This work present an approach in the didactis of teach of Spatial and Fractal Geometry, to facilitate the interation of students with education. The adopted approach to Spatial Geometry, was of the to construct the geometric fi gures as soon as present the conepts needed. The insertion of a new content, that is Fractal Geometry, contributes to the development of students, it helps to realize the application of math conepts in our daily life. One of the tools used to facilitate the teaching was the use of dynamicsoftware, that s one way that allows having a better interaction with the conepts, allowing a better understanding of the content. Through this work it is expected that Geometry teaching can be performed di erently and motivating to provide better intellectual development students.

The purpose of this dissertation is to present an introductory approach to the complex dynamics and fractal geometry, especially the Mandelbrot set. With the goal to simplify and stimulate the introduction of complex number in high school, the approach adopted was: the definition of the Mandelbrot set and its characteristics; the relationship between the Mandelbrot set and Julia set; how to find by using the Mandelbrot set. One of the tools used to help the teaching was Geogeobra, a dynamic software that allows the student to build the Mandelbrot set. Through this study, it is expected to motivate the learning of complex numbers by using fractal obtained by the study of function ( ) . Obtaining, as a result, a differentiated and motivating way of learning for a better understanding and intellectual development of the students.

We obtain an analytical solution for the pressure-transient behavior of a vertically hydraulic fractured well in a heterogeneous reservoir. The heterogeneity of the reservoir is modeled by using the concept of fractal geometry. Such reservoirs are called fractal reservoirs. According to the theory of fractional calculus, a temporal fractional derivative is applied to incorporate the memory properties of the fractal reservoir. The effect of different parameters on the computed wellbore pressure is fully investigated by various synthetic examples.

Madrid, tal como outras grandes cidades, não foi pensada e gerada como um todo. Pelo contrário, ela resulta de um conjunto de diferentes contextos e tecidos urbanos, a partir dos quais, se originam espaços de transição entre eles, e que, frequentemente, se reflectem em espaços fragmentados, com piores acessibilidades e descaracterizados, portanto, incapazes de gerar espaço urbano de qualidade e referência, ainda que, por vezes, tais espaços gozem de enormes potencialidades. O contexto urbano escolhido para o desenvolvimento deste trabalho enquadra-se nas características, acima descritas, tendo como objectivo principal consolidar a malha urbana envolvente à estação de Chamartín, Madrid, qualificando o espaço urbano e as acessibilidades pedonais entre a zona envolvente e a estação. Para tal, desenvolvemos um objecto arquitectónico adaptado às necessidades e características do lugar, que possa valorizar o meio envolvente, assim como, tirar partido de todas as vatagens dos serviços de mobilidade proporcionados pela estação intermodal de Charmartín. Nesse sentido, criamos um complexo edificado constituído por distintos equipamentos destinados às artes, à cultura e à ciência, constituído por salas de teatro, anfiteatros, salas de exibições, auditórios, biblioteca, salões de eventos, espaços de ensino, residências temporárias, entre outros. O programa arquitectónico proposto, associado a um grande fluxo de utilizadores e cargas, permitiu-nos tirar partido do serviço de transporte público existente, facilitando os acessos aos equipamentos e, esperando um impacto positivo que, através dos nossos estudos e análises apresentados neste trabalho, podemos concluir pelo impacto da implantação de um equipamento arquitectónico destas características, necessário e adequado ao meio urbano em que se insere, com capacidade para o abarcar e através dele se regenerar.

Here we present a research about fractals, its history, mathematical concepts and applications. Some constructions were developed using some basic concepts of the Theory of Numbers, Trigonometry and Linear Algebra that can be applied in High School. We presented some activities that can be applied in High School’s classes in order to develop mathematical concepts such as geometric transformations, Geometry, Trigonometry, Logarithm, notions of limit among other topics. Keywords: Fractal Geometry, Geometric Transformation, Logarithm, Pascal Triangle, Arrays, Linear Algebra, High School.

In this work, we study Fractal Geometry, characterizing some classic fractals and presenting some historical aspects related to this theory. Furthermore, we want to highlight that Fractal Geometry allows us to observe an interesting connection among the construction of mathematical elements and some objects present in the nature. An interesting aspect is that some related concepts can be approached in elementary and high school

This research concerns the teaching and learning of Mathematics in the presence of computational tools, particularly in relation to the construction of fractals. The main aim of the work is to contribute to the analysis of the computational transposition process (Balacheff, 1994) of Fractal Geometry in four computational learning environments. The study entails an analysis of the nature of the meanings that each of the educational software environments affords for particular mathematical notions, that is, its epistemological domain of validity . Additionally the idea of microworld is used in relation to Dynamic Geometry environments and those based of Turtle Geometry . The theoretical approach adopted warrants and guides the methodological choices, which involve the qualitative analysis of the set of chosen computational environments. The analyses show that the construction of fractals in two of these environments made use of the concept of the Chaos Game. In these two cases, the formal structures of both softwares present tools based on geometrical transformations of the plane, although with distinct kinds of control, especially in terms of the extent to which relationships between tools and the concepts involved are made explicit. The results also enable the proposal of teaching situations concerning the construction of fractals in these environments, aimed at providing a context for the exploration of notions related to geometrical transformations of the plane and their geometrical and algebraic representations

The continuous process of the universe expansion is triggering global cooling, energy loss and its “degeneration”. Delineating an analogy between the process of expansion or “aging” of the universe and neurodegeneration, such as occurs in Alzheimer’s disease, is a difficult and complex task. However, assuming that we are an integral part of the universe, sharing the same atoms and particles as well as the same laws that govern it, we can make use of philosophical, scientific and technological tools to help us better understand both the universal and neurodegenerative phenomena. The complexity and chaos theories and fractal geometry bring light on these issues and provide a better understanding about this inseparable, self-creative, self-organized “disorder” which involves all complex systems. In addition, fractal geometry has been described as a useful tool to improve the description of neurons’ image. However, fractal dimension alone does not completely specify the morphology of a cell, in fact it is a statistical parameter to identify and differentiate the neuronal cell types. Studies have shown that fractal dimension is correlated with cognitive impairment, and that the fractal dimension analysis can be a useful method for assessing the Alzheimer’s disease progression. Within this context, the purpose of this manuscript is to perform a systemic reflection on what may be associated with Alzheimer’s disease in the light of concepts of physics, mathematics and biological sciences.

Fractals seemed to have permeated most of scientific fields, including ecology. In fact, biodiversity and ecological processes are affected by spatial complexity, and fractals can help understand patterns at multiple scales. In this paper we evaluated the main quantitative trends and the profile of ecological publications using fractals. Publication about fractals in Ecology experienced a high increase in the last two decades and most articles were published in high visible ecological journals. However, studies were authored mainly by Americans, and researchers from developing countries had a minor contribution. In addition, studies were highly biased towards terrestrial environments, and empirical approaches were preferred.