1894.] 



On Homogeneous Division of Space. 



PC, AE, FQ, BD. These twelve lines are all in air, and they are the 

 edges of our curved-faced parallelepiped. To describe its faces take 

 points infinitely near to one another along the line PC (straight or 

 curved as may be) : and take the corresponding points in BD. Join 

 these pairs of corresponding points by lines in air infinitely near to 

 one another in succession. These lines give us the face PBDC. 

 Corresponding points in AE, FQ, and corresponding lines between 

 them give us the parallel face AFQE. Similarly we find the other 

 two pairs of the parallel faces of the parallelepiped. If the solids 

 touch one another anywhere, either at points or throughout finite 

 areas, we are to reckon the interface between them as air in respect 

 to our present rules. 



6. We have thus found the most general possible parallelepipedal 

 partitioning for any given homogeneous assemblage of solids. Pre- 

 cisely similar rules give the corresponding result for any possible 

 partitioning if we first choose the twenty-four corners of the tetra- 

 kaidekahedron by finding six tetrahedrons and giving them arbitrary 

 translatory motions according to the rule of 4. To make this clear 

 it is only now necessary to remark that the four corners of each 

 tetrahedron are essentially corresponding points, and that if one of 

 them is in air all of them are in air, whatever translatory motion we 

 give to the tetrahedron. 



7. The transition from the parallelepiped to the tetrakaideka- 

 hedron described in 4 will be now readily understood, if we pause to 

 consider the vastly simpler two-dimensional case of transition from a 

 parallelogram to a hexagon. This is illustrated in figs. 1 and 2 ; with 

 heavy lines in each case for the sides of the hexagon, and light lines 

 for the six of its diagonals which are sides of constructional triangles. 

 The four diagrams show different relative positions in one plane of 

 two equal homochirally similar triangles ABC, A'B'C' ; oppositely 



FIG. 1. 



A' 





