BREDER: FISH SCHOOLS AS OPERATIONAL STRUCTURES 



Figure l.-Two space lattices in perspective, each with a single 

 cell shown as a solid. A. The cubic lattice. B. The rhomboidal 

 lattice. The arrow indicates the manner of transformation by 

 which the cubic lattice becomes the rhomboidal lattice. 



the transformation from cube to rhombohedron. 

 All the angles in this rhombic lattice are either 60° 

 or 120°, SO transformed from the cubic lattice with 

 only angles of 90°. On the floor of the cubic lattice 

 in Figure lA, the nearest points to the central one, 

 in the same plane, are four in number. These are 

 connected to each other by a dotted line. On the 

 floor of the rhombohedron in Figure IB, the sides 

 of which have internal angles of 60° and 120°, the 

 nearest points to the central point include four at 

 the corners of the dotted parallelogram plus two 

 more, indicated by the dark points. These define a 

 regular hexagon because the parellelograms are 

 composed of two equilateral triangles. 



If models of identical fishes are stationed with 

 their centers at each lattice point, and if all the 

 models are in parallel orientation, the group 

 superficially resembles a fish school. It becomes 

 immediately apparent however that such a lattice 

 of fishes has characteristics that are never seen in 

 a school. If they had ever been seen in such a 

 formation, their appearance would have been so 

 striking that the details of the regimentation 

 would have been recorded long ago. In such a 

 school, viewed from above, fish would be seen in 

 horizontal files and these files would be swimming 

 ahead in rows transverse to their direction of 

 travel. Viewed from the side, each fish within the 

 school would have another directly above and 

 another directly below it, forming columns, except 

 the two fish marking the upper and lower limits of 

 the school in each vertical column of fishes. These 

 two would be without another fish above and 

 below, respectively. Thus we can temporarily put 

 this unschoollike lattice aside. 



Fish models positioned at the points of the 

 rhombic lattice do not show the peculiar features 

 seen in the cubic lattice, but have a more distinct 

 resemblance to fish schools. It is difficult to deny 



that schooling fishes, in most situations, are indeed 

 approximating this configuration, the details of 

 which will be discussed later. 



Turning now from space lattices to the packing 

 of space, it is easy to arrive at the above rhombic 

 lattice by a very different route. As a preliminary 

 mathematical simplification, fishes and the im- 

 mediately surrounding water that envelops each 

 fish individually in a school shall be equated to 

 spheres, the centers of which are located on the 

 axis of the fish midway between the end of the 

 snout and the tip of the tail. Here it is necessary to 

 describe some of the less obvious geometrical 

 features of a mass of spheres packed together as 

 closely as possible. A single layer of identical 

 spheres on a plane surface packed at maximum 

 density may be represented on paper by an 

 equivalent packing of circles (Figure 2). A hexa- 

 gon may be circumscribed about each circle, one 

 of which is shown in the lower left corner. 



Figure 2.-The closest possible packing of a single layer of 

 identical spheres or circles, showing the relationships to hexa- 

 gons and their six equalateral triangles as well as the disposi- 

 tion of a single diameter in each circle when drawn radiating 

 from the center of the circle with the circumscribed hexagon. 



The individual diameters of each circle as shown 

 in Figure 2 lie along radiating lines emanating 

 from the common center of the hexagons.^ Those 

 lying on the radials passing through the apices of 

 the larger hexagon are continuous lines (major 

 axes), while those passing through the equivalent 

 points on the smaller hexagon are dashed lines 

 (minor axes). If these diameters are all permitted 

 to become parallel to one another, a very different 



^Although simple, this geometric treatment of transforma- 

 tions of related diameters of packed circles or spheres is 

 evidently original here, or at least no approach to this treatment 

 has been found. No formal proofs are necessarv as the usage here 

 is simple enough to be self-evident and would be irrelevant to 

 present purposes. 



473 



