BREDER: FISH SCHOOLS AS OPERATIONAL STRUCTURES 



and to other minor distortions where contacts are 

 made with other balls, all proportional to the 

 amount of pressure and its direction. The pattern 

 of lattice considered here as closest to the spatial 

 distribution commonly shown by schooling fishes 

 can be reached by very simple transformations. 



The calculations that equated the diameters of 

 the spheres to the fishes' lengths can be altered. 

 Here the lengths are changed but the positions of 

 fishes in space remain the same. 



A change that evidently does occur regularly 

 involves altering the angles in the quadrilateral 

 mesh composed of two triangles as illustrated in 

 Figure 9, where A and B represent the quadri- 

 laterals in Figure 1, and C represents a quadrila- 

 teral that has been used by Weihs (1973a) in 

 connection with his studies on vortex streets. It is 

 called simply a "diamond" by that author. His 

 model resulted from considerations of energy 

 saving requirements. The Weihs (1973a) diamond 

 can be used as a very convenient basic unit or celP 

 characteristic of the fish school lattices, without 

 altering any of concepts discussed here. At this 

 writing, all known changes from the conditions of 

 regular geometrical figures are on the side of 

 increased differences between the two pairs of 

 angles of the diamond. No instances have been 

 found in real fish schools that would lie between 

 case A and B of Figures 1 and 9, unless the 

 widespread separations which have been con- 

 sidered as degenerating schools are included. All 

 other variations found are on the far side of B 

 except for the data of Pitcher (1973), which is 

 precisely at B. In Figure 9, A shows the square 

 pattern with 90°, B shows the 60°, 120° rhombus, 

 and C shows a rhombus with 30°, 150° which 

 depicts a condition frequently seen in fish schools 

 and is, as already indicated, the Weihs (1973a) 

 diamond. Carrying this angular reduction further, 

 the end is reached as the side to side distance 

 between fishes is reduced to zero, so that the total 

 length of the figure becomes a single line equal to 

 twice the length of a side of the diamond. At the 

 other end of this series of quadrilaterals, an 

 increase beyond 90° produces another series. In 



^In most schooling fishes two individuals, if isolated from the 

 others, will swim together side by side or with one diagonally 

 ahead of the other. If three fish are so isolated, they will normally 

 form a pattern of three points of a diamond. In this case there is 

 usually much more shifting around than in the case of two, while 

 four fish tend to form a diamond. It has been a common practice 

 for workers in this field to consider these cases of very small 

 schools. From groups of less than four, it is impossible to make 

 any reasonable estimate of the shape of the diamond. Some 

 judgment can normally be obtained from a group of four, 

 although even that might vary somewhat from a school. 



Figure 9. -Three quadrilaterals (lattice elements) as related to 

 "diameters" or "fish lengths." See text for full explanation. 



this case the final result is also a single line, equal 



to twice the length of a side but at right angles to 

 the one reached at the other extreme of the series, 

 as described above. 



Figure 10 shows how these matters relate to the 

 hexagons and how the quadrilaterals relate to an 

 entire school. Each small circle in the upper row of 

 three diagrams represents the midpoint of each 

 fish. The four fishes, each on a diamond point, are 

 represented by heavy horizontal lines represent- 

 ing the fish lengths. The direction of swimming is 

 understood to be from left to right. All the others, 

 shown only by the small circles, are moving paral- 

 lel to and in the same common direction as the four 

 indicated. Starting at A with a square and passing 

 to B composed of two equilateral triangles, the 

 series terminates at C with acute angles of 30°, 





Figure lO.-The relations of the three quadrilaterals shown in 

 Figure 9 to the station points in a school and to the corresponding 

 hexagons (upper row). The clear turning sectors and those 

 requiring a too close mutual approach are shown in the lower row. 

 The latter are marked by their two radii and arc by a heavy solid 

 line. Their axes, the lines of contact, are marked by dashed radii. 

 See text for full explanation. 



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