Orientador: Prof. Dr. Márcio Fabiano da Silva / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, Santo André, 2018. / Nesta dissertação exploramos o número de ouro, ou razão áurea F presente em diferentes contextos como segmento dividido em média e extrema razão, o retângulo áureo, os triângulos áureos e o pentágono regular. Em especial, os triângulos áureos de um pentágono regular, chamados de triângulos centrais de ouro e a obtenção da razão áurea através das relações das áreas dos triângulos; através das relações entre os incentros, ortocentros, circuncentros e lados dos triângulos. Construímos círculos de 9 pontos e destacamos a presença da razão áurea, em sua maioria, na reta central dos triângulos de ouro, limitadas pelo vértice comum e o ponto médio de sua base isósceles e o centro de cada circunferência de 9 pontos. Apresentamos ainda, propostas de atividades, em sala de aula, para abordar de maneira diferenciada com os alunos o número de ouro F, acompanhadas de suas resoluções presentes em um apêndice. / In this dissertation we explore the number of gold, or golden ratio F present in different contexts as segment divided into average and extreme ratio, golden rectangle, golden triangles and regular pentagon. In particular, the golden triangles of a regular pentagon, called central golden triangles and the attainment of the golden ratio through the relations of the triangle areas; through the relationships between the incenters, orthocentres, circumcenters, and sides of the triangles. We construct a circle of 9 points and we emphasize the presence of the golden ratio, for the most part, in the central line of the golden triangles, limited by the common vertex and the midpoint of its isosceles base and the center of each circumference of 9 points. We also present proposals for activities in the classroom to approach students in a differentiated way the number of gold F, accompanied by their resolutions present in an appendix.

The bias against attractiveness is fairly implicit and furthermore, powerfully impacts people’s subsequent impressions of and behaviors toward others (Cash, Gillen, & Burns, 1977; Dion et al., 1972). Pallet, Link and Lee (2010) examined the effect of various facial spatial configurations on attractiveness and found that raters rated faces as most attractive when the eyeto-mouth ratio approximated 36% of the face length (the "golden ratio"), which coincides with the measurements of an average and thus more attractive face. The present study examined the extent to which the distance of these objectively measured facial features affected mentors’ perceptions of their protégés, the subsequent mentoring given to them, and the protégés’ own behavior (e.g. seek feedback, request specific information).The gender composition of the mentor-protégé dyad was expected to moderate these relationships. I also examined whether, given the expected effects of facial measurements, withholding access to visual cues would affect mentor perceptions and behavior. Participants were 118 mentor/protégé dyads from a large Southeastern university who volunteered to participate in a formal online peer mentoring program. After seeing their protégés’ profiles (and for those in the experimental condition, a picture), mentors chatted with their protégés once a week for 30 minutes for a total of 4 weeks. Results indicated that protégés with facial features moderately distant from the golden ratio were perceived as more similar by mentors in same-gender dyads and received greater mentoring than did protégés closest and farthest from the golden ratio. In opposite-gender dyads, however, mentors reported greater similarity toward those that were farthest from the golden ratio but provided the greatest mentoring to those closest to the golden ratio. The relationship iv between facial measurements and protégé proactivity was moderated by whether or not their mentor had access to their picture. While protégés closest to the ratio were more proactive in the picture condition, those that were farthest from it were more proactive in the non-picture condition. Proactivity was as expected associated with greater levels of mentoring, which was ultimately related to a more fulfilled and beneficial relationship for protégés (i.e. less stress, greater self-efficacy and satisfaction). The results of this study indicate that facial measurements are associated with both differences in mentor and in protégé behavior and that the specific nature of these relationships differs as a function of gender composition. Implications for practice and theory will be discussed

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This thesis deals with complex processes, relationships and phenomena which go beyond the assumed and accepted materialistic perception of the world in general and the process of architectural design specifically. It clarifies and actually redefine the sustainability from a point of higher universal principles forming our environment. At the beginning this thesis presents an extensive knowledge base of existing holistic design systems including Feng-shui, sacred geometry, geomancy or numerology. The purpose is to build and establish a solid foundation for understanding and further research as well as objectively interpreting lesser known topics as a whole. On the basis of intersecting information through the above mentioned topics we specify the hypothesis which proposes the pre-existence of one unifying design matrix of harmonic structures in architecture. The structure of its verification takes us from an analysis of the science of human perception to systems of self-similar contextual references of animated and inanimated forms. These systems of emergent form and flow are basically known as fractals, and could be expressed in both mathematical and geometrical languages. The thesis research then consists of analysing chosen examples of urban and architectural scale in sense of fractality, symbolism and geometrical matrices. We include and integrate the research of associated and relevant phenomena in pedagogical practice, and a case study of the application of fractals in development of a chosen town. From the results of this thesis we abstract five non-dogmatic guidelines or tenets for architectural design; which are supported by experimental verification on some of the author´s buildings. These tenets stand as pillars of implosive architecture. This kind of architecture in context of transcendental overlaps means a possibility of how to bring our anthropogenic environment closer to the natural and harmonic code of the Universe.

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The thesis has been written on the basis of main architectural concepts and their application in real life within a historical context investigation. Main architectural concepts are represented in a harmony of architectural composition that deals with relations among form, structure and space in the unique art work. Genius loci play very important role in the architectural creation - it represents a special and extra-ordinary urban locality and its architectural value in the historical, geographical and cultural context. The origin of Ostravice village within the historical frame concept of Moravia domain is the topic of this thesis. Other historical documents gave information about Bruneswerde as the Brno Castle. That means Brno Castle had to be founded not in Brno (as it has been consid-ered for many years) but it was located in Beskydy at Ostravice. The sacred city – Civitas Dei had been located in Bruneswerde region in early ages as the centre of European culture, education and spiritual life. Civitas Dei – divine Jerusalem in the transcription of St. Augustine's book De civitate Dei – is not only glorification of God and religious fantasy. It is also one of significant clues for recognition of historical architecture. The depiction of unknown settlement from the book of unde-fined origin is a superb testimony about extinct architectural works. They are legible from cadastral maps. Brno Castle - residence of nobility and power served shelter to St. Vojtěch, St. Prokop, St. Václav and St. Ludmila as it is obvious for the mentioned picture. Three major temples, three com-position axes symbolized by three towers on coins are in analogical relation to Brno triangle of four saints - the Saint family of Brno temples. Powerful ambitions of Brno City principals and clergy are inscribed into the urban structure in the way of composition relations which are legible to them who devoted themselves to the mystery of harmony. The absence of historical continuity affects personal attitudes and also identity of the whole community. The architecture truly reflects the past state of polis and it is eloquent even after its death. The architectural composition relations influence the natural environment for long time, longer than the architectural work existence. The geometric order of Renaissance Brno existence has not been in attention of architects so far. The features were discovered in characteristic paintings by Albrecht Durer. They are evidently secret works of the genius. A meaningful collaborator and follower in the extensive project was also his friend Jan Čert from Brno and lately from Vienna. His noble genealogy played a significant role in history of Silesia and Moravia for centuries. It is tendency to consider him as Austrian or even German architect. It is because of the fact that the genealogy tree of his noble family had roots in Moravia. It is supposed that Durer with Jan Čert´s support created the extraordinarily monumental architectural and urban works in Brno. A remarkable consensus in the urban composition of two squares and transition of traditional urban structure of Ostravice Civitas Dei into the modern Brno is also confirmed due to the identification of noble creators and owners, who were at the foundation, transformation and extinction of elements of the Moravian urban structure.