Construções com régua e compasso
Ano de defesa: | 2018 |
---|---|
Autor(a) principal: | |
Orientador(a): | |
Banca de defesa: | , , |
Tipo de documento: | Dissertação |
Tipo de acesso: | Acesso aberto |
Idioma: | por |
Instituição de defesa: |
Universidade Federal Rural de Pernambuco
|
Programa de Pós-Graduação: |
Programa de Pós-Graduação em Matemática (PROFMAT)
|
Departamento: |
Departamento de Matemática
|
País: |
Brasil
|
Palavras-chave em Português: | |
Área do conhecimento CNPq: | |
Link de acesso: | http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/7884 |
Resumo: | The problems of constructions have always occupied a prominent position in geometry. Only with the use of ruler and compass we can carry out a huge diversity of constructions and in all these constructions, the ruler is used only to draw straight lines. Although the greeks use other instruments, the classical restriction to the use of only the ruler and the compass was a matter of great importance to them. Among the problems of constructions with ruler and compass, that of constructing a regular polygon of n sides is probably of greater interest. The constructions of the equilateral triangle, the square, the regular pentagon and the regular hexagon have been known since Antiquity and occupy (or have occupied) position in the study of geometry in schools. However, for some regular polygons this construction (only with the use of ruler and compass) is not possible. As example we can mention the regular heptagon. There are other construction problems that deserve prominent position and for which a construction with ruler and compass is not possible. As examples we can mention the three classic problems of the Greeks: the duplication of the cube (or the construction of the edge of a cube whose volume is the double of that of a cube of a given angle), the trisection of any angle (or the construction of dividing an arbitrary angle in three equal parts) and the quadrature of the circle (or the construction of a square with area equal to the area of a given circle). The importance of studying these problems lies in the fact that they cannot be solved with ruler and compass only, although these instruments are used to solve many other construction problems. The attempt to find a solution to these problems influenced greek geometry, leading to important discoveries. As examples of these discoveries we can mention: the conic sections, some cubic and quartic curves and several transcendent curves. Subsequently a result of great importance was the development of the theory of equations related to domains of rationality, algebraic numbers and group theory. We note that attempting to solve problems like these without solution has resulted in one of the most significant developments in mathematics. The purpose of this dissertation is to show some classic constructions, as the case of the regular polygon of seventeen sides, and to deal with the impossibility of construction by means of ruler and compass, such as: duplication of a cube, trisection of an angle, quadrature of a circle (in this case we will only indicate the proof) and constructions of regular polygons. |
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NEVES, Rodrigo José GondimNEVES, Rodrigo José GondimSILVA, Bárbara Costa daBEDREGAL, Roberto CallejasMIGUEL, Marcos José2019-03-18T14:05:04Z2018-10-19MIGUEL, Marcos José. Construções com régua e compasso. 2018. 101 f. Dissertação (Programa de Pós-Graduação em Matemática (PROFMAT)) - Universidade Federal Rural de Pernambuco, Recife.http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/7884The problems of constructions have always occupied a prominent position in geometry. Only with the use of ruler and compass we can carry out a huge diversity of constructions and in all these constructions, the ruler is used only to draw straight lines. Although the greeks use other instruments, the classical restriction to the use of only the ruler and the compass was a matter of great importance to them. Among the problems of constructions with ruler and compass, that of constructing a regular polygon of n sides is probably of greater interest. The constructions of the equilateral triangle, the square, the regular pentagon and the regular hexagon have been known since Antiquity and occupy (or have occupied) position in the study of geometry in schools. However, for some regular polygons this construction (only with the use of ruler and compass) is not possible. As example we can mention the regular heptagon. There are other construction problems that deserve prominent position and for which a construction with ruler and compass is not possible. As examples we can mention the three classic problems of the Greeks: the duplication of the cube (or the construction of the edge of a cube whose volume is the double of that of a cube of a given angle), the trisection of any angle (or the construction of dividing an arbitrary angle in three equal parts) and the quadrature of the circle (or the construction of a square with area equal to the area of a given circle). The importance of studying these problems lies in the fact that they cannot be solved with ruler and compass only, although these instruments are used to solve many other construction problems. The attempt to find a solution to these problems influenced greek geometry, leading to important discoveries. As examples of these discoveries we can mention: the conic sections, some cubic and quartic curves and several transcendent curves. Subsequently a result of great importance was the development of the theory of equations related to domains of rationality, algebraic numbers and group theory. We note that attempting to solve problems like these without solution has resulted in one of the most significant developments in mathematics. The purpose of this dissertation is to show some classic constructions, as the case of the regular polygon of seventeen sides, and to deal with the impossibility of construction by means of ruler and compass, such as: duplication of a cube, trisection of an angle, quadrature of a circle (in this case we will only indicate the proof) and constructions of regular polygons.Os problemas de construções sempre ocuparam posição de destaque na Geometria. Apenas com o uso de régua e compasso podemos executar uma diversidade enorme de construções e em todas essas construções, a régua é utilizada apenas para traçar retas. Apesar de os gregos utilizarem outros instrumentos, a restrição clássica à utilização apenas da régua e do compasso era para eles um assunto de grande importância. Entre os problemas de construção com régua e compasso, o de construir um polígono regular de n lados é provavelmente o de maior interesse. As construções do triângulo equilátero, do quadrado, do pentágono regular e do hexágono regular são conhecidas desde a Antiguidade e ocupam (ou já ocuparam) posição de destaque no estudo da geometria nas escolas. No entanto, para alguns polígonos regulares essas construções (apenas com o uso de régua e compasso) não são possíveis. Como exemplo inicial podemos citar o heptágono regular. Existem outros problemas de construções que merecem posição de destaque e para os quais uma construção com régua e compasso não é possível. Como exemplos podemos citar os três problemas clássicos dos gregos: a duplica ção do cubo (ou a construção da aresta de um cubo cujo volume é o dobro do de um cubo de aresta dada), a trissecção de um ângulo qualquer (ou a construção de dividir um ângulo arbitrário dado, em três partes iguais) e a quadratura do círculo (ou a construção de um quadrado com área igual à área de um círculo dado). A importância do estudo desses problemas reside no fato de que eles não podem ser resolvidos com régua e compasso apenas, apesar desses instrumentos serem utilizados para resolver muitos outros problemas de construção. A tentativa de encontrar solução para esses problemas in uenciou a geometria grega, levando a descobertas importantes. Como exemplos dessas descobertas podemos citar as secções cônicas, algumas curvas cúbicas e quárticas e várias curvas transcendentes. Posteriormente um resultado de grande importância foi o desenvolvimento da teoria das equações ligadas a domínios de racionalidade, números algébricos e teoria dos grupos. Notamos com isso que a tentativa de resolver problemas como esses sem solução resultou em um dos mais signi cativos desenvolvimentos da Matemática. A proposta dessa dissertação é mostrar algumas construções clássicas, como a do heptadecágono regular e tratar da impossibilidade da construção por meio de régua e compasso, tais como duplicação de um cubo, trissecção de um ângulo, quadratura de um círculo (nesse caso faremos apenas a indicação da prova) e construções de polígonos regulares.Submitted by Mario BC (mario@bc.ufrpe.br) on 2019-03-18T14:05:04Z No. of bitstreams: 1 Marcos Jose Miguel.pdf: 1148839 bytes, checksum: 13ebebdc0fd7e279ef641aa277261992 (MD5)Made available in DSpace on 2019-03-18T14:05:04Z (GMT). No. of bitstreams: 1 Marcos Jose Miguel.pdf: 1148839 bytes, checksum: 13ebebdc0fd7e279ef641aa277261992 (MD5) Previous issue date: 2018-10-19application/pdfporUniversidade Federal Rural de PernambucoPrograma de Pós-Graduação em Matemática (PROFMAT)UFRPEBrasilDepartamento de MatemáticaConstrução geométricaConstrução com réguaConstrução com compassoCIENCIAS EXATAS E DA TERRA::MATEMATICAConstruções com régua e compassoinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesis7256355350190039125600600600-6155401143231123537-7090823417984401694info:eu-repo/semantics/openAccessreponame:Biblioteca Digital de Teses e Dissertações da UFRPEinstname:Universidade Federal Rural de Pernambuco (UFRPE)instacron:UFRPEORIGINALMarcos Jose Miguel.pdfMarcos Jose Miguel.pdfapplication/pdf1148839http://www.tede2.ufrpe.br:8080/tede2/bitstream/tede2/7884/2/Marcos+Jose+Miguel.pdf13ebebdc0fd7e279ef641aa277261992MD52LICENSElicense.txtlicense.txttext/plain; charset=utf-82165http://www.tede2.ufrpe.br:8080/tede2/bitstream/tede2/7884/1/license.txtbd3efa91386c1718a7f26a329fdcb468MD51tede2/78842019-03-18 11:05:04.041oai:tede2: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Biblioteca Digital de Teses e Dissertaçõeshttp://www.tede2.ufrpe.br:8080/tede/PUBhttp://www.tede2.ufrpe.br:8080/oai/requestbdtd@ufrpe.br ||bdtd@ufrpe.bropendoar:2019-03-18T14:05:04Biblioteca Digital de Teses e Dissertações da UFRPE - Universidade Federal Rural de Pernambuco (UFRPE)false |
dc.title.por.fl_str_mv |
Construções com régua e compasso |
title |
Construções com régua e compasso |
spellingShingle |
Construções com régua e compasso MIGUEL, Marcos José Construção geométrica Construção com régua Construção com compasso CIENCIAS EXATAS E DA TERRA::MATEMATICA |
title_short |
Construções com régua e compasso |
title_full |
Construções com régua e compasso |
title_fullStr |
Construções com régua e compasso |
title_full_unstemmed |
Construções com régua e compasso |
title_sort |
Construções com régua e compasso |
author |
MIGUEL, Marcos José |
author_facet |
MIGUEL, Marcos José |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
NEVES, Rodrigo José Gondim |
dc.contributor.referee1.fl_str_mv |
NEVES, Rodrigo José Gondim |
dc.contributor.referee2.fl_str_mv |
SILVA, Bárbara Costa da |
dc.contributor.referee3.fl_str_mv |
BEDREGAL, Roberto Callejas |
dc.contributor.author.fl_str_mv |
MIGUEL, Marcos José |
contributor_str_mv |
NEVES, Rodrigo José Gondim NEVES, Rodrigo José Gondim SILVA, Bárbara Costa da BEDREGAL, Roberto Callejas |
dc.subject.por.fl_str_mv |
Construção geométrica Construção com régua Construção com compasso |
topic |
Construção geométrica Construção com régua Construção com compasso CIENCIAS EXATAS E DA TERRA::MATEMATICA |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::MATEMATICA |
description |
The problems of constructions have always occupied a prominent position in geometry. Only with the use of ruler and compass we can carry out a huge diversity of constructions and in all these constructions, the ruler is used only to draw straight lines. Although the greeks use other instruments, the classical restriction to the use of only the ruler and the compass was a matter of great importance to them. Among the problems of constructions with ruler and compass, that of constructing a regular polygon of n sides is probably of greater interest. The constructions of the equilateral triangle, the square, the regular pentagon and the regular hexagon have been known since Antiquity and occupy (or have occupied) position in the study of geometry in schools. However, for some regular polygons this construction (only with the use of ruler and compass) is not possible. As example we can mention the regular heptagon. There are other construction problems that deserve prominent position and for which a construction with ruler and compass is not possible. As examples we can mention the three classic problems of the Greeks: the duplication of the cube (or the construction of the edge of a cube whose volume is the double of that of a cube of a given angle), the trisection of any angle (or the construction of dividing an arbitrary angle in three equal parts) and the quadrature of the circle (or the construction of a square with area equal to the area of a given circle). The importance of studying these problems lies in the fact that they cannot be solved with ruler and compass only, although these instruments are used to solve many other construction problems. The attempt to find a solution to these problems influenced greek geometry, leading to important discoveries. As examples of these discoveries we can mention: the conic sections, some cubic and quartic curves and several transcendent curves. Subsequently a result of great importance was the development of the theory of equations related to domains of rationality, algebraic numbers and group theory. We note that attempting to solve problems like these without solution has resulted in one of the most significant developments in mathematics. The purpose of this dissertation is to show some classic constructions, as the case of the regular polygon of seventeen sides, and to deal with the impossibility of construction by means of ruler and compass, such as: duplication of a cube, trisection of an angle, quadrature of a circle (in this case we will only indicate the proof) and constructions of regular polygons. |
publishDate |
2018 |
dc.date.issued.fl_str_mv |
2018-10-19 |
dc.date.accessioned.fl_str_mv |
2019-03-18T14:05:04Z |
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dc.identifier.citation.fl_str_mv |
MIGUEL, Marcos José. Construções com régua e compasso. 2018. 101 f. Dissertação (Programa de Pós-Graduação em Matemática (PROFMAT)) - Universidade Federal Rural de Pernambuco, Recife. |
dc.identifier.uri.fl_str_mv |
http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/7884 |
identifier_str_mv |
MIGUEL, Marcos José. Construções com régua e compasso. 2018. 101 f. Dissertação (Programa de Pós-Graduação em Matemática (PROFMAT)) - Universidade Federal Rural de Pernambuco, Recife. |
url |
http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/7884 |
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