Relevant bibliographies by topics / Geometry Axioms

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Academic literature on the topic 'Geometry Axioms'

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Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Geometry Axioms"

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Richter, William, Adam Grabowski, and Jesse Alama. "Tarski Geometry Axioms." Formalized Mathematics 22, no. 2 (June 30, 2014): 167–76. http://dx.doi.org/10.2478/forma-2014-0017.

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Summary This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. Specifically, we obtain Gupta’s amazing proof which implies Hilbert’s axiom I1 that two points determine a line. The primary Mizar coding was heavily influenced by [9] on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski’s geometry related to real planes were constructed.
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Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms – Part II." Formalized Mathematics 24, no. 2 (June 1, 2016): 157–66. http://dx.doi.org/10.1515/forma-2016-0012.

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Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].
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Grigoryan, Yu. "Axioms of Heterogeneous Geometry." Cybernetics and Systems Analysis 55, no. 4 (July 2019): 539–46. http://dx.doi.org/10.1007/s10559-019-00162-3.

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BEESON, MICHAEL, PIERRE BOUTRY, and JULIEN NARBOUX. "HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY." Bulletin of Symbolic Logic 21, no. 2 (June 2015): 111–22. http://dx.doi.org/10.1017/bsl.2015.6.

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AbstractWe use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.
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Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms. Part III." Formalized Mathematics 25, no. 4 (December 20, 2017): 289–313. http://dx.doi.org/10.1515/forma-2017-0028.

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Summary In the article, we continue the formalization of the work devoted to Tarski’s geometry - the book “Metamathematische Methoden in der Geometrie” by W. Schwabhäuser, W. Szmielew, and A. Tarski. After we prepared some introductory formal framework in our two previous Mizar articles, we focus on the regular translation of underlying items faithfully following the abovementioned book (our encoding covers first seven chapters). Our development utilizes also other formalization efforts of the same topic, e.g. Isabelle/HOL by Makarios, Metamath or even proof objects obtained directly from Prover9. In addition, using the native Mizar constructions (cluster registrations) the propositions (“Satz”) are reformulated under weaker conditions, i.e. by using fewer axioms or by proposing an alternative version that uses just another axioms (ex. Satz 2.1 or Satz 2.2).
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von Plato, Jan. "The axioms of constructive geometry." Annals of Pure and Applied Logic 76, no. 2 (December 1995): 169–200. http://dx.doi.org/10.1016/0168-0072(95)00005-2.

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Tarski, Alfred, and Steven Givant. "Tarski's System of Geometry." Bulletin of Symbolic Logic 5, no. 2 (June 1999): 175–214. http://dx.doi.org/10.2307/421089.

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AbstractThis paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.
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Gottlieb, Alex D., and Joseph Lipman. "Group-Theoretic Axioms For Projective Geometry." Canadian Journal of Mathematics 43, no. 1 (February 1, 1991): 89–107. http://dx.doi.org/10.4153/cjm-1991-006-2.

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AbstractWe show that a certain category 𝓖 whose objects are pairs G ⊃ H of groups subject to simple axioms is equivalent to the category of ≧ 2-dimensional vector spaces and injective semi-linear maps; and deduce via the "Fundamental Theorem of Projective Geometry" that the category of ≧ 2-dimensional projective spaces is equivalent to the quotient of a suitable subcategory of 𝓖 by the least equivalence relation which identifies conjugation by any element of H with the identity automorphism of G.
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Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms. Part IV – Right Angle." Formalized Mathematics 27, no. 1 (April 1, 2019): 75–85. http://dx.doi.org/10.2478/forma-2019-0008.

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Summary In the article, we continue [7] the formalization of the work devoted to Tarski’s geometry – the book “Metamathematische Methoden in der Geometrie” (SST for short) by W. Schwabhäuser, W. Szmielew, and A. Tarski [14], [9], [10]. We use the Mizar system to systematically formalize Chapter 8 of the SST book. We define the notion of right angle and prove some of its basic properties, a theory of intersecting lines (including orthogonality). Using the notion of perpendicular foot, we prove the existence of the midpoint (Satz 8.22), which will be used in the form of the Mizar functor (as the uniqueness can be easily shown) in Chapter 10. In the last section we give some lemmas proven by means of Otter during Tarski Formalization Project by M. Beeson (the so-called Section 8A of SST).
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BEESON, MICHAEL. "CONSTRUCTIVE GEOMETRY AND THE PARALLEL POSTULATE." Bulletin of Symbolic Logic 22, no. 1 (March 2016): 1–104. http://dx.doi.org/10.1017/bsl.2015.41.

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AbstractEuclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions.
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Dissertations / Theses on the topic "Geometry Axioms"

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Thorgeirsson, Sverrir. "Hyperbolic geometry: history, models, and axioms." Thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503.

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Ward, Peter James. "Euclid's Elements, from Hilbert's Axioms." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1354311965.

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Toniolo, Luciano Santos. "Cônicas em modelos físicos." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-24102018-151118/.

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Este trabalho é um estudo realizado em torno das principais curvas cônicas estudadas por alunos do ensino básico: parábola, elipse e hipérbole. A ideia central do trabalho é a autosuficiência, pois apresentamos todas as ferramentas matemáticas necessárias para o entedimento desses entes e suas aplicações, desde os axiomas iniciais da geometria plana até as definições formais das cônicas e demonstrações de suas propriedades. Espera-se que uma pessoa não especializada em matemática, ao ler o trabalho, entenda toda a matemática no entorno das aplicações dessas cônicas.
This work is a study carried out around the main conic curves studied by elementary school students: parabola, ellipse and hyperbola. The main idea of this work is to be self-contained, starting from the basic axioms from the geometry and after we present formal definitions, properties and applications of conics in the everyday life. It is expected that a person that is not a specialist in mathematics, are able to read and understand all the mathematics in the surroundings of the applications of these conics.
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Portela, Antonio Edilson Cardoso. "Noções de geometria projetiva." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25586.

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PORTELA, Antonio Edilson Cardoso. Noções de geometria projetiva. 2017. 58 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.
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Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de ANTONIO EDILSON CARDOSO PORTELA, para que o mesmo realize algumas correções na formatação do trabalho. 1- SUMÁRIO ( A formatação do sumário está incorreta, primeiro, retire o último ponto final que aparece após a numeração dos capítulos e seções (Ex.: 3.1. Axioma....; deve ser corrigido para: 3.1 Axioma.....), o alinhamento dos títulos deve seguir o modelo abaixo 1 INTRODUÇÃO.....................00 2 O ESPAÇO...........................00 3 GEOMETRIA........................00 3.1 Axiomas...............................00 REFERÊNCIAS...................00 (OBS.: não altere a formatação do negrito, pois já estava correta) 2- TITULO DOS CAPÍTULOS E SEÇÕES ( retire o ponto final que aparece após o último dígito da numeração dos capítulos e seções, seguindo o modelo do sumário. Retire o recuo de parágrafo dos títulos das seções. Ex.: 3.1 Axioma.......) 3- REFERÊNCIAS ( substitua o termo REFERÊNCIAS BIBLIOGRÁFICAS apenas por REFERÊNCIAS, com fonte n 12, negrito e centralizado. Retire a numeração progressiva que aparece nos itens da referência. Atenciosamente, on 2017-09-06T17:56:50Z (GMT)
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In this work, initially, some results of Linear Algebra are presented, in particular the study of the Vector Space R^n, which becomes, together with Analytical Geometry, the language used in the chapters that follow. We present a study from an axiomatic point of view, from the perspectives of Hilbert's axioms and we elaborate models of planes for the Euclidean, Elliptic and Projective Geometries. The validity of the Incidence and Order axioms for Euclidean Geometry is verified. In R^3, an approach is made to the study of the plane and the unitary sphere, highlighting the elliptical line obtained by the intersection of these sets, thus making an approach to the Elliptic Geometry. With the concepts and definitions studied in the Vector Space R^n, Three-dimensional Space and in the Euclidean and Elliptic Geometries we will approach the study of Projective Geometry, demonstrating propositions and verifying its axioms.
Neste trabalho, inicialmente, apresenta-se alguns resultados da Álgebra Linear, em especial o estudo do Espaço Vetorial R^n, que passa a ser, juntamente com a Geometria Analítica, a linguagem empregada nos capítulos que se seguem. Apresentamos um estudo de um ponto de vista axiomático, sob a ótica dos axiomas de Hilbert e elaboramos modelos de planos para as Geometrias Euclidiana, Elíptica e Projetiva. É verificada a validade dos axiomas de Incidência e Ordem para a Geometria Euclidiana. No R^3, é feita uma abordagem do estudo de plano e da esfera unitária, destacando a reta elíptica obtida pela interseção destes conjuntos, passando assim a fazer uma abordagem da Geometria Elíptica. Com os conceitos e definições estudadas no Espaço Vetorial R^n, Espaço tridimensional e nas Geometrias Euclidiana e Elíptica, abordaremos o estudo da Geometria Projetiva, demonstrando proposições e verificando os seus axiomas.
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Pejlare, Johanna. "On Axioms and Images in the History of Mathematics." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-8345.

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This dissertation deals with aspects of axiomatization, intuition and visualization in the history of mathematics. Particular focus is put on the end of the 19th century, before David Hilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consists of three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931) and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. A thorough analysis of his foundational work is made as well as an investigation into his general view on science and mathematics. Furthermore, his thoughts on geometry and its nature and what consequences his view has for how he proceeds in developing the axiomatic system, is studied. In the second paper different aspects of visualizations in mathematics are investigated. In particular, it is argued that the meaning of a visualization is not revealed by the visualization and that a visualization can be problematic to a person if this person, due to a limited knowledge or limited experience, has a simplified view of what the picture represents. A historical study considers the discussion on the role of intuition in mathematics which followed in the wake of Karl Weierstrass' (1815–1897) construction of a nowhere differentiable function in 1872. In the third paper certain aspects of the thinking of the two scientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It is investigated how Klein and Hertz related to the idea of naïve images and visual thinking shortly before the development of modern axiomatics. Klein in several of his writings emphasized his belief that intuition plays an important part in mathematics. Hertz argued that we form images in our mind when we experience the world, but these images may contain elements that do not exist in nature.
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Freitas, Brasilio Alves. "Introdução à geometria euclidiana axiomática com o geogebra." Universidade Federal de Juiz de Fora, 2013. https://repositorio.ufjf.br/jspui/handle/ufjf/1188.

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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Por conhecer a grande dificuldade dos alunos de Ensino Médio, da rede pública Estadual de Minas Gerais, em relação aos conceitos, demostrações e deduções básicas da Geometria Euclidiana plana, foi elaborado um pequeno roteiro de estudo dos axiomas que regem esses conteúdos e também uma introdução às construções geométricas básicas, utilizando os instrumentos euclidianos e o software gratuito GeoGebra. O desenvolvimento do trabalho trouxe como objetivo dotar os alunos do Ensino Fundamental, cursando oitavo ano (antiga sétima série), de uma compreensão gradual e intuitiva da geometria euclidiana plana, buscando, de forma fundamentada fixar os aspectos conceituais básicos que são extremamente necessários para estudos mais aprofundados em cursos posteriores. As atividades propostas no capítulo 4 foram criadas com o intuito de que o aluno, percorrendo os conceitos mostrados no capítulo 2, tenha oportunidade de abstrair-se literalmente e ou com recursos algébricos em um processo de demonstração das propriedades de diversas figuras geométricas.
Knowing the great hardship high school students of Minas Gerais public school system have concerning the basic concepts, demonstrations and deductions of the Euclidean Geometry, a small study guide of the axioms that rule these contents was made, and also an introduction to the basic geometry constructions using the Euclidean instruments and the free software GeoGebra. The work’s development brought as a goal to endow the middle school students, attending the eight year (the old seventh grade), a gradual and intuitive understanding of the Euclidian Geometry, trying to fix the basic conceptual aspects that are deeply necessary for further studies. The proposed activities on chapter four intend to give the student, going trough the concepts shown on chapter two, the opportunity to abstract on a descriptive way and/or use algebraic resources in a process of demonstration of many geometrical forms.
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SOUZA, Carlos Bino de. "Geometria hiperbólica : consistência do modelo de disco de Poincaré." Universidade Federal Rural de Pernambuco, 2015. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/6695.

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Euclid wrote a book in 13 volumes called Elements where systematized all the mathematical knowledge of his time. In this work, the 5 postulates of Euclidean geometry were presented. For several years, the 5th Postulate was frequently asked, this inquiries it was discovered that there are several other possible geometries, including hyperbolic geometry. Beltrimi proved that hyperbolic geometry is consistent if Euclidean geometry is consistent. Hilbert showed that Euclidean geometry is consistent if the arithmetic is consistent and presented an axiomatic system that capped the gaps in Euclid’s axiomatic system. Poincaré created a model, called the Poincaré disk, to represent the plan of hyperbolic geometry. The objective of this work is to show that the Poincaré disk model is consistent with reference Axioms Hilbert, replacing only the Axioms of Parallel to "On a point outside a line passes through the two parallel straight lines given", by constructions of Euclidean geometry.
Euclides escreveu uma obra em 13 volumes chamada de Elementos onde sistematizava todo o conhecimento matemático do seu tempo. Nesta obra, foram apresentados os 5 postulados da Geometria Euclidiana. Durante vários anos, o 5o Postulado foi muito questionado, desses questionamentos descobriu-se a existência de várias outras Geometrias possíveis, entre elas a Geometria Hiperbólica. Beltrimi provou que a Geometria Hiperbólica é consistente se a Geometria Euclidiana é consistente. Hilbert mostrou que a Geometria Euclidiana é consistente se a Aritmética é consistente e apresentou um sistema axiomático que preencheu as lacunas do sistema axiomático de Euclides. Poincaré criou um Modelo, chamado de Disco de Poincaré, para representar o plano da Geometria Hiperbólica. O objetivo deste trabalho é mostrar que o Modelo de Disco de poincaré é consistente, tomando como referência os Axiomas de Hilbert, substituindo apenas os Axiomas das Paralelas para "Por um ponto fora de uma reta passam duas retas paralelas à reta dada", através de construções da Geometria Euclidiana.
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Barreto, Carlos Alberto. "A geometria do origami como ferramenta para o ensino da geometria euclidiana na educação básica." Mestrado Profissional em Matemática, 2013. https://ri.ufs.br/handle/riufs/6503.

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The purpose of this monograph is to study the geometry of origami and its applications in Euclidean Geometry as a tool that contributes to the teaching of Geometry in Basic Education. Provide a brief history of origami and its arrival in Brazil and as a result we present the axioms that define the simple movements that can be performed using points and straight lines in a plane. We also study the classic problems of doubling the cube and the trisection of the angle, showing that they are possible to be solved through the Geometry of Origami. We show then the Origami applications for studies of flat Euclidean space, emphasizing the study of Plato polyhedra. We finished the job by showing how we developed the Origami Project - Mathematics and Art in the State College John XXIII .
O objetivo desta monografia é fazer o estudo da Geometria do Origami e de suas aplicações na Geometria Euclidiana como instrumento que contribua para o ensino da Geometria na Educação Básica. Fornecemos um pequeno histórico do Origami e de sua chegada ao Brasil e na sequência apresentamos os axiomas que definem os movimentos simples que podem ser realizados utilizando pontos e retas num plano. Estudamos também os problemas clássicos da duplicação do cubo e da trissecção do ângulo, mostrando que são possíveis de ser resolvidos por meio da Geometria do Origami. Mostramos, então, aplicações do Origami para estudos de Geometria Euclidiana plana e espacial, dando ênfase ao estudo dos poliedros de Platão. Encerramos o trabalho, mostrando como foi desenvolvido o Projeto Origami - Matemática e Arte no Colégio Estadual João XXIII .
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Bassan, André Roberto. "Observações sobre geometria sintética /." Rio Claro, 2015. http://hdl.handle.net/11449/132066.

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Orientador: Alice Kimie Miwa Libardi
Banca: Sérgio Roberto Nobre
Banca: Edson de Oliveira
Resumo: O objetivo deste trabalho é apresentar alguns resultados da Geometria Euclidiana no plano, que são vistos no ensino fundamental e médio sob ponto de vista sintético, ou seja, não serão assumidos os axiomas métricos. Como aplicação faremos algumas construções, usando as ferramentas desenvolvidas
Abstract: The objective of this work is to present some results of Euclidean geometry which are given in elementary and high school from the synthetic point of view, that is we will not assume the metric axioms. As an application we will make some constructions using the developed tools
Mestre
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Freitas, Aline Claro de [UNESP]. "Origami: o uso como instrumento alternativo no ensino da geometria." Universidade Estadual Paulista (UNESP), 2016. http://hdl.handle.net/11449/134280.

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Frente à realidade do ensino contemporâneo que demanda a necessidade de diversificar o uso de estratégias de ensino, pretendemos propor uma abordagem, por meio de material concreto e que pode tornar-se bastante significativa no ensino da matemática. Este trabalho discute sobre a história, aplicações clássicas e utilização do origami em sala de aula. Após uma breve apresentação histórica sobre o origami, apresentamos uma abordagem axiomática deste instrumento. Dois dos três famosos problemas matemáticos gregos da antiguidade que não podem ser solucionados através da régua e compasso: trissecção do ângulo e duplicação do cubo encontram uma solução por meio das técnicas de origami. Além disso, apresentamos sugestões de roteiros de aulas e a atividade aplicada em sala de aula que obteve resultado satisfatório.
Faced with the reality of contemporary teaching that demands the need to diversify the use of teaching strategies, we intend to propose an approach through concrete material and can become quite significant in mathematics education. This monograph discusses about the history, classic applications and use origami in the classroom. After a brief historical introduction about origami, we present an axiomatic approach of this instrument. Two of the three famous Greek mathematical problems of antiquity that can’t be solved by ruler and compass: trisection angle and doubling the cube find a solution through of origami techniques. In addition, we present suggestions classes scripts and the activitie applied in the classroom that obtained satisfactory result.
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Books on the topic "Geometry Axioms"

1

Whitehead, Alfred North. Axioms of projective geometry. [Place of publication not identified]: Nabu Press, 2010.

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Xiao-shan, Gao, and Chang Ching-chung 1936-, eds. Machine proofs in geometry: Automated production of readable proofs for geometry theorems. Singapore: World Scientific, 1994.

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Goetsch, James Robert. Vico's axioms: The geometry of the human world. New Haven: Yale University Press, 1995.

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Fetisov, A. I. Proof in geometry. Mineola, N.Y: Dover Publications, 2006.

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Zahlbereiche, naive Mengenlehre, Axiomatik der Geometrie: Grundlagenfragen. Bremen: Universitätsdruckerei Bremen, 2007.

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Erro, Luis Enrique. Axioma: El pensamiento matemático contemporáneo. México: Dirección de Difusión Cultural, Departamento Editorial, 1985.

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Borceux, Francis. An Axiomatic Approach to Geometry: Geometric Trilogy I. Springer, 2016.

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Whitehead, Alfred North. The Axioms of Descriptive Geometry. BiblioBazaar, 2009.

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Whitehead, Alfred North. The Axioms Of Descriptive Geometry. Kessinger Publishing, LLC, 2007.

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The Axioms of Descriptive Geometry. Dover Publications, 2004.

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Book chapters on the topic "Geometry Axioms"

1

Ramsay, Arlan, and Robert D. Richtmyer. "Axioms for Plane Geometry." In Introduction to Hyperbolic Geometry, 9–29. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4757-5585-5_2.

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Sossinsky, A. "Hilbert’s axioms for plane geometry." In The Student Mathematical Library, 271–82. Providence, Rhode Island: American Mathematical Society, 2012. http://dx.doi.org/10.1090/stml/064/19.

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Anglin, W. S., and J. Lambek. "Non-Euclidean Geometry and Hilbert’s Axioms." In The Heritage of Thales, 89–92. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-0803-7_18.

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Biagioli, Francesca. "Axioms, Hypotheses, and Definitions." In Space, Number, and Geometry from Helmholtz to Cassirer, 51–80. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-31779-3_3.

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Ramsay, Arlan, and Robert D. Richtmyer. "Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models." In Introduction to Hyperbolic Geometry, 202–17. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4757-5585-5_8.

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Specht, Edward John, Harold Trainer Jones, Keith G. Calkins, and Donald H. Rhoads. "Consistency and Independence of Axioms; Other Matters Involving Models." In Euclidean Geometry and its Subgeometries, 413–516. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-23775-6_21.

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Scheid, Harald, and Wolfgang Schwarz. "Axiome der Geometrie." In Elemente der Geometrie, 321–36. Berlin, Heidelberg: Springer Berlin Heidelberg, 2016. http://dx.doi.org/10.1007/978-3-662-50323-2_9.

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Scheid, Harald, and Wolfgang Schwarz. "Axiome der Geometrie." In Elemente der Geometrie, 245–60. Heidelberg: Spektrum Akademischer Verlag, 2007. http://dx.doi.org/10.1007/978-3-8274-3126-4_8.

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Scheid, Harald, and Wolfgang Schwarz. "Axiome der Geometrie." In Elemente der Geometrie, 245–60. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-662-49551-3_8.

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Busemann, Herbert, and B. B. Phadke. "Minkowskian Geometry, Convexity Conditionsand The Parallel Axiom." In Selected Works II, 671–87. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-65624-3_47.

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Conference papers on the topic "Geometry Axioms"

1

Petrov, Alexander P., and L. V. Kuzmin. "Visual space geometry derived from occlusion axioms." In Optical Tools for Manufacturing and Advanced Automation, edited by Robert A. Melter and Angela Y. Wu. SPIE, 1993. http://dx.doi.org/10.1117/12.165001.

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Arahovitis, Ioannis L. "DNA: THE AXIOMS OF CELL GEOMETRY THE DIFFERENTIAL GEOMETRY OF ITS FUNCTION THE AFFINE GEOMETRY OF ITS STRUCTURE." In Proceedings of the 9th International Workshop on Mathematical Methods in Scattering Theory and Biomedical Engineering. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814322034_0011.

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Pimentel, Elaine. "Proof systems for Geometric theories (PROGEO)." In Workshop Brasileiro de Lógica. Sociedade Brasileira de Computação - SBC, 2020. http://dx.doi.org/10.5753/wbl.2020.11459.

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We plan to study the problem of finding conservative extensions of first order logics. In this project we intend to establish a systematic procedure for adding geometric theories in both intuitionistic and classical logics, as well as to extend this procedure to bipolar axioms, a generalization of the set of geometric axioms. This way, we obtain proof systems for several mathematical theories, such as lattices, algebra and projective geometry, being able to reason about such theories using automated deduction.
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Schleich, Benjamin, Michael Walter, Sandro Wartzack, Nabil Anwer, and Luc Mathieu. "A Comprehensive Framework for Skin Model Simulation." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-82204.

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The need for geometrical variations management is an important issue in design, manufacturing and all other phases of product development. Two main axioms cover geometrical variations, namely the axiom of manufacturing imprecision and the axiom of measurement uncertainty. Therefore, this paper reviews common models for the description of non-ideal geometry (shape with geometric deviations) and shows how the random field theory can be applied to create more realistic skin models (a model which comprises these geometric deviations). Furthermore, methods to estimate and to express the underlying random field from a sample population are shown. These can be used to create and simulate random shapes considering systematic and random deviations observed through measurement or gathered from manufacturing process simulations. The proposed approach incorporates given information from manufacturing process simulations or prototypes. Based on these information, skin model samples are created which can represent the “realistic” part in assembly simulations or other geometrical analyses. This can help to identify the optimal tolerance sets within every stage of the product development process. The efficiency of the introduced approaches is shown in a case study.
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Han, Jeong-Yeol, and Sukmock Lee. "Geometry for off-axis parabolic mirrors." In Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation III, edited by Roland Geyl and Ramón Navarro. SPIE, 2018. http://dx.doi.org/10.1117/12.2318256.

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Choi, Jihoon. "First axion dark matter search with toroidal geometry." In The European Physical Society Conference on High Energy Physics. Trieste, Italy: Sissa Medialab, 2017. http://dx.doi.org/10.22323/1.314.0625.

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Panta Pazos, Rube´n. "Behavior of a Sequence of Geometric Transformations for a Truncated Ellipsoid Geometry in Transport Theory." In 17th International Conference on Nuclear Engineering. ASMEDC, 2009. http://dx.doi.org/10.1115/icone17-75758.

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The neutron transport equation has been studied from different approaches, in order to solve different situations. The number of methods and computational techniques has increased recently. In this work we present the behavior of a sequence of geometric transformations evolving different transport problems in order to obtain solve a transport problem in a truncated ellipsoid geometry and subject to known boundary conditions. This scheme was depicted in 8, but now is solved for the different steps. First, it is considered a rectangle domain that consists of three regions, source, void and shield regions 5. Horseshoe domain: for that it is used the complex function: f:D→C,definedasf(z)=12ez+1ezwhereD=z∈C−0.5≤Re(z)≤0.5,−12π≤Im(z)≤12π(0.1) The geometry obtained is such that the source is at the focus of an ellipse, and the target coincides with the other focus. The boundary conditions are reflective in the left boundary and vacuum in the right boundary. Indeed, if the eccentricity is a number between 0,95 and 0,99, the distance between the source and the target ranges from 20 to 100 length units. The rotation around the symmetry axis of the horseshoe domain generates a truncated ellipsoid, such that a focus coincides with the source. In this work it is analyzed the flux in each step, giving numerical results obtained in a computer algebraic system. Applications: in nuclear medicine and others.
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Miklos, Balint, Joachim Giesen, and Mark Pauly. "Discrete scale axis representations for 3D geometry." In ACM SIGGRAPH 2010 papers. New York, New York, USA: ACM Press, 2010. http://dx.doi.org/10.1145/1833349.1778838.

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Fu, Qiang, and Zezhong C. Chen. "Efficient, Accurate Geometric Modeling for Three-Axis Sculptured Surfaces Milling." In ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/detc2010-28910.

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Efficient, accurate geometric modeling for three-axis sculptured surfaces milling is quite challenging due to complexity of workpiece geometry change during machining. This paper presents an efficient, accurate approach to extracting the cutter/workpiece engagement (CWE) geometry and applying this geometry to an existing mechanistic force model in order to predict instantaneous cutting force, torque and power. In our research, a basic geometric modeling of chip removal in three-axis milling is investigated, and an effective model is proposed to represent the cutter swept profile. Computationally efficient, closed-form formulations are derived for general APT (Automatically Programmed Tools) cutter geometry. A Z-level B-Rep model is adopted to represent the in-process workpiece model, and an innovative geometric approach is used to extract the CWE geometry. Then, a mechanistic cutting force model is integrated to predict the cutting forces. As a result, a milling process simulation system is developed for three-axis virtual milling of sculptured surfaces. The developed system is experimentally verified by comparing the simulation results with actual forces measured from machining a test surface.
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Haghpassand, Khorssand, and James Oliver. "Computational Geometry for Optimal Workpiece Orientation." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0116.

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Abstract Workpiece orientation is formulated as an optimal design problem based on a discrete approximation of design surface geometry and the kinematic capabilities of the process machine tool. The primary process application addressed is three- and four-axis numerically controlled (NC) milling, although the techniques presented may be applied to machines with more general articulation. Recent developments in applied spherical geometry are employed to formulate a nonlinear optimization problem. For three-axis milling applications, by assigning a weight to each surface normal of the discrete model corresponding to the actual area it represents, the orientation is optimized such that the angle between the normals and the milling tool axis is minimized. This formulation is augmented, for four-axis milling, to incorporate limitations of the rotational degree of freedom into the optimization formulation.
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Reports on the topic "Geometry Axioms"

1

Wu, A. Y., S. K. Bhaskar, and A. Rosenfeld. Computation of Geometric Properties from the Medial Axis Transform in 0 (n logn) Time. Fort Belvoir, VA: Defense Technical Information Center, June 1985. http://dx.doi.org/10.21236/ada158934.

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