BREDER: FISH SCHOOLS AS OPERATIONAL STRUCTURES 



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Figure 6.-The rhombohedral sphere pack of three layers as 

 viewed from above. Based on Figures IB and 2. See text for full 

 description. 



figure parts of the adjacent outlines of four 

 spheres of the first layer are shown by solid lines. 

 The dotted lines of three overlying spheres of the 

 second layer are also shown. This indicates clear 

 vertical passages through the overlying junctures 

 that permit passage through the two layers of 

 spheres (small open circles) while two are blocked 

 by overlying spheres (small dark circles). The 

 pattern is repeated throughout the system. A third 

 layer of spheres may be identical with the first, a 

 fourth layer identical with the second, and so on 

 indefinitely. This pattern preserves the integrity 

 of the vertical passages, but this need not be the 

 case. If the second layer of sphere centers ex- 

 changes the position of the black and open small 

 circles, the clear passages occur where the black 

 circles are now shown and vice versa. As any layer 

 may be so reversed the passages may be blocked in 

 many complicated patterns. The shortest possible 

 passage can be the vertical distance between the 

 level of the centers of two adjacent layers of 

 spheres, otherwise the passage may be indefinitely 

 long. 



As these planes, referred to above as layers, 

 form the faces of the generating rhombohedron 

 shown in Figure IB, these passages run in three 

 intersecting directions, as do the three planes of 

 the lattice. The passages are all interrelated, as 

 altering the relationships of the sphere centers in 

 one plane automatically alters those in the two 

 others. 



The above may be simpler to visualize by refer- 

 ring to the perspective illustration of Figure 7. 



Figure 7.-The rhombohedral packing of spheres in perspective, 

 showing only sphere centers. Two other sets of planes could be 

 drawn through these centers at angles determined by the sides of 

 the generating cell, two of which are shown between layers 4 and 

 5. See text for full explanation. 



Here it has been necessary to completely alter the 

 symbols used in Figure 6 owing to other needs. 

 Plane 1 of Figure 7 is identical with the first layer 

 of Figure 6. The hexagon of Figure 6 is shown in 

 Figure 7 as one of dotted lines. Planes 1, 2, and 3 of 

 Figure 7 represent the corresponding layers of 

 Figure 6. The two added planes, 4 and 5, show more 

 realistically the vertical passage running from A 

 to A. It has clearance through the first three planes 

 but is blocked at plane 4 and runs clear through 5. 

 Note that plane 4 is "reversed" from 2, which is the 

 reason for the blockage. The passage from B to B is 

 blocked by planes 1, 3, and 5, but not by 2 and 4. 



The indications of the rhombohedral cells by 

 dotted lines between planes 4 and 5 clearly show 

 how two additional sets of planes could be passed 

 through the points. 



A perspective view of the simpler cubic packing 

 of spheres is shown in Figure 8 for comparison 

 with Figures 1, 4, 5, and 7. Only four planes are 

 shown, as more are unnecessary. It is evident that 

 the cubic cell and consequent total right angled 

 construction precludes any of the rhombic 

 complications. 



These two systems of packing spheres are all 



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