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The axisymmetric geometries of both internal and external profiles of church bells, along with the properties of the bell metal, are of vital importance to their acoustics. The role of “golden geometry” in determining the visual characteristics of the outer profile of a modern English church bell is presented. It is shown that a golden rectangle, a golden triangle, and a golden angle are all important factors. Three angular parameters are identified as measures of these attributes of the bell’s “goldenness”. Two of these features are shown to result from an underlying regular pentagon with one side defined by the bell’s mouth. The centre of this pentagon determines the location of the bell’s shoulders. Further minor features are also shown to be determined by golden geometry. The evolution of the three angular parameters over the previous millennium is included, showing a tendency to trend quickly towards the golden values in all the cases. Ultimately, the angular parameters for the English church bell of focus here are used for comparison with other modern European bells studied previously. The bells considered all displayed at least two of the three major golden features to an agreement of better than 1%.

Background: With the shift in focus from the hard to soft tissue in orthodontic diagnosis and treatment planning, the scope of orthodontic treatment has grown beyond achieving an ideal occlusion, also to achieve ideal aesthetic facial proportions. Since time immemorial, the various horizontal and vertical proportions of an ideal face is considered to follow the golden/divine ratio. Aim: The aim of this study is to assess the vertical and horizontal proportions of the face and their relation of phi in males and females of Dakshina Kannada population. Materials and Methods: The study was carried out on 100 subjects, which included 50 males and 50 females, aged between 18 and 30 years. Photographs were taken under standardized condition and adjusted to a standardized image size (5″ × 4″). Adobe Photoshop CS5.1 software was used for making all the measurements. A total of 11 vertical and horizontal ratios were independently measured for males and females, and their relation to phi was assessed. The data were subjected to Shapiro-Wilk test for testing the normality. Homogeneity of variance assumption was tested by using Levene statistic homogeneity of variance. One sample t test was used for the comparison between phi and mean vertical and horizontal ratios in males and females. Results: In males and females of the Dakshina Kannada, a statistically significant correlation was seen between all the horizontal and vertical facial proportions and phi (p < 0.05). Between male and female groups, a significant difference was noted in the intertemporal/intercanthal ratio, interalae/nose width and Intereye-soft menton/ala-soft menton (p < 0.05). Conclusion: Golden proportion can be used as a guideline in orthodontic treatment planning. However, it should not be the decisive factor in determining the facial attractiveness.

Background The purpose of this article is to explore the claims often cited in scientific journals regarding the golden ratio, and its proposed link to beauty and idealized forms in nature, including idealized human proportions. Main body Claims made in the nineteenth century through to the modern day in the clinical literature do not appear to be supported by evidence. Short conclusions There is no convincing evidence that the golden ratio is linked to idealized human proportions or facial beauty. There is currently no evidence to support the use of the golden ratio in orthognathic or facial aesthetic/reconstructive surgical planning or analysis of results.

Introduction The navel contributes to abdominal surface identity and beauty. In Western societies, the display of the navel in women’s fashion has grown and, nowadays, women are much more concerned about its shape and position. Despite this, few studies are available on navel surface anatomy and there is no standardization regarding its proper placement in cosmetic abdominoplasty. Materials and Methods In this observational study, we analyzed navel shape and position on 81 high quality pictures, having been chosen as top 2013 bikini models by editors of mass media. An on-line survey on navel shape and position has been made via facebook.com, involving 1,682 people. Results The analysis revealed that navel position is quite variable based on the proportions analyzed with an acceptable narrow data spread of the xiphoid–umbilicus:umbilicus-abdominal mean crease ratio of 1.62 ± 0.16. The data dispersion for the other three ratios was wider, making them less reliable as references. The most appreciated navel shape was the vertical oval with superior hooding (82 %), and the less appreciated ones were the horizontal oval (29 %) and the protruding shape (47 %). When comparing navel position on the same body, the majority of participants choose the one with the navel relocated according to the golden ratio (i.e., 1.618) Conclusion The most attractive navel position is located at the xiphoid–umbilicus:umbilicus-abdominal crease golden ratio. Bony landmarks seem to be not reliable as references for proper navel positioning. The use of the Fibonacci (golden mean) caliper intraoperatively might aid in proper positioning of the navel in abdominoplasty. No Level Assigned This journal requires that authors assign a level of evidence to each submission to which Evidence-Based Medicine rankings are applicable. This excludes Review Articles, Book Reviews, and manuscripts that concern Basic Science, Animal Studies, Cadaver Studies, and Experimental Studies. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors http://www.springer.com/00266 .

The aim of this study is to demonstrate the use and the effectiveness of cephalometry and golden proportions analysis of the face in planning prosthetic treatments in totally edentulous patients. In order to apply this method, latero-lateral and posterior-anterior X-rays must be performed in addition to the common procedure. Two main concerns for totally edentulous patients are the establishment of the vertical dimension and the new position of the occlusal plane. The divine proportion analysis was carried out by the use of a golden divider. The prosthetic protocol was divided into three steps and a case was selected for better understanding. Referring to the golden relations, if the distance from the chin to the wing of the nose is 1.0, the distance from the nose to eye is 0.618. This proportion is useful and effective in determining the correct prosthetic vertical dimension. The incisal margin of the lower incisor must be positioned between Point A (A) and protuberance menti (Pm) according to the gold ratio 0.618 of the total height A-Pm. Posteriorly the occlusal plane must be placed 2 mm below the divine occlusal plane (traced from the incisal margin of lower incisors to Xi point). A prosthesis made in accordance with cephalometric parameters and divine proportions of the face helps to improve the patient’s aesthetics, function and social personality.

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.

Objectives The aim of our study was to investigate whether the parameters of ideal angles, ratios, and divine proportions that have been previously mentioned in the literature influence the aesthetic evaluation of Turkish adolescents. Materials and Methods A total of sixty patients (30 males and 30 females) were enrolled in this study. Ages of the subjects ranged from 9 to 17. Pre-treatment extra-oral photographs were taken to evaluate facial aesthetics. Two distinct panels consisting of 50 orthodontists and 50 laypersons were created for scoring the photographs of the patients. Scoring was performed using the VAS scale. Twenty-seven ratios and 19 divine proportions were measured in frontal photographs, and 26 angles were measured in profile photographs. Results Pearson correlation was used to determine the relationship between the photograph analysis measurements and VAS scores, and then regression analyses were performed to disclose to what extent significant values may warrant the term beauty. Conclusion As a result of our investigations, none of the golden proportions was associated with facial aesthetics according to both orthodontists and laypersons. According to other angles and ratios that were measured, it was determined that orthodontists noticed the sagittal position of the lower jaw, the distance between the eyes and length of the face, whereas laypersons noticed only the distance between the eyes and length of the face. Level of Evidence V This journal requires that the authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

Objective: To evaluate the facial divine proportion and its relationship with facial attractiveness in North Indian population. Materials and Methods: For evaluation of various facial proportions, standardized frontal facial photographs of total 300 subjects between 18 and 30 years of age were obtained. Black and white copies of these photographs were presented in front of an evaluation jury for assigning scores of facial attractiveness and finally 130 attractive subjects were selected. These subjects were divided into two groups, Group I (attractive females n = 65) and Group II (attractive males n = 65) and they were further analyzed for various parameters of facial proportions. Unpaired Student′s t-test was used to compare both groups. Results: Group I showed that five of seven vertical facial proportions were close to divine proportion (1.618) whereas only two vertical proportions in Group II were close to it. Transverse facial proportions in both groups deviated more from divine proportion (1.618) and were closer to silver proportion (1.414). Conclusions: Most of the facial proportions of attractive females in the North-Indian population were close to the divine proportion. Thus, facial divine proportion could be an important factor in the perception of facial attractiveness of North-Indian attractive females.

Background In 2000, the author started observing legs considered to be attractive. The goal was to have an ideal aesthetic model and compare the disparity between this model and a patient’s reality. This could prove helpful during leg sculpturing to get closer to this ideal. Postoperatively, the result could then be compared to the ideal curves of the model legs and any remaining deviations from the ideal curves could be pointed out and eventually corrected in a second session. The lack of anthropometric studies of legs from the knee to the ankle led the author to select and study attractive legs to find out the common denominators of their beauty. Method The study consisted in analyzing the features that make legs look attractive. The legs of models in magazines were scanned and inserted into a PowerPoint program. The legs of live models, Barbie dolls, and athletes were photographed. Artistic drawings by Leonardo da Vinci were reviewed and Greek sculptures studied. Sculptures from the National Archaeological Museum of Athens were photographed and included in the PowerPoint program. Results and Conclusion This study shows that the first criterion for beautiful legs is the straightness of the leg column. Not a single attractive leg was found to deviate from the vertical, and each was in absolute continuity with the thigh. The second criterion is the similarity of curve distribution and progression from knee to ankle. Level of Evidence V This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors at www.springer.com/00266 .

Beauty is the phenomenon of experiencing pleasure, through the perception of balance. According to some authors, attractive faces have ideal proportions that are related to the Divine Proportion (1.618:1). The purpose of this study was to evaluate the Divine Proportion in the human profile and investigate the effects of this ratio on the perception of the beauty of the profile. We conducted a cross-sectional study to determine the mentioned aims. In this cross-sectional study, fifty females in the age range of 20-25 years, with acceptable profiles were selected. Standardized profile silhouettes were each given a score by 20 judges (consisting of 10 men and 10 women) using the visual analog scale. Five landmarks were determined and five ratios were measured on the silhouettes by using the adobe photoshop program. The Student's t test was used to compare profile proportions of the subjects. Statistical analysis showed none of the proportions had the mean of 1.618, but in subjects having higher esthetic scores, trichion (Te)-soft tissue menton (Me):Tr-subnasale (Sn) and Tr-Me:soft tissue nasion (N)-Me, with the mean of 1.58 were closer to Divine Proportion. The results suggested the perception of beauty is influenced by the Divine Proportions and Tr-Me:Tr-Sn and Tr-Me:N-Me are the most influential ratios in the perception of the beauty of profile; however, if the Divine Proportion is to be used in treatment planning, it should be used along with other factors.