470 



NATURE 



[March 15, 1894 



nearest to Q. The rhombohedron of which rQ is body-diagonal 

 and PA, PB, PC the edges' conterminous in p, and QD, Qt:, qf 



the edges conlermirous in Q, is cur present ihcmbohedrcn. 

 The diagram of § 9 (Fig. 7), imagined to be altered lo proper 

 proportions fcr the present case, may be locked to 

 for illustration. lis three face-diagonals through p, 

 being PD, pe, pf, are perpendicular to one another. 

 So also are Q\, qb, qc, its three face-diagonals 

 through Q. The body-diagonal of the cube PQ, being 

 half the body diagonal of the cube whose edges are 

 PD, PE, PF, is equal to PD x i ,^'3 : and pa, pb, PC are 

 also each of thtm equal to this, because A, B, c are 

 centres of other equal cubes, having p for a common 

 corner. 



§ 14. The tetrahedrons used in the model are those 

 into which the parallelepiped is cut by three planes 

 through the axial diagonal, which in this case cut one 

 another at angles of 60°. We wish to be able to 

 shift the tetrahedrons into positions coriesponding to 

 those of the triangles in Fig. i, which we could not 

 do if they, were cut out of the solid. I, therefore, 

 make a mere skeleton of each tetrahedon, consisting 

 of a piece of wire bent at two points, one-third of 

 its length from its ends, at angles of 70^^ being 

 sin~^^,^'3, in planes inclined at 60° to one another. 

 The six skeletons thus made are equal and similar, 

 three homochirals and the other three also homo- 

 chirals, their enantiomorphs. In their places in the 

 primitive parallelepiped they have their middle lines 

 •coincident in its axial diagonal PQ, and their other 

 6x2 arms coincident in three pairs in its six edges 

 thrcugh p and Q. Looking at Fig. 7 we see, for 

 example, three of the edges CP, PQ, QE, of one of 

 the tetrahedrons thus constituted ; and D'j, QP, PB, 

 three edges of its enantiomorph. In the model they 

 are put together with their middle lines at equal dis- 

 tances around the axial diagonal and their arms 

 symmetrically arranged round it. Wherever two lines 

 cross they are tied, not very tightly, together by thin 

 cord many times round, and thus we can slip them along so as to 

 ibring the six middle lines either very close together, neaily as 



NO. 1272, VOL. 49] 



they would be in the prinilive parallelepiped, or farther and 



farther out from one another so as to give, by the four corners 



of the tetrahedrons, the twenty-four corners of 



all possible configurations of the plane-faced 



space-filling tetrakaidekahedron. 



§ 15. The six skeletons being symmetrically 

 arranged around an axial line we see that each 

 arm is cut by lines of other skeletons in three 

 points. For an important configuration, let the 

 skeletons Ijc separated out from the axial line just 

 so far that each arm is divided into four equal 

 parts, by those three intersectional points. The 

 tetrakaidekahedron of which the twenty-four 

 corners are the corners of the tetrahedions thus 

 placed may conveniently be called the orthic 

 tetrakaidekahedron. It has six equal square 

 faces and eight equal equiangular and equilateral 

 hexagonal faces. It was described in § 12 of 

 my paper on "The Division of Space with 

 Minimum Partitional Area" {Philosophical 

 Magazine 1887, second half year, and Acta 

 Mathematica, vol. xi. pp. 1 21 -1 24). under the 

 name of "plane-faced isotropic tetrakaideka- 

 hedron"; but I now prefer to call it oithic, 

 because, for each of its seven pairs of parallel 

 faces, lines forming corre.'^pondiiig points in the 

 two faces are perpendicular to the faces, and the 

 planes of its three pairs of square faces are perpen- 

 dicular to one another. Fig. S represents an 

 orthogonal projection on a plane parallel to one 

 of the four v airs of hexagonal faces. The heavy 

 lines are edges of the tetrakaidekahedron. The 

 light lines are edges of the tetrahedions of § 13, 

 or parts of those edges not coincident in projec- 

 tion with the edges of the telrekaidekahtdron. 

 The figures i, i, i ; 2, 2, 2 ; . , . ; 6, 6, 6 

 show corners bel )nging respectively to the six 

 tetrahedrons, two of the four corners of each 

 being projected on one point in the diagram. 

 Fig. 9 shows, on the same scale of magnitude 

 with corresponding distinction between heavy and 

 light lines, the orthogonal projection on a plane parallel to a 

 pair of square faces. 



§ 16. If the rule of § 15 with reference to the division of each 

 arm of a skeleton tetrahedron into four equal parts by points in 



