1894.] On Homogeneous Division of Space. 11 



tetrahedrons placed on a pair of its parallel faces complete a paral- 

 lelepiped. Similarly two other parallelepipeds may be made by 

 putting together the pairs that have edges of lengths equal to QB 

 and QC respectively with those edges coincident, and finishing in 

 each case with the remaining pair of tetrahedrons. The three paral- 

 lelepipeds thus found are essentially different from one another, and 

 from the four of 10 ; and thus we have the seven parallelepipeds 

 fulfilling the statement of 9. Each of the seven parallelepipeds 

 corresponds to one and the same homogeneous distribution of points. 



12. Going back to 4, we see that, by the rule there given, we 

 find four different ways of passing to the tetrakaidekahedron from 

 any one chosen parallelepiped of a homogeneous assemblage. The 

 four different cellular systems thus found involve four different sets 

 of seven pairs of neighbours for each point. In each of these there 

 are four pairs of neighbours in rows parallel to the three quartets of 

 edges of the parallelepiped and to the chosen body-diagonal ; and 

 the other three pairs of neighbours are in three rows parallel to 

 the face- diagonal s which meet in the chosen body-diagonal. The 

 second ( 11) of the two modes of putting together tetrahedrons to 

 form a parallelepiped which we have been considering suggests a 

 second mode of dividing our primitive parallelepiped, in which we 

 should first truncate two opposite corners and then divide the octa- 

 hedron which is left, by two planes through one or other of its three 

 diagonals. The six tetrahedrons obtained by any one of the twelve 

 ways of effecting this second mode of division give, by their twenty- 

 four corners, the twenty-four corners of a space-filling tetrakaideka- 

 hedronal cell, by which our fundamental problem is solved. But every 

 solution thus obtainable is clearly obtainable by the simpler rule of 

 4, commencing with some one of the infinite number of primitive 

 parallelepipeds which we may take as representative of any homo- 

 geneous distribution of points. 



13. The communication is illustrated by a model showing the six 

 tetrahedrons derived by the rule 4 from a symmetrical kind of primi- 

 tive parallelepiped, being a rhombohedron of which the axial-diagonal 

 is equal in length to each of the edges. The homogeneous distribution 

 of points corresponding to this form 01 parallelepiped is the well- 

 known one in which every point is surrounded by eight others at the 

 corners of a cube of which it is the centre ; or, if we like to look at it 

 so, two simple cubical distributions of single points, each point of one 

 distribution being at the centre of a cube of points of the other. [To 

 understand the tactics of the single homogeneous assemblage consti- 

 tuted by these two cubic assemblages, let P be a point of one of the 

 cubic assemblages, and Q any one of its four nearest neighbours of the 

 other assemblage. Q is at the centre of a cube of which P is at one 

 corner. Let PD, PE, PF be three conterminous edges of this cube so 



