FISHERY BULLETIN: VOL. 74, NO. 3 



Figure 8.-The cubic packing of spheres, directly comparable 

 with Figure 7. See text for full explanation. 



that will be considered here, as all others are much 



looser and are not relevant to this study. The 



density of these two and the number of contacts 



that interior spheres have with others are given 



below. 



Percent of volume N umber of contacts 

 Packing occupied of each sphere 



Rhomboidal 

 Cubic 



0.740 

 0.513 



12 



6 



The number of contacts indicated here are iden- 

 tical with the number of "nearest neighbors" 

 mentioned in reference to the equivalent space 

 lattices. 



Pitcher's (1973) data on clusters of spheres 

 presented another way of explaining the com- 

 plications of close sphere packing. It emphasizes 

 the measurements from center to center, with 

 which he was working, rather than the overall 

 pattern of a larger group, which emphasizes the 

 layering effect of polarized parallel diameters 

 discussed here. 



Structure and Functioning of 

 Natural Schools 



The series of diagrams in the preceding section 

 is virtually a key to determining what, if any, 



space lattice a given school of fishes could approx- 

 imate and it clearly indicates what types of space 

 lattices do not find their embodiment in fish 

 schools. Reason and observation also indicate that 

 school-forming fishes establish their schools 

 rapidly with great unanimity of action. The 

 schools come to stability only after each individual 

 has the common orientation, all normally as close 

 together as the spatial requirements of their 

 individual propulsive acts permit. The organiza- 

 tion is strictly one formed in this manner and 

 without any of the differential behavior that more 

 complex lattices would require. 



Pitcher (1973), by purely empirical means, ar- 

 rived at the geometrical relationships of a school 

 of Phoxinus phoxinus (Linnaeus) identical with 

 the present formal lattice reached by theory. His 

 fishes fit our theoretical operations even better 

 than any of the fishes checked for this study. Our 

 material all showed some attenuation of the lattice 

 along the axis of travel, which also was the case in 

 Weihs (1973a). This may simply mean that Phox- 

 inus keeps a tighter school than any species we 

 checked, or that there is some small effect here 

 that relates to speed of fish and their absolute size. 

 Possibly, however, it may be related to a differ- 

 ence in behavior between a school swimming 

 ahead in quiet water and one holding a stationary 

 position in flowing water, as did Pitcher's fish. In 

 the latter, optical fixation on fellow fishes and 

 some background feature is possible, but in the 

 former, fixation is only possible on other members 

 of the school as the background apparently drifts 

 backward. If this effect does modify the spacing of 

 the fishes, stationary schools in fast flowing rivers 

 where backgrounds are visible should more closely 

 approach the theoretical. 



Spacing of Fishes 



Using the preceding examination of lattices and 

 the packing of spheres, a preliminary comparison 

 with fish schools may start by continuing the 

 equating of fishes in a school to the diameters of 

 the packed spheres. Schooling fishes should not be 

 expected to space themselves exactly as spheres 

 and they do not do so in precise detail, see Pitcher 

 (1973), but a basic resemblance exists. 



If the rigid sphere of geometry be mentally 

 replaced by a soft rubber ball, the approximation 

 comes closer to that of a fish embedded in a school 

 of its fellows. Thus a group of such balls, when 

 packed together, are subjected to slight flattening 



476 



