Hidden Time-Patterns in Cyclic Human Movements: A Matter of Temporal Fibonacci Sequence Generation and Harmonization

Cristiano Maria Verrelli ^1* Lucio Caprioli ² Marco Iosa ³

• ¹ Department of Electronic Engineering, University of Rome Tor Vergata, Roma, Italy
• ² Department of Industrial Engineering, University of Rome Tor Vergata, Rome, Lazio, Italy
• ³ Department of Psychology, Faculty of Medicine and Psychology, Sapienza University of Rome, Italy., Rome, Sicily, Italy

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 that not necessarily equal 0 and 1. By taking inspiration from their use in symmetric walking, generalized Fibonacci sequences of finite length have been very recently adopted to extend the resulting original walking gait characterization to completely automatized gestures in elite swimmers and tennis-players, by accordingly associating the durations of the subphases of the gesture to the elements of such sequences. This can happen 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 of lateral/frontal movements of the entire body in tennis forehand execution or no wind in the serve shot. Now, in such sequences of subphase 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 subphase 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 like 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.