1. Introduction
Fibonacci numbers play crucial roles in combinatorial mathematics and elementary number theory. Although such numbers have been investigated for centuries, they continue to intrigue mathematicians and researchers in many areas of human endeavor (as can be seen in [
1] and the references therein), while providing new tools for expanding the frontiers of mathematical study. In respect to this, the
golden ratio , namely the positive solution to the equation
, appears in some very fundamental relationships involving numbers, with one of the most basic occurrences of the golden ratio involving the use of two seed values and a simple Fibonacci-like additive recursion relationship.
Walking and running are human gait modes exhibiting different mechanics and energetics. A double support phase, i.e., both limbs are in ground contact, identifies the walking gait, whereas a double float phase, i.e., no limb is in ground contact, identifies the running gait. However, both physiological (symmetric and recursive) human walking and running are characterized, from a temporal point of view, by only four specific time intervals, associated with the durations of gait cycle, swing, stance and double support (double float) phases. More precisely, physiological symmetric walking (running) is classically recognized to exhibit:
A stance (swing) duration, (), being close to 62% of gait cycle duration ;
A swing (stance) duration being close to 38% of gait cycle duration;
A double support (double float) duration, (), being consequently close to 24% of gait cycle duration.
As recently formally recognized in [
2], the above sequence
can be viewed as a slight approximation of the special sequence in walking:
and its conceptual counterpart in running:
The above sequence
is nothing but a generalized four-length Fibonacci sequence [
3] (with the real numbers
,
as seeds), generally defined, for any values
and
, by (
is the negative solution to
):
satisfying the recurrence:
Regarding
: (i) the general Fibonacci sequence structure is rooted in the following duration constraints of symmetric walking (and running, respectively):
(respectively,
),
; (ii) the specific
-dependent (temporally harmonic) self-similar structure relies on the special chain of ratios
(respectively,
) all being equal to
. Indeed,
,
,
. In other words, conditions in (i)–(ii) apparently induce a repeated self-proportional partition (namely self-similarity), in accordance with the fact that two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. This way, walking and running implicitly involve a fractal nature, in which the structure of the larger scale resembles the structure of the sub-unit and in which one of the simplest ways of transformation, i.e., the new domain composed of two previous ones, is highlighted.
Very recent experimental and theoretical analyses—inspired by the aforementioned cyclic human movements in walking and running (see also [
4,
5]—have found harmonic structures to even appear in front crawl swimming [
6] at a middle/long distance pace, at which the presence of redundant movements is reduced
via in-phase synchronization with the induced waves in the water. With respect to this, the recent paper [
6] not only provides a mathematical framework and experimental consistency for recognizing preliminary evidence, for elite front-crawl swimmers swimming at a middle/long distance pace, a
recovery phase duration that is very close to the fundamental unit
(where
is equal to the duration of the front crawl stroke divided by 12) but it also illustrates the existence of harmonic structures—in both their simple and enhanced versions—that elite swimmers seek to reproduce in order to improve their performance. We also refer the reader to [
7,
8] for a different scenario involving the dimensionless Strouhal number.
A more technically advanced swimming stroke is the butterfly. Simultaneous strokes, such as breaststroke and butterfly, are in fact considered to be highly technical due to the complex coordination of the arm and leg actions [
9]. In particular, in butterfly, the out-of-water arms recovery is facilitated by the leg undulation. To be effective, however, the kick must appear as a consequence of a body wave-like cefalo–caudal undulation motion [
10]. Having a glide time with the arms extended forward at the top of the stroke is certainly a strategy used to reduce the energetic cost (metabolic power/velocity) for long-distance swims [
11]. This way, the head, trunk and upper limbs are profiled in a streamlined position in order to glide, and consequently, provide some rest at each stroke during a long butterfly swim (some aquatic animals such as giant cormorants, penguins, and dolphins improve the metabolic efficiency of swimming by adopting locomotion patterns with alternating periods of propulsion and gliding [
10]). However, this is not effective for achieving a higher stroke rate [
10] and for avoiding great instantaneous velocity fluctuations. On the other hand, when velocity and stroke rate increase, coordination becomes closer to an in-phase mode [
12,
13], just like in human locomotion and in quadrupedal coordination. Indeed, the significant skill effect in [
10] indicates that elite swimmers—whom are the object of analysis in this paper—have stronger arm/leg synchronization than the sub-elite swimmers: elite swimmers adopt a shorter glide to overcome great forward resistance and generate higher forces during the arm pull; sub-elite swimmers often compensate for coordination mistakes by applying greater force. With respect to this, we also refer the reader to [
14]—for devices performing propulsion analysis in swimming; [
15]—for a quantitative evaluation of phases of turns during competition; and [
16]—for the role of the hip movement in the stroke mechanics.
The aim of this paper was to extend, for the first time in the literature—to the best of our knowledge—the aforementioned findings regarding human walking and running at comfortable speed and front crawl swimming strokes at a middle/long distance pace: a harmonically self-similar temporal partition, which relies on the generalized Fibonacci sequence and the golden ratio, is formally defined for the highly complex and upper and lower-limbs-coordinated butterfly stroke. Quantitative indices, named -bonacci butterfly stroke numbers, are accordingly proposed to assess such a hidden time-harmonic and self-similar structure being subtly exhibited by elite swimmers at middle distance pace.
3. Experimental Analysis
The feasibility of the preceding analysis is here illustrated by the dedicated analysis of butterfly stroke training sessions for: (i) seven international-level swimmers, namely IL1 (male, 31 y, 190 cm, 80 kg), IL2 (female, 31 y, 170 cm, 65 kg), IL3 (female, 27 y, 168 cm, 58 kg), IL4 (male, 20 y, 196 cm, 80 kg), IL5 (male, 19 y, 180 cm, 73 kg), IL6 (male, 19 y, 193 cm, 85 kg), IL7 (female, 27 y, 173 cm, 66 kg); (ii) two national-level swimmers, namely, NL1 (female, 19 y, 166 cm, 55 kg), NL2 (male, 25 y, 185 cm, 95 kg), all of them swimming at their own
middle distance pace. In particular, the above international-level swimmers and national-level swimmers are reported, within the corresponding sets, in order of physical shape (measured as race performance capabilities) at the moment of data acquisition. While the international-level swimmers IL1–IL7 compete at major international events on a regular basis and hold national/international records, the national-level swimmers NL1–NL2 are national medalists who compete on a regular basis of major national events (with NL1 being close to becoming an international-level swimmer). The analysis was performed by using high frame rate videos (100 for IL1, NL2; 120 fps for IL2–IL7, NL1) of stable strokes
via the 2D BioMovie ERGO system at
http://www.infolabmedia.eu/ (accessed on 7 June 2021).
3.1. Phase Durations and Interlimb Coordination
Phase and delay durations for all the swimmers IL1–IL7, NL1–NL2 are reported in
Table 1,
Table 2,
Table 3 and
Table 4:
-(almost) kick-to-kick temporal symmetric repetitive butterfly strokes S (under constraints
) are exhibited (Kick-to-kick temporal symmetry appears to be almost verified for IL1–IL7, NL1–NL2, with the modulus of the difference between
and
belonging to the set
ms. Constraint
even appears to be almost verified for IL1–IL7, NL1–NL2, with the modulus of the difference between
and
belonging to the set
ms.), characterized by the
- and NTD-values reported in
Table 3 and
Table 4. In accordance with the evidence of [
10] on elite swimmers, all of such relatively small NTDs define
highly coordinated strokes. All nine swimmers’ strokes exhibit a negative delay
(leading to a lag time in glide position) and a positive delay
. On the other hand, in contrast to the international-level swimmers IL1, IL3–IL6, the international-level swimmers IL2, IL7 and the national-level swimmers NL1, NL2 are characterized by a small negative superposition of two contradictory actions (
). Furthermore,
for NL1, whereas
is negative for IL1–IL7 and NL2.
3.2. Self-Similarity Analysis
The aggregate phase percentage values for the international-level swimmers IL1–IL7 and the national-level swimmers NL1–NL2 are reported in
Table 5 and
Table 6, along with the values for the indices
and
in (23) and (24). Comments are in order:
Rather small values are obtained for IL1–IL7, with IL1’s one being the smallest, owing to the strict closeness of the corresponding phase percentage values to , , , , and ;
Relatively large reductions in the self-similarity and enhanced self-similarity magnitudes (especially of the latter) appear for the national-level swimmers NL1–NL2 when compared to the international-level swimmers IL1–IL7;
The - and - values turn out to reproduce the order of physical shape within the two swimmers’ set;
IL5 even presents an -value () that is close to the one () characterizing the strongly enhanced self-similar structure of Remark ();
The slight differences in the phase durations of
Table 1 and
Table 2 (compare, for instance, IL2 to NL1, or IL4 to IL5, or IL2 to IL3), which lead to the differences in self-similarity magnitudes of
Table 5 and
Table 6, have been successfully identified via the high frame rate analysis used in this paper, with the self-similarity information complementing the delay partition values of
Table 3 and
Table 4;
Larger percentage reductions in enhanced self-similarity (with respect to self-similarity) are exhibited by IL2, IL3, IL5—when compared to IL1, IL4, IL5, IL6, NL1, NL2—so that the - values for IL2–IL3 and IL5 tend to thicken (more than the -ones) towards the - values for IL4 and IL6, respectively.