Orientador: Prof. Dr. Antonio Cândido Faleiros / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, 2014. / No curriculo escolar o valor que Pi assume e de 3,14. Pois, para os calculos utilizados pelos alunos, este valor e mais que suficiente. Mas o que e Pi? Como podemos determinar o valor de Pi? Como foi calculado em tempos antigos? Como pode o valor ser encontrado hoje usando a tecnologia mais moderna? Estas sao algumas das questoes que vamos explorar partindo de um breve historico da evolucao do Pi, seguido de uma variedade de metodos para chegar ao seu valor com mais casas decimais precisas e finalizando com duas sugestoes de atividades para encontrar um valor aproximado para o Pi. A primeira, usando o Metodo de Monte-Carlo com gotas de chuva, que pode ser utilizada a partir do Ensino Fundamental II, e a segunda para o Ensino Medio, temos uma adaptacao do Metodo de Arquimedes com poligonos regulares inscritos no circulo. Ambas sao desenvolvidas utilizando o software Microsoft Excel. / The central idea of this paper is it to present argumentation for the importance of the study of the number Pi. In the school curriculum Pi takes the value of 3.14. For a student¡¦s purposes this value is more than adequate. However, what is ¿à? How do we determine the value of ¿à? How was it calculated in ancient times? How can the value be found today using the most modern technology? These are some of questions we will explore starting from a brief history of the evolution of Pi, followed by a variety of methods in order to find its value with precisely decimal places and ending with two suggested activities for finding an approximate value. The first one that uses the Monte-Carlo Method with rain drops can be utilized from Primary School II on and the other, for Secondary School, we have an adaptation of Archimedes' Method using regular polygon inscribed in the circle. Both of them are realised using the Microsoft Excel software.

This thesis examines the stone-carved architectural spiral fluted column from second-millennium B.C. Mesopotamia to the fourth-century A.D. Roman Empire, and establishes its relationship to technological devices such as water screws, screw presses, and other machines. Evidence from literary sources and archaeological records shows the increasing architectural use of the helical spiral during that time, particularly in structures such as theatres, nymphaea, colonnades and decorative gateways. The use of spiral designs on coins, sarcophagi, pottery and wall paintings is also discussed. The thesis presents: the mathematics of the spiral as applied in Mesopotamian architecture; spiral use in the Aegean Bronze and Iron Ages and the Greek and Roman worlds; and its use in technology and mechanical devices, specifically those of Archimedes and Hero. The conclusion summarises the evidence, demonstrating that the construction of the spiral fluted column evolved from that of the Archimedean water screw. / Graduate / 2015-08-20 / 0324 / 0346 / 0579 / ghenders@uvic.ca

Cette thèse porte sur l’étude des microcentrales hydroélectriques à vis d’Archimède. Il s’agit dans ces travaux de proposer une solution alternative à l’hydroélectricité à petite échelle, en exploitant les ressources inutilisées telles que les petites rivières ou les cours d’eau. Ces microcentrales inversent le principe de fonctionnement de la pompe à vis d’Archimède. Elles exploitent la puissance hydraulique disponible dans les usines hydroélectriques de très basses chutes. A partir de l’état de l’art, un modèle numérique est établi pour estimer les rendements, la production d’énergie et la rentabilité pour un fonctionnement à vitesse variable des microcentrales hydroélectriques à vis d’Archimède. Les différents résultats théoriques et de simulation ont été validés expérimentalement sur le prototype de l’INSA de Strasbourg, développé dans le cadre de cette thèse. Enfin, des actions sont menées sur le prototype expérimental afin de maximiser le transfert d’énergie et de limiter les pertes. Pour finir, une stratégie MPPT (Maximum Power Point Tacking) très spécifique est développée et est actuellement en cours d’implémentation. / This work focuses on a small hydro plant which uses the Archimedes screw. This is an alternative solution to smallscale hydropower as it employs unused resources such as small rivers or streams. Archimedes screw plants reverse the pump use principle and employ the available stream power for energy production in very low head application. Based on the state-of-the-art, a numerical model is established to estimate efficiencies, energy production and profitability of variable speed operations for a small hydro plant using Archimedes screw. Several theoretical results and simulations are proposed. We validated them experimentally on the prototype of INSA Strasbourg, developed within the framework of this thesis. Finally, actions carried out on the experimental prototype led to maximizing the transfer of energy and limiting losses. Then a very specific MPPT (Maximum Power Point Tracking) control strategy is developed and is currently being implemented.

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, 2017. / Este trabalho tem por objetivo apresentar como a Geometria e o Infinito estão relacionados, tomando como ponto de partida e referência a História da Matemática. Apresentamos como os pitagóricos tiveram contato com este tema, o que na sequência culminou na conhecida "Crise dos Incomensuráveis", resultando no declínio e fim da escola pitagórica. Após esse período, na "Academia de Platão", com o matemático Eudoxo, a "Crise dos Incomensuráveis" foi solucionada. Eudoxo também forneceu apoio teórico, com o Método da Exaustão, para as descobertas de Arquimedes em relação ao cálculo da área do círculo. Primeiramente, apresentamos e discutimos o que é o Infinito, tratando de temas como o Infinito Real e o Infinito Potencial. Analisamos, com exemplos, os diferentes tipos de infinitos que existem, a partir de conjuntos infinitos e suas propriedades. Em seguida, passamos a explorar como Arquimedes conseguiu encontrar um algoritmo capaz de calcular, com uma excelente aproximação, a área do círculo, gerando, como consequência, um método eficiente para o cálculo do número p. Também mostramos como as médias geométricas e as médias harmônicas foram utilizadas na descoberta das relações entre as áreas e os perímetros dos polígonos regulares inscritos e circunscritos no círculo. Ao final, propomos atividades didáticas relacionadas com o tema dessa dissertação para professores e estudantes de Matemática do ensino básico. / This work aims to present how Geometry and Infinity are related. Taking as a its starting point and reference the History of Mathematics. We present how the Pythagoreans came into contact with this theme. What then has culminated in the era known as "The Crisis of the incommensurable". Resulting in the decline and end of the Pythagorean School. After this period, at the "Academy of Plato", with the mathematician Eudoxus, the "Crisis of the Incommensurable"was solved. Eudoxus, in addition, provided theoretical support, with the Exhaustion Method, for Archimedes¿ discoveries in relation to the calculation of the area of the circle. First, we introduce and discuss what the Infinite is. Dealing with themes such as the Real Infinity and the Infinite Potential. We analyze, with examples, the different types of infinities that exist, from infinite sets and their properties. Then we come to understand how Archimedes was able to find an algorithm capable of calculating, with an excellent approximation, the area of the circle. Generating, therefore, an efficient method for calculating the p number. We also show how the geometric averages and the harmonic averages were used in the discovery of the relations between the areas and the perimeters of the regular polygons inscribed and circumscribed in the circle. At the end, we propose didactic activities related to the theme of this dissertation for teachers and students of Mathematics.

This thesis examines the stone-carved architectural spiral fluted column from second-millennium B.C. Mesopotamia to the fourth-century A.D. Roman Empire, and establishes its relationship to technological devices such as water screws, screw presses, and other machines. Evidence from literary sources and archaeological records shows the increasing architectural use of the helical spiral during that time, particularly in structures such as theatres, nymphaea, colonnades and decorative gateways. The use of spiral designs on coins, sarcophagi, pottery and wall paintings is also discussed. The thesis presents: the mathematics of the spiral as applied in Mesopotamian architecture; spiral use in the Aegean Bronze and Iron Ages and the Greek and Roman worlds; and its use in technology and mechanical devices, specifically those of Archimedes and Hero. The conclusion summarises the evidence, demonstrating that the construction of the spiral fluted column evolved from that of the Archimedean water screw. / Graduate / 2018-08-20 / 0324 / 0346 / 0579 / ghenders@uvic.ca

Title: Platonic and Archimedean solids and their properties in teaching of mathematics at secondary schools Author: Eva Dohnalová Department: Department of Didactics of Mathematics Supervisor: doc. RNDr. Jarmila Robová, CSc. Abstract: This work is an extension of my bachelor work and it is intended for all people interested in regular and semiregular polyhedra geometry. It is a comprehensive text which summarizes brief history, description and classification of regular and semiregular polyhedra. The work contains proofs of Descartes' and Euler's theorems and proofs about number of regular and semiregular polyhedra. It can be also used as a didactic aid in the instruction of regular and semiregular solids at secondary schools. This text is supplemented by illustrative pictures made in GeoGebra and Cabri3D. Keywords: Regular polyhedra, platonic solids, Platon, semiregular polyhedra, Archimedean solids, Archimedes, dulaism, Descartes' theorem, Euler's theorem.