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The Greek Method of Exhaustion: Leading the Way to Modern IntegrationDeSouza, Chelsea E. 27 June 2012 (has links)
No description available.
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Επίλυση ελλειπτικών προβλημάτων σε κανονικά πολύγωνα με χρήση γνωστών μεθόδων, καθώς και μεθόδων που προκύπτουν από νέες μαθηματικές αναλύσεις του προβλήματος. / Numerical solution of elliptic boundery value problems in regular polygons using well established methods as well as new thansformations recently developed.Κανδύλη, Αναστασία 16 May 2007 (has links)
Η παρούσα διπλωματική αναφέρεται σε ελλειπτικά προβλήματα συνοριακών συνθηκών σε κανονικά πολύγωνα, εστιάζοντας στην αρκετά γενική εξίσωση Helmholtz. Θα εφαρμοσθούν οι γνωστές υπολογιστικές μέθοδοι επίλυσης ελλειπτικών προβλημάτων (όπως η παρεμβολή με τμηματικά κυβικά πολυώνυμα) και θα αναπτυχθούν και μέθοδοι που προκύπτουν από νέες μαθηματικές αναλύσεις του προβλήματος. / In this work we deal with elliptic boundary value problems which are defined in regular polygons. The numerical results presented in the defence are derived using well established methods, such as the finite differemces and the 2d collocation, as well as a new method introduced recently which appears to yield nice results.
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A construção do pentágono regular segundo EuclidesSilva, Alex Cristophe Cruz da 16 July 2013 (has links)
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Previous issue date: 2013-07-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we present some constructions of the regular pentagon, the main
one is a construction of Euclid found in his book The Elements. We also present
some applications of this construction. / Neste trabalho, apresentamos algumas construções do pentágono regular, sendo
a principal delas uma construção de Euclides encontrada no seu livro Os Elementos.
Apresentamos, também, algumas aplicações desta construção.
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CritÃrio para a construtibilidade de polÃgonos regulares por rÃgua e compasso e nÃmeros construtÃveis / Criterion for constructibility of regular polygons by ruler and compass and constructible numbersAislan Sirino Lopes 17 May 2014 (has links)
CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Este trabalho aborda construÃÃes geomÃtricas elementares e de polÃgonos regulares realizadas com rÃgua nÃo graduada e compasso respeitando as regras ou operaÃÃes elementares usadas na Antiguidade pelos gregos. Tais construÃÃes serÃo inicialmente tratadas de uma forma puramente geomÃtrica e, a fim de encontrar um critÃrio que possa determinar a possibilidade de construÃÃo de polÃgonos regulares, passarÃo a ser discutidas por um viÃs algÃbrico. Este tratamento algÃbrico evidenciarà uma relaÃÃo entre a geometria e a Ãlgebra, em especial, a relaÃÃo entre os vÃrtices de um polÃgono regular e as raÃzes de polinÃmios de uma variÃvel com coeficientes racionais. Este tratamento algÃbrico nos levarà naturalmente ao conceito de construtibilidade de nÃmeros e pontos no plano de um corpo, o que exigirà o uso de extensÃes algÃbricas de corpos, e os critÃrios para a construtibi- lidade destes nos levarà a um critÃrio de construtibilidade dos polÃgonos pretendidos. / This work discusses basic geometric constructions and constructions of regular polygons with ruler and compass made respecting the rules or elementary operations used by the ancient Greeks. Such constructs are initially treated in a purely geometric form and, in order to find a criterion that can determine the possibility of construction of regular polygons, will be discussed by an algebraic bias. This algebraic treatment will show a relationship between
geometry and algebra, in particular, the relationship between the vertices of a regular polygon and the roots of polynomials in a variable with rational coefficients. This algebraic treatment leads us naturally to the concept of constructibility of numbers and points in a field, which
will require the use of algebraic field extensions, and the criteria for the constructibility of these leads to a criterion for constructibility of polygons.
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Poliedros de Platão como estratégia no ensino da geometria espacialNogueira, Simone Paes Gonçalves January 2014 (has links)
Orientador: Prof. Dr. André Ricardo Oliveira da Fonseca / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2014. / Our work aims to make a brief study on polyhedrons, focusing specially
on solid platonics. First, we will present the historical moment in which this
topic was discussed, as well as mention the mathematicians who contributed
to the first studies about it. Then, we will explain what are regular polygons,
dihedral angle and regular polyhedron. We will also discuss the reasons why
there are only five solid platonics and we will demonstrate the Euler Characteristics,
through induction. We will provide sample activities, which can be
used in classrooms, in order to in
uence positivetly the learning process of
students. Therefore, such students will be able to better learn and understand
the content, rather than just decorating the \formulas". We will also
show an intuitive idea of calculating the area and volumes of solid platonics,
which is something rarely demonstrated in textbooks. Further on, we will
demonstrate how this topic is presented by the National Curriculum Parameters
\Parâmetros Curriculares Nacionais (PCN)", and relate it to how it
is developed and and taught since the first years of schools until the second
year of High School, time in which this topic is more deeply studied. There
are sample questions, which can be found in national examinations, such
as Saresp (São Paulo's government exam) and ENEM (Federal government
exam). Throughout this work you will be able to see imagens that were taken
during a project envolving students from a second High School year, which
was taken place a public school.
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Ladrilhamentos irregulares, discos extremos e grafos de balão / Irregular tiling, extremes discs and graphs of balloonBatista, Frederico Ventura 28 February 2012 (has links)
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Previous issue date: 2012-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation aims to study two topics related to modern topology and geometry. The first of these themes is dedicated to the study of packaging and record covering spheres in the hyperbolic plane, in which we treat the study results due to Bavard (1996) [3]. The second issue that was addressed refers to the study of edges pairing for irregular polygons. In this part we try to expose an example, created during our studies, for a pairing that generates a tiling of the hyperbolic plane by an irregular polygon. Also use the techniques developed by Mercio Botelho Faria, Catarina Mendes de Jesus and Panteleón D. R. Sanchez in [14] to obtain matching of edges of regular polygons through surgeries in surfaces associated with trivalent graphs. / Esta dissertação tem como objetivo o estudo de dois temas ligados a topologia e a geometria moderna. O primeiro destes temas é dedicado ao estudo de empacotamento e coberturas de discos do plano hiperbólico, no qual tratamos de estudar resultados devidos a Bavard (1996) [3]. Já o segundo tema que foi abordado se refere ao estudo de emparelhamento de arestas para polígonos irregulares. Nesta parte tratamos de expor um exemplo, criado durante nossos estudos, para um emparelhamento que gera um ladrilhamento do plano hiperbólico por um polígono irregular. Além disso utilizamos as técnicas desenvolvidas por Mercio Botelho Faria, Catarina Mendes de Jesus e Panteleón D. R. Sanchez em [14] para obtermos emparelhamentos de arestas de polígonos regulares por meio de cirurgias em superfícies associadas a grafos trivalentes.
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[en] COMPLEXITY IN EUCLIDEAN PLANE GEOMETRY / [pt] COMPLEXIDADE EM GEOMETRIA EUCLIDIANA PLANASILVANA MARINI RODRIGUES LOPES 25 February 2003 (has links)
[pt] Consideramos duas formas de complexidade em geometria
euclidiana plana.Na primeira, problemas são descritos
algebricamente, e a complexidade é cotada essencialmente
pelo grau de um polinômio. Como consequência, mostramos
que
vários resultados gerais e familiares em geometria podem
ser demonstrados a partir da simples verificação de dois
ou
três casos particulares. A segunda forma faz uso da
descrição sintática dos teoremas, que permite uma
quantificação da complexidade em termos lógicos (número
de
quantificadores e átomos de uma fórmula). Inspirados por
esta última abordagem, são descritos alguns procedimentos
de demonstração automática. Alguns grupos habituais de
operções em geometria são apresentados com a intenção de
simplificar as duas abordagens.Através do estudo de
técnicas mais avançadas em matemática trazemos novos
pontos de vista a assuntos estudados no ensino médio. / [en] Two forms of complexity in Euclidean plane geometry are
considered. In the first one, problems are described
algebraically, and the complexity level is measured
essentially by the degree of a polynomial. As a
consequence, many familiar and general results in geometry
can be proved by inspecting two or three special cases. The
second form uses the syntactic description of a theorem
allowing for a quanti.cation of the complexity in logic
terms (number of quantifiers and atoms in the formula).
Inspired by this approach, some procedures in mechanized
proofs are described. We also present some traditional
groups of operations in geometry which simplify the two
approaches. The study of more advanced techniques in
mathematics sheds new light on standard high school topics.
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Émergence et évolution des objets mathématiques en Situation Didactique de Recherche de Problème : le cas des pavages archimédiens du plan / Emergence and evolution of mathematical objects, during a “ Didactical Situation of a Problem Solving ” : the Case of Archimedean tilings of the planeFront, Mathias 27 November 2015 (has links)
Étudier l'émergence de savoirs lors de situations didactiques non finalisées par un savoir préfabriqué et pré-pensé nécessite un bouleversement des points de vue, aussi bien épistémologique que didactique. C'est pourquoi, pour l'étude de situations didactiques pour lesquelles le problème est l'essence, nous développons une nouvelle approche historique et repensons des outils pour les analyses didactiques. Nous proposons alors, pour un problème particulier, l'exploration des pavages archimédiens du plan, une enquête historique centrée sur l'activité du savant cherchant et sur l'influence de la relation aux objets dans la recherche. De ce point de vue, l'étude des travaux de Johannes Kepler à la recherche d'une harmonie du monde est particulièrement instructive. Nous proposons également, pour l'analyse des savoirs émergents en situation didactique, une utilisation d'outils liés à la sémiotique qui permet de mettre en évidence la dynamique de l'évolution des objets mathématiques. Nous pouvons finalement conclure quant à la possibilité de construire et mettre en œuvre des ≪ Situations Didactiques de Recherche de Problème ≫ assurant l'engagement du sujet dans la recherche, l'émergence et le développement d'objets mathématiques, la genèse de savoirs. L'étude nous conforte dans la nécessité d'une approche pragmatique des situations et la pertinence d'un regard différent sur les savoirs à l'école / The study of the emergence of knowledges in teaching situations not finalized by a prefabricated and pre-thought knowledge requires an upheaval of point of view, epistemological as well as didactic. For the study of learning situations in which the problem is the essence, we develop a new historical approach and we rethink the tools for didactic analyzes. We propose, then, for a particular problem, exploration of Archimedean tilings of the plane, a historical inquiry centered on the activity of the scientist in the process of research and on the influence of the relationship with objects. From this perspective, the study of Johannes Kepler’s work in search of a world harmony is particularly instructive. We also propose, for the analysis of the emerging knowledge in teaching situations, to use tools related to semiotics, which allows to highlight the dynamic of evolution of mathematical objects. We can finally conclude on the opportunity to build and implement “Didactic Situations of Problem Solving”, which ensure the commitment of the subject in the research, the emergence and development of mathematical objects, the genesis of knowledges. The study reinforces the necessity of a pragmatic approach of situations and the relevance of a different look at the knowledge at school
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Visualisation, navigation and mathematical perception: a visual notation for rational numbers mod1Tolmie, Julie, julie.tolmie@techbc.ca January 2000 (has links)
There are three main results in this dissertation.
The first result is the construction of an abstract visual space for rational
numbers mod1, based on the visual primitives, colour, and rational radial
direction. Mathematics is performed in this visual notation by defining
increasingly refined visual objects from these primitives. In particular,
the existence of the Farey tree enumeration of rational numbers mod1
is identified in the texture of a two-dimensional animation.
¶
The second result is a new enumeration of the rational numbers mod1,
obtained, and expressed, in abstract visual space, as the visual object
coset waves of coset fans on the torus. Its geometry is shown to encode
a countably infinite tree structure, whose branches are cosets, nZ+m,
where n, m (and k) are integers. These cosets are in geometrical 1-1
correspondence with sequences kn+m, (of denominators) of rational
numbers, and with visual subobjects of the torus called coset fans.
¶
The third result is an enumeration in time of the visual hierarchy of the
discrete buds of the Mandelbrot boundary by coset waves of coset fans.
It is constructed by embedding the circular Farey tree geometrically into
the empty internal region of the Mandelbrot set. In particular, coset fans
attached to points of the (internal) binary tree index countably infinite
sequences of buds on the (external) Mandelbrot boundary.
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