Fig 2Fig 1

When Kepler choose the title “Harmonices Mundi” (The Harmony of the Word) for his book of 1619 reporting the discover of the third law of planetary motion, he aimed at discussing physical phenomena through the lens of an intrinsic harmony at the basis of geometrical forms and physical systems. Also human movements often show an intrinsic harmony, as noted by the neurophysiologist Luria, who introduced the concept of “kinetic melody”. Kinetic melodies are dynamic patterns of movements at the basis of a vast repertoire of possible human movements such as walking, speaking, reaching, running, etc. containing an intrinsic harmony and fluidity. Some pathologies can disrupt these patterns, and neurorehabilitation should aim at their recovery, as suggested by Oliver Sacks who talked about the need for patients of a restoration of movements based on their natural rhythm embedded into kinetic melodies.

Walking is probably the most important human motor act that is perfectly embedded in a kinetic melody. In healthy subjects, the symmetry of our body is reflected in symmetric kinematic and kinetic patterns that were performed by lower limbs in counterphase. Also upper limbs are moved in symmetric manner but in counterphase each other and with respect to ipsilateral lower limbs.

For symmetry, it is obvious that one foot starts a stride (id est, its gait cycle) in the moment in which the other foot is in the middle of its own stride, id est at 50% of its gait cycle. It implies that the trunk performed the same movement twice in the sagittal plane during each stride. According to this gait structure, it has been introduced a parameter called Harmonic Ratio that is the ratio between the sum of the amplitudes of even harmonics and that of odds harmonics of a signal (for example trunk acceleration or displacement) evaluated along antero-posterior or cranio-caudal body axis (along medio-lateral axis, it has been computed as the ratio between odds and even harmonics). The Harmonic Ratio has been identified as an important prognostic factor of the risk of fall in elderly. For its name, this parameter has been often related to the harmony of walking, but in this case “harmonic” mainly refers to the fact that it has been computed by the harmonics obtained by means of Fast Fourier Transformation. The harmony that this parameter can assess is the symmetry of two consecutive steps contained into a stride, a feature possible only if each steps cover the 50% of the entire gait cycle.

The gait cycle has been characterized also by another percentage, that is the duration of the stance phase, in which the foot makes contact with the ground, with respect to the swing phase, in which the foot advances in the air. Much of the literature agrees that stance covers the 60-62% of a physiological gait cycle, and swing the remaining 40-38%. This is considerable as an important benchmark of human gait, and an alteration of stance to swing proportion has usually been identified as a sign of pathological gait. Despite it, the reasons for which this proportion is a so reliable parameter of physiological gait remained for years poorly investigated.

Fig 2

In 2013, we noted that the ratio between stance and swing coincides with the golden ratio. This is an irrational number, symbolically indicated with f that is = (1+50.5)/2 (about 1.618034). This number has been related to the problem reported by Euclid in III century b.C. to cut a given straight line so that the proportion between the shorter part to the longer one is the same as the longer part to the whole. The greek letter f was chosen to indicate this number in honor of the sculptor Phidia, who supervised the construction of the Parthenon, the façade of which is a golden rectangle, i.e. a rectangle having lengths of sides in the proportion of f. Henceforth, mathematicians, physicists, biologists, architects, and artists have been interested in the intrinsic symmetric properties of this number, that was also called “divine proportion” during the Renaissance, from the title of a book (“De divina proportione”) written by Luca Pacioli and illustrated by Leonardo da Vinci. Could Kepler be not fashinated by a so “harmonic” number? In 1611, he discovered that f is the asymptotic limit of the ratio of consecutive Fibonacci numbers. He stated that geometry has two great treasures: one is the theorem of Pythagoras, and the other the division of a line into extreme and mean ratio, and he combined them in the Kepler’s triangle.

f is the simplest example of fractal mathematics, where the term “fractal” refers to structures composed of subunits, in which the larger-scale structure resembles the subunit structure, in accordance to a property called self-similarity.

A wide variety of seemingly disparate physical and biological systems, such as leaf disposition on plant stems and seed arrangement on flower heads, spiral structures of galaxies and mollusks, quantum phase transitions, nucleotide frequencies, cell and shell growth, show similar harmonic characteristics related to f. In human sciences, f has been observed with regard to body proportions and aesthetic preferences.

The fact that stance/swing approximates f, implies an intrinsic fractal frame hidden below the orchestrated repetitive structure of physiological gait that can be represented with the following equation:

Screen Shot 2016-03-02 at 2.26.35 pm

As already known since ancient Greek eras, also anthropometry has proportions approximately close to f. In another study we found that the artificial alteration of body segment proportions (obtained with special shoes with high outsoles) affected the gait ratio (stance/swing), whereas no changes occurred during walking with the adjunction of extra masses on body segments. It suggests that the relationship between anthropometric and gait golden proportion may be linked to the pendular mechanism of walking, being the oscillation period of a pendulum dependent only on length and not on mass.

However, the gait ratio is altered in pathological conditions without any alterations in anthropometric proportions. In a clinical study, we found a significant alteration of gait ratio in patients with Parkinson’s disease. This alteration resulted quite resistant to dopamine administration. The results reported in that study seemed to suggest that the loss of gait kinetic melody of patients with PD may be related to deficits in the cerebellum – globus pallidus – spinal central pattern generator pathway, that is involved into the perception and production of harmonic features, and less sensitive to dopamine than neural circuits involved in the other motor tasks. The kinetic melody of walking may be related to the production of internal cues that can be altered in a pathology such as Parkinson’s Disease, whereas for healthy subjects the equilibrium point could be the same for anthropometry and gait and represent a sort of resonant frequency at which biomechanics and motor control converge. Further studies should investigate the hypothesis that human anthropometry phylogenetically evolved towards proportions favouring the harmony of walking. I am pretty sure that this fascinating hypothesis could have intrigued even Kepler.

 

References

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

Iosa M, Morone G, Bini F, Fusco A, Paolucci S, Marinozzi F. The connection between anthropometry and gait harmony unveiled through the lens of the golden ratio. Neurosci Lett. 2016;612:138-44.

Iosa M, Morone G, Fusco A, Marchetti F, Caltagirone C, Paolucci S, Peppe A. Loss of fractal gait harmony in Parkinson’s Disease. Clin Neurophysiol. 2016;127(2):1540-6.

 

About the author:

Marco Iosa

Clinical Laboratory of
Experimental Neurorehabilitation
Fondazione Santa Lucia I.R.C.C.S., Rome

GOLDEN GAIT: A bridge between golden ratio and harmony of walking (by Marco Iosa)

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