ABSTRACT Fifteen glands from clinically healthy bovine calves were evaluated using geometric morphometry methods. The right and left lobes were studied separately, comparing their differences in size and shape. The results reflected the presence of fluctuating asymmetry for the shape; that is, a deviation from the symmetry between lobes expected for this bilateral structure. No directional asymmetry was found. It is assumed that the degree of asymmetry is completely physiological; That is, fluctuating asymmetry, commonly used as a measure of developmental instability, would not represent in the case studied an inability of the gland to maintain its normal physiology.

RESUMEN Se evaluaron 15 glándulas de terneros, clínicamente sanos, mediante métodos de morfometría geométrica. Se estudiaron por separado los lóbulos derecho e izquierdo, comparando sus diferencias en tamaño y forma. Los resultados reflejaron la presencia de asimetría fluctuante para la forma; es decir, una desviación de la simetría entre lóbulos esperada para esta estructura bilateral. No se encontró asimetría direccional. Se asume que el grado de asimetría resulta totalmente fisiológica; o sea, que la asimetría fluctuante, comúnmente usada como medida de la inestabilidad del desarrollo, no representaría en el caso estudiado una incapacidad de la glándula para mantener su normal fisiología.

Abstract Seeking to define more robust and executable structures, this article presents a computational framework for the optimal design of steel truss frames, which integrates two software of great prominence in structural engineering, MATLAB and ANSYS. Computational interfaces were created to enable the automation of the iterative optimization process. The optimization algorithm and computational interfaces were developed on the MATLAB platform, while structural analyses were performed with the ANSYS Mechanical APDL platform. Some details of the computational implementation are presented herein. The objective is to minimize the cost of structures by determining the optimal positioning of nodal coordinates and the optimal choice of commercially available structural profiles - shape and size optimization, respectively. For shape optimization, a computational model was defined to automatically generate the geometry of the structure, maintaining some intrinsic relations of symmetry and collinearity between elements. The design constraints are critical nodal displacements, stresses, and slenderness of elements, following the requirements of ABNT NBR 8800:2008. To exemplify the application of the proposed methodology, this article presents the optimal design of trusses for roofs of industrial sheds, considering its non-linear geometric behavior, typical of this type of structure.

Resumen Introducción: La sínfisis del mentón aporta equilibrio, simetría y armonía al rostro. Por ser una estructura anatómica mandibular muy importante y altamente modificable en cirugía estética, se considera indispensable conocer su morfología y así obtener una correcta interacción en el diagnóstico, pronóstico y terapéutica médico-odontológica. Objetivo: Determinar la morfología sínfisis del mentón en jóvenes adultos de la ciudad de Quito- Ecuador, año 2019, mediante la identificación de las longitudes sagitales y coronales. Metodología: Este estudio tuvo un enfoque cuantitativo, tipo descriptivo-observacional, con temporalidad retrospectiva. Se calibraron, analizaron y trazaron 384 radiografías digitales laterales de adultos jóvenes de la ciudad de Quito-Ecuador, año 2019 en el software AutoCAD-2019. Resultados: Existió una elevada frecuencia de la forma geométrica cuadrada (60%) con respecto a la rectangular (40%), adicionalmente se identificó que la figura cuadrada es más frecuente en individuos de sexo femenino y sujetos clase I (56%) y II (70%), además de todos los patrones de crecimiento vertical hipodivergentes (58%), normodivergentes (64%) e hiperdivergentes (57%). Conclusiones: La forma cuadrada de la sínfisis del mentón es mayor que la forma rectangular. Por la originalidad de los resultados, se sugiere realizar nuevos estudios que correlacionen la morfología mentoniana con las diferentes estructuras cráneo-maxilofaciales.

Abstract Introduction: The chin symphysis contributes to the balance, symmetry and harmony of the face. Since it is an important anatomical jaw structure and highly modifiable in aesthetic surgery, it is considered crucial to know its morphology and, thus, obtain a correct interaction in the diagnosis, prognosis and medical-dental therapy. Objective: To determine the symphysis morphology of the chin in young adults in the city of Quito, Ecuador, in the year 2019, through the identification of sagittal and coronal lengths. Methodology: This study had a quantitative approach, descriptive-observational type, with retrospective temporality. We calibrated, analyzed and plotted 384 lateral digital radiographs of young adults from the city of Quito-Ecuador, year 2019 in the software AutoCAD-2019. Results: There was a high frequency of the square geometric shape (60%) in comparison to the rectangular one (40%). Additionally, it was identified that the square figure is more frequent in female individuals and class I (56%) and II (70%) subjects, in addition to all the hypodivergent (58%), normodivergent (64%) and hyperdivergent (57%) vertical growth patterns. Conclusions: The square shape of the chin symphysis is larger than the rectangular shape. Due to the originality of the results, it is suggested to carry out new studies that correlate chin morphology with the different cranio-maxillofacial structures.

Abstract We aim at finding, from purely theoretical analysis, the behavior that the refractive index should have within a cylindrical waveguide so that the radiation entering the system in a definite way is guided through it. Based on the criterion we have set in a previous article applying the Fermat’s extremal principle, in the framework of the geometrical optics, we depict the radiation confinement regions for refractive index profiles often used in the construction of waveguides, one step, multi-step and parabolic, by drawing upon the Legendre transform space as an intermediate resource in the process. We have also studied the possibility of performing the reverse path: for a wanted confinement region, to find the parameters defining the refractive index profile of the waveguide to be built. We conclude that such a process is possible as long as we know the shape of the profile. Under such restriction, our analysis allows us to deduce the characteristics that the guide should have so that the radiation entering with a given angle and at a certain distance from its axis remains confined. The technique can be used in design processes as a resource to limit the parameters that characterize the system.

Abstract We analyze the behavior of an optical focusing system, used by Richard Feynman in his famous physics classes. The system is a biconvex optical lens with symmetry of revolution around an axis, defined by the property that the rays that start from a point source, located in a focus on the axis, converge to an image point on the other side of the lens, so that the light waves on them arrive simultaneously at this last point. This ensures that the waves of a given frequency, traveling according to different paths from one focus to the other through air and glass, complete the same number of cycles when they reach the image point and therefore interfere in phase with each other. In this work we explicitly show that the profile of the Feynman lens must be hyperbolic, a form that for paraxial rays eliminates the spherical aberration (but not the chromatic aberration) of the lens.

Resume Analizamos el comportamiento de un sistema óptico de enfoque, utilizado por Richard Feynman en sus famosas clases de física. El sistema es una lente óptica biconvexa con simetría de revolución en torno a un eje, definida por la propiedad de que los rayos que parten de una fuente puntual, ubicada en un foco sobre el eje, convergen a un punto imagen al otro lado de la lente, de modo que las ondas luminosas sobre ellos llegan simultáneamente a este último punto. Esto asegura que las ondas de una frecuencia dada viajando según diferentes caminos desde un foco al otro a través de aire y vidrio, completan el mismo número de ciclos cuando alcanzan el punto imagen y por lo tanto interfieren en fase entre sí. En este trabajo mostramos explícitamente que el perfil de la lente de Feynman debe ser hiperbólico, forma que para rayos paraxiales elimina la aberración esférica (pero no la aberración cromática) de la lente.

The developmental stability of an organism is reflected in its ability to produce an ‘ideal’ form under a particular set of conditions. Bilateral structures in bilaterally symmetrical organisms offer a precise symmetry against which departures may be compared. The tool mostly used to estimate the development sta-bility is fluctuating asymmetry, which considers small random deviations that occur between the left and right sides of a bilateral trait. Fluctuating asymmetry is considered as the only form of asymmetry that can serve as a useful indicator of environmental/genetic stress. We summarized four decades of studies where fluctuating asymmetry was used to assess the effects of environmental stress in small mammals. This group of species has been widely used in ecological studies to infer environmental disturbances because of its wide range of characteristics. We selected 27 articles that were compiled with Google Scholar (Mountain View, CA) using “fluctuating asymmetry” and “small mammals” as key words, written in English and with ecological objectives. We focused our analyses on the approaches used to evaluate fluctuating asymmetry (linear measurements or geometric morphometrics), the stress factor (natural or anthropogenic), the region where the study was devel-oped, the number of traits used in the studies and the data sources, including measures obtained from samples of barn owl pellets, scientific collections and captured animals. The review shows the importance of including fluctuating asymmetry in ecological studies as a reliable, cheap and fast biological indicator of the effect of environmental stress on mammals.

La esta-bilidad del desarrollo de un organismo se refleja en la capacidad que posee de producir una forma “ideal” bajo un conjunto particular de condiciones. Las estructuras bilaterales en organismos con simetría bilateral ofrecen una simetría precisa sobre la cual se pueden comparar desviaciones. La herramienta más utilizada para estimar la estabilidad del desarrollo es la asimetría fluctuante, la cual considera las pequeñas desviaciones aleatorias que ocurren entre los lados derecho e izquierdo de rasgos bilaterales, y es la única asimetría adecuada como indicador de estrés ambiental/genético. Resumimos cuatro décadas de estudios en donde la asimetría fluctuante fue utilizada para evaluar el efecto de estrés ambiental sobre pequeños mamíferos. Este grupo de especies ha sido ampliamente utilizado en estudios ecológicos para inferir perturbaciones ambientales debido a sus variadas características. Se seleccionaron 27 artículos compilados con Google Académico, utilizando “asimetría fluctuante” y “pequeños mamíferos” como palabras claves, escritas en inglés y con objetivos ecológicos. Centramos nuestro análisis en los enfoques utilizados para evaluar la asimetría fluctuante (medidas lineales o morfometría geomé trica), el factor de estrés (natural o antropogénico), la región donde se desarrolló el estudio, el número de rasgos utilizado en los estudios y las fuentes de datos (muestras de pellets de lechuza, colecciones científicas y capturas directa de animales). La revisión muestra la importancia de incluir la asimetría fluctuante en estudios ecológicos como un indicador biológico confiable, económico y rápido del efecto del estrés ambiental sobre los mamíferos.

Abstract In this paper, energy absorption and deformation capacity of circular thin-walled members with elliptical cut-outs are investigated both numerically and experimentally. Thin-walled members possess the uniform height, thickness, average cross sectional area, material and volume are subjected to axial quasi-static loading. To conduct such tests, special fixture arrangement is designed for placing the specimen in the compression loading machine. The deformation mechanisms and the corresponding collapse mode along with its energy absorption of the thin-walled tubes were investigated in detail for various aspect ratios (0.315, 0.523, 0.854, 1, 3.375 and 4.08). The explicit finite element code ABAQUS was then employed to perform the numerical studies inview of mitigating the influence of cutout shape, location and symmetry on energy absorption and crush characteristics. In aspect ratio of 4.08, whose major axis length of 24.5 mm observed maximum crash force efficiency (CFE) of 14% possessing symmetric discontinuity. The results of experimental and simulations are in good agreement and shows that the location and symmetry of cutouts had considerable effect on collapse crushing behaviour.

Abstract In this paper we consider a particle moving on a curved surface. From a variational principle, we write the equation of motion and the constraining force, both in terms of the Darboux frame adapted to the trajectory, that involves geometric information of the surface. By deformation of the trajectory on the surface, the constraining force and equation of motion of the perturbation are obtained. We show that the transversal deformation follows a generalized Raychaudhuri equation that contains extrinsic information besides the geodesic curvature. Results in the case of surface with axial symmetry can be parametrized in terms of the angular momenta.

Resumen En este artículo consideramos el movimiento de una partícula sobre una superficie curvada. De un principio variacional, encontramos las ecuaciones de movimiento y la fuerza de constricción, en el marco de Darboux adaptado a la trayectoria. Deformando estas ecuaciones, encontramos que la perturbación transversal satisface una ecuación de Raychaudhuri generalizada que contiene información extrínseca además de la curvatura geodésica. En el caso de superficies con simetría axial, los resultados se pueden parametrizar en términos del momento angular.

Abstract Introduction Early detection of suspicious skin lesions is critical to prevent skin malignancies, particularly the melanoma, which is the most dangerous form of human skin cancer. In the last decade, image processing techniques have been an increasingly important tool for early detection and mathematical models play a relevant role in mapping the progression of lesions. Methods This work presents an algorithm to describe the evolution of the border of the skin lesion based on two main measurable markers: the symmetry and the geometric growth path of the lesion. The proposed methodology involves two dermoscopic images of the same melanocytic lesion obtained at different moments in time. By applying a mathematical model based on planar linear transformations, measurable parameters related to symmetry and growth are extracted. Results With this information one may compare the actual evolution in the lesion with the outcomes from the geometric model. First, this method was tested on predefined images whose growth was controlled and the symmetry known which were used for validation. Then the methodology was tested in real dermoscopic melanoma images in which the parameters of the mathematical model revealed symmetry and growth rates consistent with a typical melanoma behavior. Conclusions The method developed proved to show very accurate information about the target growth markers (variation on the growth along the border, the deformation and the symmetry of the lesion trough the time). All the results, validated by the expected phantom outputs, were similar to the ones on the real images.

Pre-Inca Diaguita decorative patterns present a visual logic, characterized by the use of complex symmetries, optical illusions, an endless variety of simple geometric forms, horror vacui, hypnotic attraction and other techniques. These aspects, as well as the association of this visual art with an animal alter ego (jaguar) and evidence of the use of hallucinogenics, suggest a cultural link with specific ethnographic (shipibo-conibo) and archaeological (mojocoya) visual art. We propose that in this case we are dealing with visual "technologies of enchantment" (Gell 1998) that are deemed to have social agency and captivate the viewer with extremely complex, apparently animated non-mimetic abstract patterns. In ethnographic contexts, similar decorative patterns play an important role in shamanic healing strategies.

El arte diaguita preincaico presenta una lógica visual caracterizada por el empleo de complejas simetrías, ilusión de movimiento y vibración, infinita variabilidad a partir de elementos geométricos acotados, horror vacui, atracción hipnótica, entre otros. Estas características, así como la asociación de este arte a un alter ego animal (jaguar) y la evidencia de consumo de alucinógenos, permiten plantear un vínculo cultural con cierto arte visual de naturaleza arqueológica (mojocoya) y etnográfica (shipibo-conibo). Planteamos que, en este caso, se trata de "tecnologías de encantamiento" (Gell 1998) que tienen agencia y cautivan al espectador, pues los patrones abstractos poseen características de animación no mimètica de difícil comprensión. En contextos etnográficos, estos patrones decorativos cumplen un importante rol en estrategias de sanación chamánicas.

Este artigo apresenta um panorama acerca da didática da matemática e da simetria, além de uma proposta de atividades. Baseando-se em teorias que envolvem a didática da matemática, elaborou-se uma sequência didática fazendo uso de materiais acessíveis, possível de ser realizada em turmas numerosas. Paralelamente, pela Etnomatemática, percebe-se que ao longo da história, os padrões simétricos relacionam-se com o respectivo momento cultural. Nessa pesquisa, explora-se uma abordagem das transformações geométricas para o Ensino Fundamental. Por meio da aplicação e análise da sequência didática detalhada nesse trabalho, acredita-se ter contribuído para que surjam outras possibilidades, considerando sempre as especificidades de cada turma.

This article presents an overview about the didactics of mathematics and symmetry, as well as an activity proposal. Based on theories involving didactics of mathematics, an instructional sequence using accessible materials, capable of being accomplished in numerous classrooms, was designed. At the same time, by means of ethnomathematics, it is noticed that, throughout history, symmetrical patterns are related to the respective cultural moment. In this search, an approach of geometric transformations for K-12 education is explored. Through this elaboration, application, and analysis of an instructional sequence, we believe that we have contributed for the emergence of other possibilities, always considering the specificities of each class.

INTRODUCTION: This work aims to pre-operatively manufacture custom-made low-cost implants and physical models (‘biomodels’) of fractured skulls. The pre-DOI: operative manufacturing of biomodels and implants allows physicians to study and plan surgery with a greater possibility of achieving the expected result. Customization contributes to both the esthetic and functional outcome of the implant because it considers the anatomy of each patient, while the low cost allows a greater number of people to potentially benefit. METHODS: From CT images of a fractured skull, a CAD model of the skull (biomodel) and a restorative implant were constructed digitally. The biomodel was then physically constructed with 3D Printing, and Incremental Sheet Forming (ISF) was used to manufacture the implant from a sheet of pure grade 2 titanium. Before cutting the implant’s final shape from a pre-formed sheet, heat treatment was performed to avoid deformations caused by residual stresses generated during the ISF process. RESULTS: A comparison of the dimensions of the implant and its respective CAD biomodel revealed geometric discrepancies that can affect both functional and aesthetic efficiency. Nevertheless, the final shape preserved symmetry between the right and left sides of the skull. Electron microscopy analysis did not indicate the presence of elements other than pure titanium. CONCLUSION: Dimensional variability can be decreased with changes in the manufacturing process (i.e., forming and cutting) and the heating ramp. Despite biomedical characteristics, there was no contamination of the implant by harmful chemical elements. 3D Printing was effective in making the biomodel, enabling pre-operative planning and improving physician-patient communication. Current results indicate that ISF is a process that can be used to obtain custom-made implants.

The aim of this study was to investigate the skills and deficiencies of senior university students enrolled in a mathematics education program with the concepts of symmetry and rotation of geometric figures. The study was conducted with 32 prospective teachers in the Department of Education at a public university located in Western Turkey. This descriptive study was designed with a case study. A structured interview technique was used for data collection. A measurement tool consisting of 12 drawing problems testing symmetry (5 problems) and rotation (7 problems) ability was used in the study. Descriptive statistical methods were used for data analysis. The drawings were analyzed individually, and the students' mistakes and deficiencies were categorized. According to the results of the study, prospective elementary mathematics teachers did not have difficulty in drawing the symmetry of an object or in determining the symmetry axis. However, while they could rotate the figure when a rotation axis was provided, they failed to rotate it in the absence of an axis. In addition, prospective elementary mathematics teachers generally failed to find the center of the given rotated figures. The results of the present study were consistent with results in the literature.

El objetivo de este estudio fue investigar las habilidades y dificultades de los estudiantes de un curso de formación de profesores de matemáticas en relación con los conceptos de simetría y rotación de figuras geométricas. El estudio se llevó a cabo con 32 futuros profesores en un departamento de educación de una universidad pública ubicada en el oeste de Turquía. Éste es un estudio descriptivo que está diseñado con un estudio de caso. Para la recopilación de los datos se aplicó una técnica de entrevista estructurada. En el estudio se utilizó una herramienta de medición consistente de doce problemas de dibujo, cinco sobre la capacidad en la simetría y siete en la de rotación. Para el análisis de datos se emplearon métodos de estadística descriptiva. Los dibujos fueron analizados de forma individual y se clasificaron los errores y las deficiencias de los estudiantes. De acuerdo con los resultados del estudio, los futuros profesores de matemáticas elementales no tuvieron dificultad en dibujar la simetría de un objeto o en hallar el eje de simetría. Sin embargo, mientras que ellos pudieron rotar la figura cuando se proporcionaba un eje de rotación, fallaron al rotar el objeto cuando no se le dio dicho eje. Además se observó que generalmente se equivocaban al localizar el centro de las figuras rotadas. Los resultados del presente estudio fueron consistentes con los de la literatura.

Este artigo tem como objetivo apresentar uma possibilidade de trabalho com o tema simetria na sala de aula do Ensino Fundamental. A proposta baseia-se na pesquisa desenvolvida no Mestrado Profissional em Ensino de Ciências e Matemática realizada segundo uma abordagem qualitativa. Na pesquisa, foi realizado um trabalho de campo que consistiu no desenvolvimento de uma sequência didática planejada sob a perspectiva do ensino de Matemática através da resolução de problemas com alunos de uma turma de sétimo ano do Ensino Fundamental. Na elaboração da sequência didática, foram levados em consideração aspectos como a intuição e a visualização, relevantes ao estudo da geometria. Durante a realização da sequência didática os diálogos entre os alunos e professor foram gravados e, posteriormente, transcritos. Para a análise dos dados adotaram-se os procedimentos da fenomenologia. Na interpretação dos dados vimos que o ensino de simetria através da resolução de problemas potencializa a aprendizagem. Mostra-se ainda que, em um ambiente de investigação, os alunos são capazes de identificar propriedades, argumentar sobre as características geométricas percebidas e justificar suas afirmações.

This paper presents an alternative way of working with the theme of symmetry in the elementary school classroom. The proposal is based on qualitative research developed in the Professional Masters degree program in Science and Mathematics Teaching. We conducted field-work consisting of applying a sequence of activities for students in the seventh grade. The sequence was developed from the perspective of mathematics teaching using problem solving, taking into consideration aspects relevant to the study of geometry, such as intuition and visualization. In carrying out the activities, the dialogues between students and teacher were recorded and later transcribed. For data analysis we used the procedures of phenomenology. When interpreting the data, we observed that the teaching of symmetry using problem-solving enhances learning. We also found that, in an investigative environment, students are able to identify properties, argue about the geometric characteristics, and justify their opinions.

This article reports on regularities observed in learners' preconceptions of reflective symmetry. Literature suggests that the very existence of such regularities indicates a gap between what learners know and what they need to know. Such a gap inhibits further understanding and application, and hence needed to be investigated. A total of 235 Grade 11 learners, from 13 high schools that participate in the First Rand Foundation-funded Mathematics Education project in the Eastern Cape, responded to a task on reflective symmetry. Our framework for analysing the responses was based on the taxonomy of structure of the observed learning outcome. The results indicated that 85% of learner responses reflect a motion understanding of reflections, where learners considered geometric figures as physical motions on top of the plane. While this understanding is useful in some cases, it is not an essential aspect of mapping understanding, which is critical for application in function notations and other analytical geometry contexts. We suggest that if this gap is to be closed, learners need to construct these reflections physically so that they may think of reflections beyond motion.