FISHERY BULLETIN: VOL. 74, NO. 3 



of their geometrical characteristics has been 

 undertaken. Much of the older literature on the 

 distribution of individuals of a population, or 

 smaller group, of animals or plants took for 

 granted that the deployment is stochastic. Clark 

 and Evans (1954) stated, "This assumption is no 

 longer a tenable one and is probably even less 

 applicable to animal populations." It is, of course, 

 doubtful if creatures with well organized locomo- 

 tor abilities and complex sensory systems are ever 

 distributed in a fully random manner. The systems 

 encountered in nature seem to be mostly those of 

 ordered arrays variously distorted by processes of 

 many kinds, sometimes obvious, but more often 

 obscure or barely discernible. Attempts to mea- 

 sure the structure of assemblages of individuals 

 have been predicated mostly on the idea of show- 

 ing the extent of their departures from theoretical 

 randomness. Since fully organized fish schools 

 have very obvious structure, it is at least equally 

 appropriate to compare them with mathematically 

 organized patterns, especially where there are 

 good theoretical reasons to expect the presence of 

 some similarity. 



Geometrical Models 



The establishment of a geometrical model of a 

 fish school is relatively simple, for whatever else a 

 fish schooF may be, it is essentially a closely 

 packed group of very similar individuals united by 

 their uniformity of orientation. A more explicit 

 definition has been given by van 01st and Hunter 

 (1970) who stated, "The principal characteristics of 

 the organization of fish schools are that the in- 

 dividuals stay together, tend to head in the same 

 direction, maintain even spacing, and the activi- 

 ties of the individuals tend to be synchronized." 

 Because of the nature of fish locomotion it is 

 necessary that a certain amount of swimming 

 room be maintained by each fish (Breder 1965, van 

 01st and Hunter 1970). Thus each fish and a "shell" 

 of water about it may be considered as a unit, and 

 a school as a packing together of these units. Such 

 structures can be handled by established math- 

 ematical procedures. The fact the fishes are all 

 moving forward and, in many instances, often 

 shifting their relative positions merely makes the 



handling of such data a little tedious, but does not 

 vitiate the basic propositions. 



One approach to the analysis of the structure of 

 a fish school, the empirical, can be made by mea- 

 suring the distance, angle, or other parameter 

 between a given fish and the other members of the 

 school. The mathematical manipulation of such 

 measurements can establish values that may serve 

 as an index to the school's organization. One's 

 imagination alone limits the selection of data. 

 Papers that have employed this type of approach 

 include Keenleyside (1955), Breder (1959, 1965), 

 Cullen et al. (1965), Hunter (1966), van 01st and 

 Hunter (1970), Symons (1971a, b), Healey and 

 Prieston (1973), Weihs (1973a), and Pitcher (1973). 

 Only Cullen et al., Symons, and Pitcher in the 

 above list attempted complete tridimensional 

 measurements. Pitcher's paper has important 

 bearing on the approach developed here on the 

 basis of abstract reasoning. It will be discussed in 

 detail later. 



A theoretical approach, equally valid, is based on 

 tridimensional geometrical concepts and con- 

 structs for purposes of comparison with fish 

 schools. Since there is an infinite variety of such 

 constructs possible, only those of some conceivable 

 application to this study are discussed here. Unlike 

 the empirical approach, there are evidently no 

 prior papers that have employed this theoretical 

 one. The following treatment has been made 

 especially explicit because of the complex rela- 

 tionships within both space lattices and space 

 packings, as some biologists who might consult 

 these pages may not have instant recall of such 

 details. 



It is necessary to introduce some elementary 

 data on tridimensional lattices that are essential 

 to an understanding of their bearing on fish 

 schools.^ The most readily visualized space lattice 

 is that in which a cube is the element or cell 

 (Figure lA). It is not the closest possible packing 

 of such points: a closer one can be obtained by 

 figuratively pushing the cubic lattice askew 

 (Figure IB) so that the special case of cubes with 

 their 90° angles become rhombohedrons with 

 other angles. The dotted arrow in Figure IB 

 indicates the amount of travel of the point in the 

 upper left front corner of the lattice in attaining 



-Definitions of this word as used here are given by Breder 

 (1959, 1967). For an extended discussion of this and other usages 

 see Shaw (1970). 



'Support of ail geometric statements made in this section may 

 be found in any formal or informal geometry text covering the 

 area concerned, such as Hilbert and Cohn-Vossen (1952) and 

 Lines (1965). 



472 



