Golden ratio and the heart, God and the science

The Golden Ratio in Pediatric Wrist Anatomy: A Divine Symmetry

Introduction: The golden ratio, which equals 1.61803…, and is usually defined by the Greek letter φ (phi), has attracted broad attention for a long time. It has been found in many phenomena in the universe including, body symmetry and locomotion. Within this context, the purpose of our study was to evaluate normal morphometric measurements of the wrist in the pediatric population and to identify if phi (φ) is part of the distal radioulnar joint. 

Methods: We retrospectively reviewed the hospital records of all skeletally immature patients requiring surgical intervention for distal radius fracture in our unit between January 2010 and January 2017. We define and describe a reproducible method to measure the ratio of the distal radial and ulnar physes.

Results: A total of 268 patients were included with a mean age of 9.41 (3-16) years and a mode of 7 years. Some 63.4% were boys -- 43.3% were right-sided injuries and 56.7% were left-sided injuries. The ratio between the total width of the radial and ulnar growth plates and the radial growth plate closely approximated φ; the mean of this ratio in all the patients included was 1.619684 (1.5848-1.6643). Most of the injuries happened in the summertime, between May and August.

Conclusion: We found that the golden ratio exists in our body to play its harmony in the pediatric wrist joint. We believe that with the support of further studies, the golden ratio might yield diagnostic and prognostic implications in the treatment of distal radius/ulnar fractures or abnormalities in this population.

Bamboo-like π-Nanotubes with Tunable Helicity and Circularly Polarized Luminescence

We report the fabrication of an exotic bamboo-like π-nanotube via the hierarchical self-assembly of a dipeptide-substituted naphthalenediimide gelator with tunable helicity and circularly polarized luminescence (CPL). It was found that in the presence of trifluoroacetic acid (TFA) the gelator molecules self-assembled into a bamboo-like π-nanotube, which is composed of truncated nanocones and CPL active. When defining the diameter ratio of the lower to upper edge of each nanocone as a parameter to express the helicity of different nanotubes, it was found that both the helicity and CPL of these nanotubes can be adjusted by the amount of TFA. Moreover, the helicity of the nanotube can be conveyed to the achiral quantum dots (QDs) and produce a hybrid nanotube/QDs CPL active materials with adjustable dissymmetry factor. This work finds a new type self-assembled bamboo-like π-nanotube and unveils their helicity and CPL control.

Golden Gait: An Optimization Theory Perspective on Human and Humanoid Walking

Human walking is a complex task which includes hundreds of muscles, bones and joints working together to deliver harmonic movements with the need of finding equilibrium between moving forward and maintaining stability. Many different computational approaches have been used to explain human walking mechanisms, from pendular model to fractal approaches. A new perspective can be gained from using the principles developed in the field of Optimization theory and in particularly the branch of Game Theory. In particular we provide a new insight into human walking showing as the trade-off between advancement and equilibrium managed during walking has the same solution of the Ultimatum game, one of the most famous paradigms of game theory, and this solution is the golden ratio. The golden ratio is an irrational number that was found in many biological and natural systems self-organized in a harmonic, asymmetric, and fractal structure. Recently, the golden ratio has also been found as the equilibrium point between two players involved into the Ultimatum Game. It has been suggested that this result can be due to the fact that the golden ratio is perceived as the fairest asymmetric solution by the two players. The golden ratio is also the most common proportion between stance and swing phase of human walking. This approach may explain the importance of harmony in human walking, and provide new perspectives for developing quantitative assessment of human walking, efficient humanoid robotic walkers, and effective neurorobots for rehabilitation.

Harmony as a convergence attractor that minimizes the energy expenditure and variability in physiological gait and the loss of harmony in cerebellar ataxia

BACKGROUND

The harmony of the human gait was recently found to be related to the golden ratio value (ϕ). The ratio between the duration of the stance and that of the swing phases of a gait cycle was in fact found to be close to ϕ, which implies that, because of the fractal property of autosimilarity of that number, the gait ratios stride/stance, stance/swing, swing/double support, were not significantly different from one another. We studied a group of patients with cerebellar ataxia to investigate how the differences between their gait ratios and the golden ratio are related to efficiency and stability of their gait, assessed by energy expenditure and stride-to-stride variability, respectively.

METHODS

The gait of 28 patients who were affected by degenerative cerebellar ataxia and of 28 healthy controls was studied using a stereophotogrammetric system. The above mentioned gait ratios, the energy expenditure estimated using the pelvis reconstructed method and the gait variability in terms of the stride length were computed, and their relationships were analyzed. Matching procedures have also been used to avoid multicollinearity biases.

FINDINGS

The gait ratio values of the patients were farther from the controls (and hence from ϕ), even in speed matched conditions (P=0.011, Cohen's D=0.76), but not when the variability and energy expenditure were matched between the two groups (Cohen's D=0.49). In patients with cerebellar ataxia, the farther the stance-swing ratio was from ϕ, the larger the total mechanical work (R²_adj=0.64). Further, a significant positive correlation was observed between the difference of the gait ratio from the golden ratio and the severity of the disease (R=0.421, P=0.026).

INTERPRETATION

Harmony of gait appears to be a benchmark of physiological gait leading to physiological energy recovery and gait reliability. Neurorehabilitation of patients with ataxia might benefit from the restoration of harmony of their locomotor patterns.