

1894.] On Homogeneous Division of Space. 3 



have only to swell out* from the untouched portions of surface of 

 each tree homogeneously until the swelling portions of surface meet 

 in the interstitial air spaces (for simplicity we are supposing the 

 earth removed, and roots, as well as leaves and twigs, to be perfectly 

 rigid). The wonderful cell-surface which we thus find is essentially 

 a case of the tetrakaidekahedronal cell, which I shall now describe 

 for any possible homogeneous assemblage of points or molecules. 



3. We shall find that the form of cell essentially consists of 

 fourteen walls, plane or not plane, generally not plane, of which 

 eight are hexagonal and six quadrilateral ; and with thirty-six edges, 

 generally curves, of meeting between the walls ; and twenty-four 

 corners where three walls meet. A cell answering this description 

 must of course be called a tetrakaidekahedron, unless we prefer to 

 call it a fourteen-walled cell. Each wall is an interface between one 

 cell and one of fourteen neighbours. Each of the thirty-six edges is a 

 line common to three neighbours. Each of the twenty-four corners is a 

 point common to four neighbours. The old-known parallelepipedal 

 partitioning is merely a very special case in which there are four neigh- 

 bours along every edge, and eight neighbours having a point in common 

 at every corner. We shall see how to pass ( 4) continuously from or to 

 this singular case, to or from a tetrakaidekahedron differing infini- 

 tesimally from it ; and, still continuously, to or from any or every 

 possible tetrakaidekahedronal partitioning. 



4. To change from a parallelepipedal to a tetrakaidekahedronal 

 cell, for one and the same homogeneous distribution of points, proceed 

 thus : Choose any one of the four body-diagonals of a parallelepiped 

 and divide the parallelepiped into six tetrahedrons by three planes each 

 through this diagonal, and one of the three pairs of parallel edges 

 which intersect it in its two ends. Give now any purely translational 

 motion to each of these six tetrahedrons. We have now the 4x6 

 corners of these tetrahedrons at twenty-four distinct points. These are 

 the corners of a tetrakaidekahedron, such as that described generally 

 in 3. The two sets of six corners, which before the movement coin- 

 cided in the two ends of the chosen diagonal, are now the corners of 

 one pair of the hexagonal faces of the tetrakaidekahedron. When we 

 look at the other twelve corners we see them as corners of other six hexa- 

 gons, and of six parallelograms, grouped together as described in 15 

 below. The movements of the six tetrahedrons may be such that the 

 groups of six corners and of four corners are in fourteen planes as we 

 shall see in 14 ; but, if they are made at random, none of the groups 

 will be in a single plane. The fourteen faces, plane or not plane, of 

 the tetrakaidekahedron are obtained by drawing arbitrarily any set 

 of surfaces to constitute four of the hexagons and three of the quadri- 

 laterals, with arbitrary curves for the edges between hexagon and 



* Compare ' Mathematical and Physical Papers, 1 TO!. 3, art. 97, 5. 



B 2 



