March 8, 1894] 



NATURE 



445 



bridgeshire County Council. The scheme was thrown out— on 

 financial grounds— by the Senate, and here it seemed likely that 

 acricuhural education would come to a standstill, had it not been 

 for the action of the County Councils of the Eastern Counties,' 

 who, with the help of certain University professors, organised 

 the Cambridge and Counties Agricultural Education Commit- 

 tee, an arrangement by which the counties supply the funds, 

 while the University members supply the teaching. Under 

 this scheme agricultural students are now receiving at Cam- 

 bridge instructton in a number of subjects bearing directly on 

 agriculture. The students are not necessarily members of the 

 University, nor is agriculture a recognised department of Uni- 

 versity study; but it has now been practically sanctioned by the 

 appointment of a University syndicate, whose duly it is to 

 superintend the examinations on which the new diploma is to 

 be granted. This procedure has a precedent in the successfully 

 established diploma in State Medicine, and cannot fail to exert 

 — both as a check and a stimulus — a wholesome effect on the 

 unofficial agricultural department. 



The first examination will be held in July. It consists of two 

 parts : Part i. embraces botany, chemistry, physiology, entomo- 

 logy, geology, engineering, and book-keeping, in so far as each 

 subject bears on agriculture. Part ii. comprises practical 

 agriculture and surveying. The examinations are open to all 

 who present themselves, and who pay the moderate fee de- 

 manded. Intending candidates may, it seems, obtain informa- 

 tion from Prof. Livein:,' (who has taken the chief share in the 

 work from the University side of the question) or from Mr. 

 Francis Darwin. 



ON HOMOGENEOUS DIVISION OF SPACE. 



§ I. n^HE homogeneous division of any volume of space 

 means the dividing of it into equal and similar 

 parts, or cells, as 1 shall call them, all sameways oriented. If 

 we take any point in the interior of one cell or on its boundary, 

 and corresponding points of all the other cells, these points form 

 a homogeneous assemblage of single points, according to 

 Bravais' admirable and important definition. •* The general 

 problem of the homogeneous partition of space may be stated 

 thus : — Given a homogeneous assemblage of single points, it is 

 required to find every possible form ol cell enclosing each of 

 them subject to the condition that it is of the same shape and 

 sameways oriented for all. An interesting application of this 

 problem is to find for a crystal (that is to say, a homogeneous 

 assemblage of groups of chemical atoms) a homogeneous 

 arrangement of partitional interfaces such that each cell con- 

 tains all the atoms of one molecule. Unless we knew the exact 

 geometrical configuration of the constituent parts of the group 

 of atoms in the crystal, or crystalline molecule as we shall call 

 it, we could not describe the partitional interfaces between one 

 molecule and its neighbour. 



Knowing as we do know for many crystals the exact geometrical 

 character of the Bravais assemblage of corresponding points of 

 its molecules, we could not be sure that any solution of the 

 partitional problem we might choose to take would give a cell 

 containing only the constituent parts of one molecule. For 

 instance, in the case of quartz, of which the crystalline mole- 

 cule is probably 3(SiO._,), a form of cell chosen at random 

 might be such that it would enclose the silicon of one molecule 

 with only some part of the oxygen belonging to it, and some of 

 the oxygen belonging to a neighbouring molecule, leaving out 

 some of its own oxygen, which would be enclosed in the cell 

 of either that neighbour or of another neighbour or other 

 neighbours. 



§ 2. This will be better understood if we consider another 

 illustration — a homogeneous assemblage of equal and similar 

 trees planted close together in any regular geometrical order on 

 a plane field either inclined or horizontal, so close together that 

 roots of different trees interpenetrate in the ground, and j 

 branches and leaves in the air. To be perfectly homogeneous 



1 The scheme is now carried on by funds supplied by the County j 

 Councils of Cambridgeshire, the Isle of Ely, Essex, Norfolk, Northants, 

 Leicestershire, Hunts, East and West Suffolk, and by a grant from the 

 Board of Agriculture. 



- A paper read before the Royal Society on Januaiy i8, by Lord 

 Kelvin, P.R.S. 



'^Journal de I'Ecolc PolytechniqHC, tome xix. cahier 33, pp. 1-12S (Paris, 

 1850), quoted and used in my "Mathematical and Physical Papers," vol. 

 iii. art. 97, p. 400. 



NO. 127 f, VOL. 49] 



every root, every twig, and every leaf of any one tree must 

 have equal and similar counterparts in every other tree. So far 

 everything is natural, except, of course, the aljsolute homo- 

 geneousness that our problem assumes ; but now, to make a 

 homogeneous assemblage of molecules in space, we must sup- 

 pose plane above plane each homogeneously planted with trees 

 at equal successive intervals of height. The interval between 

 two planes may be so large as to allow a clear space above the 

 highest plane of leaves of one plantation and below the lowest 

 plane of the ends of roots in the plantation above. We shall 

 not, however, limit ourselves to this case, and we shall suppose 

 generally that leaves of one plantation intermingle with roots 

 of the plantation above, always, however, subject to the con- 

 dition of perfect homogeneousness. Here, then, we have a 

 truly wonderful problem of geom.etry — to enclose ideally each 

 tree within a closed surface containing every twig, leaf, and 

 rootlet belonging to it, and nothing belonging to any other tree, 

 and to shape this surface so that it will coincide all round with 

 portions of similar surfaces around neighbouring trees. Won- 

 derful as it is, this is a perfectly easy problem if the trees 

 are given, and if they fulfil the condition of being perfectly 

 homogeneous. 



In fact we may begin with the actual bounding surface 

 of leaves, bark, and roots of each tree. Wherever there is 

 a contact, whether with leaves, bark, or roots of neighbour- 

 ing trees, the areas of contact form part of the required 

 cell-surface. To complete the cell-surface we 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 bj called a tetrakaidekahedron, 

 unless we prefer to call it a fourteen-walled cell. Each wall is 

 an interface between one ceil and one of fourteen neighbours. 

 Each of the thirty-six edges is a line common to three neigh- 

 bours. Each of the twenty-four corners is a point common to 

 four neighbours. The old-known parallelejjipedal partitioning 

 is merely a very special case in which there are four neighbours 

 along every edge, and eight neighbours having a point in 

 common at every corner. We shall see how to pass (§ 4) con- 

 tinuously from or to this singular case, to or from a tetrakaide- 

 kahedron differing infinitesimaliy from it ; and, still continuously, 

 to or from any or every possible tetrakaidekahedronal 

 partitioning. 



§ 4. To change from a parallelepipedal to a tetrakaideka- 

 hedronal 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 parall el edges which intersect it in its 

 two ends. Give now any purely translational motion to each of 

 these six tetrahedrons. We have now the 4 x 6 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 coincided 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 hexagons, 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 quadrilaterals, with arbitrary curves for the 

 edges between hexagon and hexagon and between hexagons and 

 quadrilaterals, and then by drawing parallel equal and similar 

 counterparts to these surfaces in the remaining four hexagonal 



1 Compare " Mathematical and Physical Papers," vol. iii. art. 97, § 5., 



