FISHERY BULLETIN: VOL. 74, NO. 3 



which is the most compressed of the three. Also 

 shown are the relations between the quadrilater- 

 als and the corresponding hexagons, as well as the 

 number of fishes in a given area. 



Continuing with Figure 10, it is obvious that the 

 direction of travel could be in any other horizontal 

 direction of swimming, than the one shown here. 

 It should be noted however, that the lattice of each 

 shows that if the fish turned so as to be parallel 

 with any edge of their parallelogram, the fishes 

 would all be brought to the nose to tail position, 

 something which does not occur. 



In the lower row of three corresponding dia- 

 grams in Figure 10 the dashed radial lines show 

 the directions of swimming that would place the 

 fish in contact. The clear spaces indicate where the 

 passages are unobstructed. The enclosed areas, 

 which surround the dotted lines of contact, meet 

 the clear areas at a point halfway between that 

 line and the centers of the clear areas, except in C 

 which is not based on a regular quadrilateral or 

 hexagon. This will be further discussed under 

 Problems of a School Turning. 



In any school, a certain minimum distance from 

 the nose of a following fish to the tail of a leading 

 fish is maintained. The evident need for this 

 separation is natatorial. Requirements differ with 

 the various types of fishes that form schools. 

 Although fishes do not leave wakes behind, as does 

 a motor-propelled ship, there is still the matter of 

 dying vortices (Rosen 1959; Breder 1965). This 

 alone could account for the need of a spatial lead. 

 Conceptually, fishes could swim satisfactorily on 

 any of the diameters shown in Figures 2 and 3, 

 except those on the major axial lines. The min- 

 imum distances between these diameters (fishes) in 

 a line occur halfway between these axes as in 

 Figure 2. It is to be noted also that the horizontal 

 rows of diameters tend to line up so that the 

 diameters are not all the same distance from each 

 other as in Figure 3A. This change continues with 

 angles less than 15° so that when these diameters 

 become horizontal they are in end-to-end contact, 

 producing a series of parallel lines. This is merely a 

 matter of the geometry of the uniform rotation of 

 the diameters. No schooling fishes would tolerate 

 this condition, but would adjust their positions to 

 lie near midway between the positions of those 

 lateral to them, as shown in the diagrams of 

 Figure 9. Compare Figure SB with Figure 9C. The 

 apparent differences between the two are entirely 

 owing to the fact that the first diagram is based on 



rigid circles, or spheres, and the second does not 

 have that heavy stricture. The three quadrilaterals 

 in Figure 9 can be considered as making a closed 

 curvilinear figure, where Figure 9A would be 

 circumscribed by a circle while Figures 9B and 9C 

 would both be circumscribed by ellipses, Figure 9C 

 being much narrower than Figure 9B. This trans- 

 formation can be brought about by increasing the 

 head-to-tail distances of the fishes in a single file 

 and decreasing the distances between adjacent 

 files. 



The greatest width between the tracks of fishes 

 swimming parallel is also at the halfway angle 

 between two successive axes, as shown in Figure 

 3A. As long as all the fishes are swimming in 

 parallel courses the distance need not vary, as seen 

 in Figure 3A. The closer this angle approaches an 

 axis, the smaller becomes the distance between the 

 parallel tracks, indicated in Figure 3B. The dis- 

 tance between fishes, head to tail, varies inversely 

 as an axis is approached. 



Still photographs cannot give the sense of a 

 regular pattern of fishes that is evident on viewing 

 a school or a motion picture. Because of these 

 conditions, in those photographs shown here 

 sufficiently open to see the fishes distinctly, they 

 appear as rather ragged groups. Thus in Figure 11 

 of Katsuwonus pelamis (Linnaeus), only frag- 

 ments of some regularity of pattern can be seen. 

 Those on the left of center show the pattern of a 

 loose school while those on the right are breaking 

 ranks for feeding. This picture, however, indicates 

 several lines of fish alignment, some running from 

 top downwards to the right and others to the left, 

 from which the relationship to the diagram in 

 Figure 7 can be seen within the limits of a still 

 picture. 



Species attaining very large size, such as Thun- 

 nus thynnus (Linnaeus), tend to have dispropor- 

 tionally greater distances between individuals 

 when large, as compared to their younger and 

 smaller sizes (see Breder 1965). Contrary to this, 

 van 01st and Hunter (1970) showed that other 

 smaller fishes {Scomber, Engraulis, Trachurus, 

 and Atherinops) tighten their ranks as they grow 

 from larvae to near adult size, some abruptly and 

 others gradually. 



Hunter (1966) presented some data on the or- 

 ganization of fish schools for purposes that do not 

 concern present interests. However these data, 

 based on motion picture analysis shown in his 

 figure 2, have a distinct bearing on some features 



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