152 



♦ KNOW^LKDGE ♦ 



[May 2, 1887. 



the method of building up, illustrated in fig. 1, we have a 

 series of layers parallel to the triangular face abc. But 

 also manlfestlj- we have a series parallel to those faces of the 

 built-up pyi-amid which have ab, bc, and AC respectively as 

 base. Hence there are four directions in which parallel 

 layers can be slid. And clearly when the faces of tetra- 

 hedrons and octahedrons agree (after sliding layers in any 

 of these four ways), we can slide strips such as the three 

 AFiiB aced, ADLC abef, bekc clhcf, in the directions of their 

 length ; while corresponding strips can be carried in other 

 three directions for each of the three plane positions 

 corresponding to the three other faces of the tetrahedron, 

 having ab, bc, and CA, respectively, as base. 



T/ii]-dli/,\vhen space is built up of the solid, considered in 

 the solution of Problem XXIV., no sliding or shifting is 

 possible, without breach of continuity. 



II.— PROBLEM OF THE TIOXEYCOMB CELL-ENDS. 



[Why Puzzle XXVI. is thus called will be seen further on.] 

 Puzzle XXVI. The reader will have noticed a slight 

 error in the statement of this puzzle. The arrangement in 

 the second box of oranges as described in the February 

 number is not symmetrical like that of the oranges in a 

 triangular pyramid. It dift'ere in that the centi-es of any 

 row of oranges are vertically over those of the row next but 

 one above and below, whereas in the triangular pj'ramid of 

 oranges the centre of an orange is vertically over the space 

 between three oranges in the layer next but one below, as 

 well as in the layer next below. It is this last arrangement 

 we have to deal with, in which the oranges are set as in the 

 first and third arrangement, so far as regards the set of 

 oranges touching any given orange. 



The only difficulty with this puzzle is in making the solu- 

 tion clea r without occupying too much space or giving more 

 time to the matter than it m.ay seem to be worth, though 

 the study of it affords excellent geometrical gymnastic. 



Figs. 2 and ;? show two ways of viewing a globe sym- 

 metrically surrounded by twelve equal globes touching it 

 and each other. The globes in the two figures are lettered 



Fig. 2. 



alike. In fig. 2 the four heavy circles are supposed to 

 represent eight globes, four in the uppermost layer, four 

 below these. The layer between contains five globes in the 

 form of a cross, the middle one being the surrounded globe 

 we are chiefly c onsidering. The four dotted circles repre- 

 sent globes necessary in building up the group, but not 

 among those which touch the middle globe. In fig. 3 the 

 three dotted globes are the lowest layer, the lightly mai-ked 

 ones the middle layer, and the heavy ones the top layer. 

 Small italic letters mark the centres of globes belonging to 

 lower layers. 



In fig. 4 the circle abcd, and in fig. 5 the circle Ae/cou, 

 are supposed to be orthogonal views of the surrounded 



globe, with the points where the several globes surrounding 

 it touch its surface. (")utline dots show points of contact 

 seen through the globe. At e, f, g, and h (fig. i) two points 



Fig. 3. 



coincide ; this is shown by circles outside dots. Fig. 2 

 accounts for the arrangement of the dots marking contact 

 points in fig. 4, while fig. 3 accounts for that of the corre- 

 .sponding points in fig. 5. The reader who cannot after a 

 little thought demonstrate this, and understand details, 

 would not care to follow a verbal demonstration. But 

 indeed the figures will explain and demonstrate by themselves 

 the relations dealt with, for all readers of geometrical 

 proclivities — and no others are likely to read the.se lines. 



Now it is clear that, when the globes all simultaneously 

 expand, the pressing surfaces will Ijecome plane, and those 

 inclosing the expanded globe abid, figs. -1 and 5, will touch 



Fio. i. 



Fis. 5. 



the sphere .\bcd at the points of original contact with neigh- 

 bouring globes. 



This premised, we see that the planes touching the sphere 

 .ABCD at E, F, G, H I both above and below the plane abcd ) 

 meet at a point in o, vertically above the centre of the 

 sphere as seen, and have the apparent forms oakb, oblc, 

 OCMD, and ODXA — which are squares — but being really 

 rhombuses having shorter diameters ab, bc, cd, and da in 

 real length, and longer diameters ok, ol, cm, and on m 

 apparent length. These longer diameters are obviously 

 slanted 45° to the line of sight (consider fig. 2, to see that 

 this must be so), hence their real length will be aKs/ 2. 

 Calling the common radius of the original globes r, the 

 longer and shorter diagonals of these rhombuses or diamonds 

 will be respectively v/2;- and 2r. We perceive, further, 

 that the plane-faces touching abcd at a, b, c, and D, are 

 foreshortened into the straight lines nk, kl, l.m, and nm, 

 which, regarded as straight lines, are diagonals of other 

 rhombuses, and are each equal to 2;'. 



Our solid figure has then eight rhombus faces, all equal, 

 viz., the four supposed to be seen in fig. 4, and the four on 

 the farther side, which, if visible, would have the same out- 

 lines in the orthogonal or perspective representation. The 

 figure has four other rhombus faces whose shape has still to 

 be determined. But we know already that each of these 

 four other faces has a diameter equal to the longer diameter 

 of the eight faces already determined. 



Fig. .5 enables us to determine the other diameter. We 

 see that the planes touching the sphere abcd, fig. 5, in the 



