and tetrahedra 

 TO) in the two different positions it is evident that it must he possible to fill an 

 octahedron Oil') and a tetrahedron Til'), both with edges p, by 0(0 and HO. We only 



mention the results here. In the case of CAP) ^ (2// 4- 1) 00) and — (//— 1) HO of 



each of the two positions are required; in the case of Tip) we want -r(p' — 1)0\ 



(T p(p + 1) (p + 2) TO) corresponding in position with Til') and j p (p— 1) (p— 2) T' 

 in the opposite position. These results verify the relations in volume 



|(2p' -f 1) 00) + %(p 2 — 1) T(i) = p' cKD, Ikp— 1) 00) + f-fp + 2) TV) = i? 7") 

 o à bo 



based on the fact that 4 TO) correspond to one OW , as they ought to do. 



In tlireedimensional space divided into 00) and TO) there is plane contact between 



two polyhedra of differeut kind. So an 00) and two 2(0 in plane contact with it on 



two opposite faces form a rhombohedron, an 00) and the eight TO) in plane contact 



with it form the well-known figure of the two equal but oppositely placed tetrahedra 



penetrating one another in an octahedron. 



