# Phi-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

- A stance (swing) duration, ${t}_{\mathrm{ST}}$ (${t}_{\mathrm{SW}}$), being close to 62% of gait cycle duration ${t}_{\mathrm{GC}}$;
- A swing (stance) duration being close to 38% of gait cycle duration;
- A double support (double float) duration, ${t}_{\mathrm{DS}}$ (${t}_{\mathrm{DF}}$), being consequently close to 24% of gait cycle duration.

## 2. Generalized Fibonacci Sequences in Butterfly Stroke

#### 2.1. Butterfly Stroke Phases

- (i)
- Entry and catch phase (between the entry of the hands into the water and the beginning of their backward movement), with duration ${t}_{\mathrm{EC}}$;
- (ii)
- Pull phase (between the beginning of the backward movement of the hands and their entry into the plane vertical to the shoulders), with duration ${t}_{\mathrm{PL}}$;
- (iii)
- Push phase (between the positioning of the hands below the shoulders and their exit from the water), with duration ${t}_{\mathrm{PS}}$;
- (iv)
- Recovery phase (between the arrival of the hands at the water level and their subsequent entry into the water), with duration ${t}_{\mathrm{RE}}$,

- (i)
- Downward phase 1 (between the high and low break-even points– at first occurrence–of the feet during the first undulation), with duration ${t}_{\mathrm{K}1}$;
- (ii)
- Upward phase 1 (between the low and high break-even points–at first occurrence–of the feet during the first undulation), with duration ${t}_{\mathrm{U}1}$;
- (iii)
- Downward phase 2 (between the high and low break-even points–at first occurrence–of the feet during the second undulation), with duration ${t}_{\mathrm{K}2}$;
- (iv)
- Upward phase 2 (between the low and high break-even points–at first occurrence–of the feet during the second undulation), with duration ${t}_{\mathrm{U}2}$.

- ${T}_{1}$, between the start of Entry and catch phase and the start of Downward phase 1;
- ${T}_{2}$, between the start of Pull phase and the start of Upward phase 1;
- ${T}_{3}$, between the start of Push phase, and the start of Downward phase 2;
- ${T}_{4}$, between the start of Recovery phase, and the start of Upward phase 2.

#### 2.2. Self-Similarity and Enhanced Self-Similarity in Butterfly Stroke

**Theorem 1.**

**(${\mathcal{F}}_{4}$):**In a highly coordinated $(a,b,c,d)$-kick-to-kick temporal symmetric repetitive butterfly stroke, the sequence:

**Proof.**

**Theorem 2.**

**(${\mathcal{F}}_{6}$):**In a highly coordinated $(a,b,c,d)$-kick-to-kick temporal symmetric repetitive butterfly stroke under $\mathcal{C}$, the sequence (7) enforced with $\{{t}_{\mathrm{K}1},{t}_{\mathrm{K}2}\}$ to the left, meaning that:

**Proof.**

**Remark 1.**

**(${\mathcal{F}}_{7}$):**In the very special case of ${T}_{1}={t}_{\mathrm{K}2}-{t}_{\mathrm{K}1}$ for the highly coordinated $(a,b,c,d)$-kick-to-kick temporal symmetric repetitive butterfly stroke, the sequence (15), once enforced with ${T}_{1}$ to the left, becomes a generalized Fibonacci sequence of length 7, with the equality ${t}_{\mathrm{U}1}/({t}_{\mathrm{K}1}+{t}_{\mathrm{K}2})=\varphi $ making the sequence ${\mathcal{F}}_{7}$ possess a strongly enhanced self-similar structure; ${T}_{1}\approx 5.5735\%$ of ${t}_{\mathrm{S}}$ is additionally obtained.

#### 2.3. Quantitative Measures of Self-Similarity and Enhanced Self-Similarities

## 3. Experimental Analysis

#### 3.1. Phase Durations and Interlimb Coordination

#### 3.2. Self-Similarity Analysis

- 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 $62\%$, $38\%$, $24\%$, $14\%$, and $10\%$;
- 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 ${\mathcal{I}}_{f,4}$- and ${\mathcal{I}}_{f,6}$- values turn out to reproduce the order of physical shape within the two swimmers’ set;
- IL5 even presents an $a={T}_{1}/{t}_{\mathrm{S}}$-value ($4.71\%$) that is close to the one ($5.57\%$) characterizing the strongly enhanced self-similar structure of Remark (${\mathcal{F}}_{7}$);
- 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 ${\mathcal{I}}_{f,6}$- values for IL2–IL3 and IL5 tend to thicken (more than the ${\mathcal{I}}_{f,4}$-ones) towards the ${\mathcal{I}}_{f,6}$- values for IL4 and IL6, respectively.

## 4. Discussion

- Theorem 1 and Theorem 2 provide partition constraints that regard the leg stroke, with the inter-delay composition linking the arm phase partition with the leg phase partition;
- The durations ${t}_{\mathrm{U}1}$ and ${t}_{\mathrm{U}2}+{T}_{1}$ play the role of ${t}_{{\mathcal{F}}_{a}}$ and ${t}_{{\mathcal{F}}_{b}}$ in the front crawl stroke in [6] and of the right and left swing (right and left stance) durations in the asymmetric walking (running) of [2,5], with the involved equality between the durations of such phases simply transposing the swing (stance)-symmetry constraint of symmetric walking (running);
- In light of the twelve-tone equal temperament-based interpretation, constraint $\mathcal{C}$ imposes that the stroke duration ${t}_{\mathrm{S}}$ of an $(a,b,c,d)$-kick-to-kick temporal symmetric repetitive butterfly stroke equals the sum of the durations of eight disjoint sub-phases, three among them with duration ${t}_{\mathrm{K}1}$ and five among them with duration ${t}_{\mathrm{K}2}$: according to [6], notes D4, E4, G4, C5 correspond, in the (suspended and restored) C-variant Cadd2-chord, to the frequencies: ${f}_{3}\left(\mathrm{D}4\right)={f}_{0}{2}^{\frac{\mathbf{3}}{12}}=293.7$ Hz, ${f}_{5}\left(\mathrm{E}4\right)={f}_{0}{2}^{\frac{\mathbf{5}}{12}}=329.6$ Hz, ${f}_{8}\left(\mathrm{G}4\right)={f}_{0}{2}^{\frac{\mathbf{8}}{12}}=391.96$ Hz, ${f}_{13}\left(\mathrm{C}5\right)={f}_{0}{2}^{\frac{\mathbf{13}}{12}}=2{f}_{1}\left(C4\right)=523.2$ Hz (${f}_{0}=246.9175$ Hz), with $n=3,5,8,13$ corresponding to elements of a Fibonacci sequence and with the ratios $5/3\approx 1.667$, $8/5=1.6$, $13/8=1.625$ of consecutive elements of such sequence quickly getting close to $\varphi $;
- The above results—again occurring in elite swimmers—confirm that, in contrast to walking, a precise swimming technique (coming from a considerable amount of practice and instruction repeating specifically precise movements for a long enough time, while producing rhythmic motor patterns through the interrelationship between cortical input, central pattern generator (CPG), and sensory feedback) is relevantly involved (the intra-cyclic variation of the horizontal velocity of the hip is larger in non-expert swimmers than their expert counterparts [20]) as recalled by [6], the fractal dimension is $1.8$–$1.9$ for highly qualified expert swimmers [20], whereas it is $1.1$–$1.4$ for on-land walking [21].

## 5. Practical Implications

## 6. Strengths and Limits

## 7. Forecast

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Chen, L.; Wang, X. The power sums involving fibonacci polynomials and their applications. Symmetry
**2019**, 11, 635. [Google Scholar] [CrossRef][Green Version] - Verrelli, C.M.; Iosa, M.; Roselli, P.; Pisani, A.; Giannini, F.; Saggio, G. Generalized finite-length Fibonacci sequences in healthy and pathological human walking: Gait recursivity, asymmetry, consistency, self-similarity, and variability. In Frontiers in Human Neuroscience; Special Issue: Rhythmic Patterns in Neuroscience and Human Physiology; to be published.
- Horadam, A.F. A generalized Fibonacci sequence. Am. Math. Mon.
**1961**, 68, 455–459. [Google Scholar] [CrossRef] - Iosa, M.; Fusco, A.; Marchetti, F.; Morone, G.; Caltagirone, C.; Paolucci, S.; Peppe, A. The golden ratio of gait harmony: Repetitive proportions of repetitive gait phases. Biomed Res. Int.
**2013**, 2013, 918642. [Google Scholar] [CrossRef][Green Version] - Marino, R.; Verrelli, C.M.; Gnucci, M. Synchronicity rectangle for temporal gait analysis: Application to Parkinson’s Disease. Biomed. Signal Process. Control
**2020**, 62, 102156. [Google Scholar] [CrossRef] - Verrelli, C.M.; Romagnoli, C.; Jackson, R.R.; Ferretti, I.; Annino, G.; Bonaiuto, V. Front crawl stroke in swimming: Ratios of phase durations and self-similarity. J. Biomech.
**2021**, 118, 110267. [Google Scholar] [CrossRef] [PubMed] - Taylor, G.K.; Nudds, R.L.; Thomas, A.L.R. Flying and swimming animals cruise at a Strouhal numer tuned for high power efficiency. Lett. Nat.
**2003**, 425, 707–711. [Google Scholar] [CrossRef] [PubMed] - Eloy, C. Optimal Strouhal number for swimming animals. J. Fluids Struct.
**2012**, 30, 215–218. [Google Scholar] [CrossRef][Green Version] - Barbosa, T.M.; Keskinen, K.L.; Fernandes, R.; Colaço, P.; Lima, A.B.; Vilas-Boas, J.P. Energy cost and intracyclic variation of the velocity of the centre of mass in butterfly stroke. Eur. J. Appl. Physiol.
**2005**, 93, 519–523. [Google Scholar] [CrossRef] - Seifert, L.; Boulesteix, L.; Chollet, D.; Vilas-Boas, J.P. Differences in spatial-temporal parameters and arm–leg coordination in butterfly stroke as a function of race pace, skill and gender. Hum. Mov. Sci.
**2008**, 27, 96–111. [Google Scholar] [CrossRef] - Di Prampero, P.E.; Pendergast, D.R.; Wilson, D.W.; Rennie, D.W. Blood Lactic Acid Concentrations in High Velocity Swimming. In Swimming Medicine IV, Proceedings of the Fourth International Congress on Swimming Medicine, Stockholm, Sweden; Eriksson, B., Furberg, B., Eds.; University Park Press: Baltimore, MD, USA, 1978; pp. 249–261. [Google Scholar]
- Chollet, D.; Seifert, L.; Boulesteix, L.; Carter, M. Arm–leg coordination in elite butterfly swimmers. Int. J. Sport Med.
**2006**, 27, 322–329. [Google Scholar] [CrossRef] - Seifert, L.; Delignieres, D.; Chollet, D. Effect of expertise on butterfly stroke coordination. J. Sport. Sci.
**2007**, 25, 131–141. [Google Scholar] [CrossRef] [PubMed] - Lanotte, N.; Annino, G.; Bifaretti, S.; Gatta, G.; Romagnoli, C.; Salvucci, A.; Bonaiuto, V. A new device for propulsion analysis in swimming. Proceedings
**2018**, 2, 285. [Google Scholar] [CrossRef][Green Version] - Tourny-Chollet, C.; Chollet, D.; Hogie, S.; Papparodopoulos, C. Kinematic analysis of butterfly turns of international and national swimmers. J. Sport. Sci.
**2002**, 20, 383–390. [Google Scholar] [CrossRef] - Mason, B.R.; Tong, Z.; Richards, R.J. A biomechanical analysis of the butterfly stroke. Excel
**1991**, 7, 6–11. [Google Scholar] - Chollet, D.; Chalies, S.; Chatard, J.C. A new index of coordination for the crawl: Description and usefulness. Int. J. Sport. Med.
**1999**, 21, 54–59. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ungerechts, B.E. A comparison of the movements of the rear parts of dolphins and butterfly swimmers. In Biomechanics and Medicine in Swimming; Hollander, A.P., Huijing, P.A., de Groot, G., Eds.; Human Kinetics: Champaign, IL, USA, 1983; pp. 215–221. [Google Scholar]
- Sanders, R.H.; Cappaert, J.M.; Devlin, R.K. Wave Characteristics of butterfly swimming. J. Biomech.
**1995**, 28, 9–16. [Google Scholar] [CrossRef] - Barbosa, T.M.; Goh, W.X.; Morais, J.E.; Costa, M.J. Variation of linear and nonlinear parameters in the swim strokes according to the level of expertise. Mot. Control
**2017**, 21, 312–326. [Google Scholar] [CrossRef] - Schiffman, J.M.; Chelidze, D.; Adams, A.; Segala, D.B.; Hasselquist, L. Nonlinear analysis of gait kinematics to track changes in oxygen consumption in prolonged load carriage walking: A pilot study. J. Biomech.
**2009**, 42, 2196–2199. [Google Scholar] [CrossRef] [PubMed] - Barbosa, T.M.; Goh, W.X.; Morais, J.E.; Costa, M.J. Comparison of classical kinematics, entropy, and fractal properties as measures of complexity of the motor system in swimming. Front. Psychol.
**2016**, 7, 1–7. [Google Scholar] [CrossRef][Green Version] - Takagi, H.; Sugimoto, S.; Nishijima, N.; Wilson, B. Differences in stroke phases, arm-leg coordination and velocity fluctuation due to event, gender and performance level in breaststroke. Sport. Biomech.
**2004**, 3, 15–27. [Google Scholar] [CrossRef] [PubMed] - Ualí, I.; Herrero, A.J.; Garatachea, N.; Marín, P.J.; Alvear-Ordenes, I.; García-López, D. Maximal strength on different resistance training rowing exercises predicts start phase performance in elite kayakers. J. Strenght Cond. Res.
**2012**, 26, 941–946. [Google Scholar] [CrossRef] [PubMed] - Landlinger, J.; Lindinger, S.; Stöggl, T.; Wagner, H.; Müller, E. Key factors and timing patterns in the tennis forehand of different skill levels. J. Sport. Sci. Med.
**2010**, 9, 643–651. [Google Scholar]

**Figure 1.**Butterfly stroke: arm and leg phases (case of ${T}_{2},{T}_{3},{T}_{4}>0$) according to [12].

${\mathit{t}}_{\mathbf{EC}}$ | ${\mathit{t}}_{\mathbf{PL}}$ | ${\mathit{t}}_{\mathbf{PS}}$ | ${\mathit{t}}_{\mathbf{RE}}$ | ${\mathit{t}}_{\mathbf{S}}$ | |
---|---|---|---|---|---|

IL1 | $0.220$ | $0.220$ | $0.270$ | $0.440$ | $1.150$ |

IL2 | $0.233$ | $0.183$ | $0.316$ | $0.333$ | $1.065$ |

IL3 | $0.308$ | $0.158$ | $0.266$ | $0.333$ | $1.065$ |

IL4 | $0.375$ | $0.200$ | $0.258$ | $0.407$ | $1.240$ |

IL5 | $0.291$ | $0.291$ | $0.250$ | $0.400$ | $1.232$ |

IL6 | $0.300$ | $0.283$ | $0.333$ | $0.367$ | $1.283$ |

IL7 | $0.234$ | $0.175$ | $0.358$ | $0.417$ | $1.184$ |

NL1 | $0.266$ | $0.166$ | $0.308$ | $0.326$ | $1.066$ |

NL2 | $0.220$ | $0.150$ | $0.200$ | $0.330$ | $0.900$ |

${\mathit{t}}_{\mathbf{K}1}$ | ${\mathit{t}}_{\mathbf{K}2}$ | ${\mathit{t}}_{\mathbf{U}1}$ | ${\mathit{t}}_{\mathbf{U}2}$ | ${\mathit{t}}_{\mathbf{S}}$ | |
---|---|---|---|---|---|

IL1 | $0.120$ | $0.160$ | $0.420$ | $0.450$ | $1.150$ |

IL2 | $0.108$ | $0.166$ | $0.400$ | $0.466$ | $1.065$ |

IL3 | $0.125$ | $0.150$ | $0.391$ | $0.383$ | $1.065$ |

IL4 | $0.133$ | $0.191$ | $0.441$ | $0.450$ | $1.240$ |

IL5 | $0.120$ | $0.141$ | $0.483$ | $0.430$ | $1.232$ |

IL6 | $0.150$ | $0.166$ | $0.517$ | $0.417$ | $1.283$ |

IL7 | $0.150$ | $0.158$ | $0.467$ | $0.475$ | $1.184$ |

NL1 | $0.108$ | $0.183$ | $0.417$ | $0.408$ | $1.066$ |

NL2 | $0.120$ | $0.140$ | $0.330$ | $0.330$ | $0.900$ |

**Table 3.**Delays and related values for swimmers IL1–IL7 and NL1–NL2: delay durations (in seconds) and NTD.

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ | NTD | |
---|---|---|---|---|---|

IL1 | $0.000$ | $-0.100$ | $0.100$ | $-0.010$ | $12.320\%$ |

IL2 | $-0.075$ | $-0.200$ | $0.017$ | $-0.133$ | $23.680\%$ |

IL3 | $0.016$ | $-0.167$ | $0.066$ | $-0.050$ | $17.570\%$ |

IL4 | $0.025$ | $-0.217$ | $0.024$ | $-0.043$ | $18.060\%$ |

IL5 | $0.058$ | $-0.113$ | $0.079$ | $-0.030$ | $12.380\%$ |

IL6 | $0.033$ | $-0.117$ | $0.117$ | $-0.050$ | $13.720\%$ |

IL7 | $-0.066$ | $-0.150$ | $0.142$ | $-0.058$ | $18.960\%$ |

NL1 | $-0.050$ | $-0.208$ | $0.043$ | $-0.082$ | $21.860\%$ |

NL2 | $-0.020$ | $-0.120$ | $0.060$ | $0.000$ | $15.070\%$ |

a | b | c | d | |
---|---|---|---|---|

IL1 | $0.00\%$ | $-8.69\%$ | $8.69\%$ | $-0.87\%$ |

IL2 | $-7.04\%$ | $-18.78\%$ | $1.60\%$ | $-12.49\%$ |

IL3 | $1.50\%$ | $-15.68\%$ | $6.20\%$ | $-4.69\%$ |

IL4 | $2.02\%$ | $-17.50\%$ | $1.94\%$ | $-3.47\%$ |

IL5 | $4.71\%$ | $-9.17\%$ | $6.41\%$ | $-2.44\%$ |

IL6 | $2.57\%$ | $-9.12\%$ | $9.12\%$ | $-3.90\%$ |

IL7 | $-5.57\%$ | $-12.67\%$ | $11.99\%$ | $-4.90\%$ |

NL1 | $-4.69\%$ | $-19.51\%$ | $4.03\%$ | $-7.69\%$ |

NL2 | $-2.22\%$ | $-13.33\%$ | $6.67\%$ | $0.00\%$ |

$({\mathit{t}}_{\mathbf{K}1}+{\mathit{t}}_{\mathbf{K}2}+{\mathit{t}}_{\mathbf{U}1})/{\mathit{t}}_{\mathbf{S}}$ | ${\mathit{t}}_{\mathbf{U}1}/{\mathit{t}}_{\mathbf{S}}$ | $({\mathit{t}}_{\mathbf{K}1}+{\mathit{t}}_{\mathbf{K}2})/{\mathit{t}}_{\mathbf{S}}$ | ${\mathit{t}}_{\mathbf{K}2}/{\mathit{t}}_{\mathbf{S}}$ | ${\mathit{t}}_{\mathbf{K}1}/{\mathit{t}}_{\mathbf{S}}$ | |
---|---|---|---|---|---|

IL1 | $60.87$ | $36.52$ | $24.35$ | $13.91$ | $10.43$ |

IL2 | $63.29$ | $37.56$ | $25.73$ | $15.59$ | $10.14$ |

IL3 | $62.54$ | $36.71$ | $25.82$ | $14.08$ | $11.74$ |

IL4 | $61.69$ | $35.56$ | $26.13$ | $15.40$ | $10.73$ |

IL5 | $60.39$ | $39.20$ | $21.19$ | $11.44$ | $9.74$ |

IL6 | $64.93$ | $40.30$ | $24.63$ | $12.94$ | $11.69$ |

IL7 | $65.46$ | $39.44$ | $26.01$ | $13.34$ | $12.67$ |

NL1 | $66.42$ | $39.12$ | $27.30$ | $17.17$ | $10.13$ |

NL2 | $65.56$ | $36.67$ | $28.89$ | $15.56$ | $13.33$ |

${\mathcal{I}}_{\mathit{f},4}$ | ${\mathcal{I}}_{\mathit{f},6}$ | |
---|---|---|

IL1 | $1.89$ | $1.94$ |

IL2 | $2.20$ | $2.72$ |

IL3 | $2.30$ | $2.88$ |

IL4 | $3.25$ | $3.61$ |

IL5 | $3.45$ | $4.31$ |

IL6 | $3.78$ | $4.27$ |

IL7 | $4.25$ | $5.06$ |

NL1 | $5.63$ | $6.46$ |

NL2 | $6.19$ | $7.20$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Verrelli, C.M.; Romagnoli, C.; Jackson, R.; Ferretti, I.; Annino, G.; Bonaiuto, V. *Phi*-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers. *Mathematics* **2021**, *9*, 1545.
https://doi.org/10.3390/math9131545

**AMA Style**

Verrelli CM, Romagnoli C, Jackson R, Ferretti I, Annino G, Bonaiuto V. *Phi*-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers. *Mathematics*. 2021; 9(13):1545.
https://doi.org/10.3390/math9131545

**Chicago/Turabian Style**

Verrelli, Cristiano Maria, Cristian Romagnoli, Roxanne Jackson, Ivo Ferretti, Giuseppe Annino, and Vincenzo Bonaiuto. 2021. "*Phi*-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers" *Mathematics* 9, no. 13: 1545.
https://doi.org/10.3390/math9131545