Front crawl stroke in swimming: Phase durations and self-similarity

Human movements, such as walking and running, are able to generate rhythmic motor patterns, with the consequent appearance of hidden time-harmonic structures. Such harmonic structures are represented (at comfortable speed) by the occurrence of the golden ratio as ratio of durations of specific walking and running gait sub-phases. Preliminary experimental evidences suggest that front crawl swimming may behave, under this point of view, like walking and running. This paper aims to demonstrate that a mathematical connection between the golden ratio and the front crawl swimming stroke actually exists, at a pace that plays the role of the comfortable speed in walking and running. Generalized Fibonacci sequences are used to this purpose. They rely on the durations of aggregate phases of the front crawl swimming stroke with a clear physical meaning, while characterizing self-similarity of front crawl strokes in its simple nature and enhanced (stronger) variant. Experimental data on front crawl swimmers illustrate the theoretical derivations, suggesting that the pace playing the role of the comfortable speed in walking and running is the middle/long-distance one, while showing that the self-similarity level increases with the swimming technique and the enhanced self-similarity is associated with the performance of top-level swimmers.