Hidden time-patterns in cyclic human movements: a matter of temporal Fibonacci sequence generation and harmonization

Fibonacci sequences are sequences of numbers whose first two elements are 0, 1, and such that, starting from the third number, every element of the sequence is the sum of the previous two. They are of finite length when the number of elements of the sequence is finite. Furthermore, Fibonacci sequences are named generalized Fibonacci sequences when they are generated by two positive integers-called seeds-that do not necessarily equal 0 and 1. This relaxation provides the analyst with larger degrees of freedom if the elements of the Fibonacci sequences have to refer to the durations of the sub-phases of a physical movement or gesture that differ from 0 and 1. Indeed, by taking inspiration from their use of symmetric walking-where the stance duration is the sum of the double support and swing durations and, in turn, the duration of the entire gait cycle is the sum of the stance and swing durations-, generalized Fibonacci sequences of finite length have been very recently adopted to extend the resulting original walking gait characterization to gestures in elite swimmers and tennis players, by accordingly associating the durations of the sub-phases of the gesture to the elements of such sequences. This holds true within movement-automatization-allowable scenarios, namely, within scenarios in which no external disturbances or additional constraints affect the natural repeatability of movements: at a comfortable speed in walking, at a medium pace in swimming, and under no need for lateral/frontal movements of the entire body in tennis forehand execution or no wind in the serve shot. Now, in such sequences of sub-phase durations of a physical movement or gesture, the golden ratio has been further found to characterize hidden self-similar patterns, namely, patterns in which all the ratios between two consecutive elements of the sequence are surprisingly equal, thus representing a harmonic and mostly aesthetical gesture that admits a perfectly self-similar sub-phase partition in terms of time durations. In such a case, the larger scale structure within the gesture resembles the smaller scale structure so that the brain can aesthetically resort to the minimum amount of information for the movement temporal design. In the framework of how cognitive factors such as working memory and executive control facilitate motor learning and adaptation, this paper addresses, for the first time in the literature, the open problem of providing a complete mathematical understanding of the automatic generation process at the root of such hidden Fibonacci sequence-based and self-similar patterns appearing in the aforementioned cyclic human movements. Data referring to walking and tennis playing are used to illustrate the effectiveness of the proposed approach.