BREDER: FISH SCHOOLS AS OPERATIONAL STRUCTURES 

 0° 0' 



//// 



// 



A 



--7 



\ 



Figure 16. -Angles of sharp-turning fish schools. A. Angles 

 compared with the rhomboidal lattice. The four solid radial lines 

 represent the collision paths of turns if the original path is 

 represented by the vertical line marked 0°. This direction of a 

 fish's path is indicated by the arrowhead. The dashed radials 

 marking the end of each arc separate the clear sectors, without 

 arcs, from the occluded. The 11 short radial line segments 

 represent the new path of the fishes after they have made their 

 sharp turn. The numerical values of the angles are given in Table 

 2B. The same fish turns compared with the cubic lattice. Here the 

 fish paths are not limited to the clear sectors. See text for full 

 explanation. 



interfering with each other's swimming. Actual 

 turns of various species keep well away from the 

 critical angle. Which particular clear space is 

 selected is evidently determined, at least in part, 

 by the strength of the deflection-causing stimulus. 

 As such a turn is completed, the fish again start to 

 swim in an essentially straight line while they 

 regain the positions that were somewhat dis- 

 turbed in turning and the Weihs (1973a) diamond 

 appears again. Thus the outlined sectors in Figure 

 16A become "forbidden" paths. Since the diagram 

 in this figure is purely a geometrical construction 

 with the occluded and clear sections having equal 

 areas, this is not to say that some intrusion into 

 the outlined sectors is impossible. The axis of the 

 occluded sectors is the worst position for turning 

 and that of the clear sector the best, the areas 

 between grading gradually from one condition to 

 the other. The dotted radii are halfway between 

 the center lines of the clear and the occluded areas. 

 The turns made by real fish schools, measured by 

 motion picture analysis, and shown in Figure 16A 

 and Table 2A indicate the absence of intrusion 

 into the enclosed areas. 

 This examination of the sharp turnings of fish 



schools would not have shown these features if 

 they had been organized on some pattern other 

 than that of the hexagonal lattice. If they had been 

 organized on the square lattice, shown in Figure 4, 

 there would have been at least some in the "for- 

 bidden" sectors, as is shown in Figure 16B where 

 the same data on turning angles have been placed 

 on a diagram based on the square mesh. Here the 

 same data show less preferential behavior on the 

 part of the fishes toward the clear sectors. All the 

 schools, in the hexagonal case, stayed within the 

 boundaries of the clear sectors (Figure 16A) while 

 only 64-1-% did in the square case (Figure 16B). 

 Also the intrusion into the occluded sections 

 increased with the increasing angle between the 

 initial course and the new one. These two items are 

 additional reasons for considering the lattice to be 

 basically hexagonal. 



A typical turn of the sort discussed is shown in 

 Figure 17 and in Table 2B. This drawing is based 

 on a series of seven motion picture frames (0.44 s). 

 The sequences are of a tight school, the angles 

 between the straight paths, before and after the 

 turn, are based on the mean paths of the fishes. 

 Only a few of the individual fishes are shown in 

 Figure 17 to indicate the nature of the turn at that 

 point. Not shown are the many fishes constituting 

 the bulk of the school. 



There is also another type of sharp turn that is 

 not mentioned in the preceding description. It can 

 lead to considerable confusion because superficial- 

 ly it is readily confounded with the foregoing type. 

 It differs primarily in not being concerned with 

 angular limitations, which apparently can be 

 ignored only at the expense of making the turn 



^^ 



^^ 



^^ 



Figure 17.-A sharp turn of Selar crnmenophthaimus. Only the 

 paths before and after the turn are indicated and a few of the 

 turning fishes. The directions of the two paths are indicated by 

 arrowheads. See text for full explanation. 



485 



