Logarithmic spiral components in locomotor patterns of fish

Abstract

A logarithmic spiral with lengths of radii at 90° having a ratio of 1:0.618 recurred frequently in the locomotor patterns of nurse shark and goldfish in various experimental conditions. The evidence indicated that bilateral sense organs may play a role in steering logarithmic spiral movements.

Deep-sea foraging pathways: an analysis of randomness and resource exploitation

“…Both logarithmic (Kleerekoper et al 1973;von Buddenbrock 1917) and helical spiraling (Nageli 1860;Jennings 1901;Schaeffer 1920Schaeffer , 1926Schaeffer , 1927Bullington 1925;Streeter 1906;Lee 1894;Dunkelberger 1926;Schaeffer 1928) have been observed in a variety of terrestrial and aquatic organisms. Kleerekoper et al (1973) found nurse sharks to employ a logarithmic spi ral search pattern to locate food once sensed and to produce occasionally a locomotory pathway of up to four spirals separated by "nondescript" pathways. Although the guiding mechanisms are not understood, reference is made to Schaeffer's (1920) contention that spiraling is the basic path pattern of moving organisms, modified by sensory input, and Jenning's (1901) observation that single-celled organisms, by moving along a spiral path, maintain a general straight line of locomotion.…”

Shortest paths for finned, winged, legged, and wheeled vehicles with side-looking sensors

“…For example, in Kleerekoper et al (1973) a study of the locomotor patterns of nurse sharks indicated that the approach by these animals to a source of diffusing odor or light was along logarithmic spirals, i.e. equiangular spirals in which the lengths of radii at 90 • are in the golden ratio =( 1 − √ 5) /2 (see Figure 1).…”

“…equiangular spirals in which the lengths of radii at 90 • are in the golden ratio =( 1 − √ 5) /2 (see Figure 1). Kleerekoper et al (1973) state that "bilateral sense organs may play a role in steering logarithmic spiral movements", although evidence is inconclusive as to what "the mechanisms by which spiral movements are brought about" are. A conclusion that laterality of sense organs (both visual and olfactory) and the limited aperture of their receptive fields determine logarithmic spiral movements is not warranted by experimental observations: indeed, goldfishes continue moving in spirals even when blinded (Kleerekoper et al 1973).…”

“…Kleerekoper et al (1973) state that "bilateral sense organs may play a role in steering logarithmic spiral movements", although evidence is inconclusive as to what "the mechanisms by which spiral movements are brought about" are. A conclusion that laterality of sense organs (both visual and olfactory) and the limited aperture of their receptive fields determine logarithmic spiral movements is not warranted by experimental observations: indeed, goldfishes continue moving in spirals even when blinded (Kleerekoper et al 1973). However, the same authors conclude that "the significance of logarithmic spiralling in directed locomotion toward a stimulus source gains importance" in relation to a model (credited to F Brouwer in Kleerekoper et al (1973)), which states that "any target-seeking device which approaches the target in a direct pathway which, at all times, deviates with a constant angle from a straight line connecting the vehicle with the target, will describe a logarithmic spiral around the target.…”